Special Algebra for Special Relativity: Second Edition: Proposed Theory of Non-Finite Numbers [2 ed.]

Citation preview

SPECIAL ALGEBRA FOR

SPECIAL RELATIVITY

SECOND EDITION

2 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

The other book by Paul C Daiber: ALIEN INVASION MATH STORY

SPECIAL ALGEBRA FOR

SPECIAL RELATIVITY Proposed Theory of Non-Finite Numbers

Paul C Daiber

SECOND EDITION

4 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Copyright © 2020 by Paul C Daiber All rights reserved. This book or any portion thereof may not be reproduced or used in any manner whatsoever without the express written permission of the publisher except for the use of brief quotations in a book review, scholarly journal or other critical document. Daiber, Paul C, 1960 – Special Algebra for Special Relativity p. cm. Includes index. Paperback ISBN (see book cover) 1. Special Relativity, Electricity, Waves, Algebra, Mathematics, Infinity, Math, Abstraction I. Title.

A special thank you to amazon for making it so simple to publish a book

For My Wife Sue

Table of Contents

Finite Precision for Numbers ........................................................................ix Chapter 1 – Numbers .................................................................................. 1 1.1 Process from Descartes ......................................................................... 1 1.2 Geometric-Vectors ................................................................................ 2 1.3 Quaternions ......................................................................................... 4 1.4 Translation Back to Geometry.............................................................. 19 1.5 Singular-Label-Numbers ...................................................................... 20 1.6 Exercises ............................................................................................ 21 Chapter 2 – Particles ................................................................................. 33 2.1 Hypercomplex-Plane ........................................................................... 33 2.2 Inertial Reference Frames ................................................................... 36 2.3 The Unspecified-Speed-Parameter ....................................................... 38 2.4 Compound-Label-Numbers and Components ........................................ 39 2.5 Adding Hyperbolic-Angles.................................................................... 42 2.6 Energy, Time Dilation, Length Contraction ........................................... 46 2.7 Space-Like and Time-Like Invariants .................................................... 48 2.8 Electric Current Density ....................................................................... 52 2.9 Motion Faster than Light ..................................................................... 55 2.10 Anti-Matter ........................................................................................ 64 2.11 Distributed Material Theory ............................................................... 72

2.12 Exercises .......................................................................................... 83 Chapter 3 – Fields ..................................................................................... 89 3.1 Geometric-Vector Notation .................................................................. 89 3.2 All-Number Notation ........................................................................... 94 3.3 Gauges and Super-Potentials ............................................................. 105 3.4 Lorentz Transformation ..................................................................... 108 3.5 Biot-Savart Law ................................................................................ 116 3.6 Electric Energy-Momentum of an Electron .......................................... 119 3.7 Maxwell’s Wave Equation .................................................................. 126 3.8 Forces Using Geometric-Vector Notation............................................. 132 3.9 Force Density Invariant ..................................................................... 133 3.10 Area and Volume Differential Operators ............................................ 143 3.11 Exercises ........................................................................................ 151 Chapter 4 – Waves .................................................................................. 159 4.1 Differential Operator ......................................................................... 159 4.2 Development of the Dirac Equation .................................................... 162 4.3 Solutions to the Dirac Equation .......................................................... 166 4.4 Particle Properties ............................................................................. 169 4.5 Two Alternative Arrangements ........................................................... 173 4.6 Lorentz Transformation of a Dirac Spinor ........................................... 175 4.7 Exercises .......................................................................................... 181 Chapter 5 – Proposed Theory ................................................................... 187 5.1 Local-Real Numbers .......................................................................... 187 5.2 Cantor’s Theory of Infinite Sets.......................................................... 196 5.3 Algebra Field for Local-Real Numbers ................................................. 204 5.4 Lorentz Transformation with Non-Finite Numbers ............................... 209 5.5 Dirac Equation Development.............................................................. 229

5.6 Force Density Using the Complex-Conjugate ...................................... 237 5.7 Spin of a Photon ............................................................................... 245 5.8 Exercises .......................................................................................... 248 Appendix A – Octonions and Sedonions .................................................... 257 Appendix B – Spooky Action at a Distance ................................................ 277 Appendix C – Discovering an Abstraction .................................................. 289 The Storybook ........................................................................................ 295 Glossary ................................................................................................. 296 Index ..................................................................................................... 307 Back Cover ............................................................................................. 310

Preface The first four chapters of Special Algebra for Special Relativity present an all-number mathematical structure for Special Relativity. The fifth chapter restricts a measurable quantity to finite precision by limiting place-value digits to a maximum before and after the decimal point. For example, a physically real square of unit length on a side has only finite precision for each unit length and, likewise, the “2” diagonal also has only finite precision. The ideal of unit length by use of the integer one is not geometrically possible in the physical world, and neither is the ideal of a perfectly precise “2” irrational number. The finite imprecision larger than the measurable rational number is the division reciprocal of the finite imprecision smaller than the rational number. In Special Relativity the larger and smaller imprecision are added to the time-space hyperbolic angle “” (that relates to speed by “v = c*tanh”) using a Lorentz Transformation. The small magnitude imprecision is trivial. The large magnitude imprecision models electromagnetism because Maxwell’s Equations are derived from the Dirac Equation. Precision improves with time. The mathematics for predicting electric current density from the Dirac Spinor results in the electromagnetic field force density invariant, and it includes the empirically derived energy density and Poynting Vector. The union of those empirically derived models is new and suggests quantities in our geometric world actually do have finite precision, and that finite precision numbers apply to the more modern theories of physics.

ix SYNOPSIS – FINITE PRECISION FOR NUMBERS

Finite Precision for Numbers The proposed Theory of Non-Finite Numbers assigns finite precision to numbers. The count of place-value digits in a number, both before and after the decimal point, is limited to Aristotle’s potential infinity, a finite number. The remaining finite imprecise portion conforms to probability theory and is the non-finite number that applies to existing mathematical models of physics. The example application is the hundred-year-old Dirac Equation. A Lorentz Transformation that adds a finite imprecision term both larger and smaller than the finite rational number time-space hyperbolic angle results in Maxwell’s Equations and a uniting of the measured Poynting Vector and electromagnetic energy density into the measured force density invariant. That unity of different measured electromagnetic phenomena into one theoretical model is something new, and it suggests numbers as quantities in our physical world actually do have finite precision. The same technique of assigning finite precision can next be applied to more modern mathematical models of physics. 







The algebra field for rational numbers requires • • •

Any sum or product of two rational numbers be a rational number, as well as negatives and reciprocals (except division by zero) Zero and one be included as identity elements Commutative and associative properties for addition and multiplication as well as the distributive property of multiplication over addition

An irrational number, for example “2” or “log23”, can be included in the rational numbers’ algebra field because none of the above criteria are violated. To prove “2” is irrational, set “2 = p/q” from which “p2 = 2*q2”, from which “p” must be even and “q” must be even, and that observation is inconsistent with the ability of either “p” or “q” to be odd because both

x SPECIAL ALGEBRA FOR SPECIAL RELATIVITY can be divided by “2” until one is odd. The inconsistent observations mean no ratio “p/q” of natural numbers “{1, 2, 3, …}” possibly equals “2”. To prove “log23 = p/q” is irrational, derive “2^(log23) = 2^(p/q)” and then “2p = 3q” and observe no natural numbers “{1, 2, 3, …}” for “p” and “q” apply because an even number “2p” cannot equal an odd number “3q”. The irrational numbers with the rational numbers form the set of real numbers, and, therefore, the algebra field as defined for rational numbers also applies to real numbers. Cantor defined real numbers in the late 1800’s by stating real numbers had a quantity “2^N0” over any finite or infinite (N0) interval. “N0” (called “aleph null”) was forced to be a positive actual infinity through his Continuum Hypothesis: No set has a quantity between “N0” and “2^N0”. Attempt to prove “2” and “log23” are included in Cantor’s set of real numbers. If “2” equals the ratio of two infinity numbers, both are “2^N0” and so, maybe, both are even and cannot be odd, and therefore it appears “2” is excluded. If “log23” equals a ratio of two infinity numbers, then “2^(2^N0)” equates to “3^(2^N0)”, but there is no algebra for “3^(2^N0)” because the later developed algebra field theory applies only to finite numbers. Rather than conclude “2” and “log23” are not included in Cantor’s set of real numbers, we wonder if we are mistakenly forcing “N0” to be finite in the proofs. Cantor’s general approach of organizing numbers into sets was structured into Axiomatic Set Theory. The Axiom of Infinity addressed only Aristotle’s potential infinity, a finite number, and not a positive actual infinity. To insert an actual infinity into Axiomatic Set Theory, Cantor’s Continuum Hypothesis was added as another axiom, effectively. Also, Axiomatic Set Theory had no axiom that addressed the reciprocal of zero. Propose a reciprocal-of-zero axiom to specify what calculations are not possible: No operation that includes a reciprocal of the integer zero can result in a finite number. Per the proposed axiom, “0/0” and “1/0 - 1/0” are not permitted operations because they cannot result in a finite number. The positive or negative feature of a zero is not specified, and therefore “1/0 + 1/0” has no result and does not necessarily equal “2*(1/0)”. The operations “1/0 + 7 = 1/0” and “(1/0)*7 = 1/0” are accepted. Also “2^(1/0) = 3^(1/0) = 0 or 1/0”. “1/0” is the absolute maximum magnitude of numbers and is both or either positive and negative.

xi SYNOPSIS – FINITE PRECISION FOR NUMBERS “1/0” satisfies “p” and “q” in proofs of irrationality. For “2”, “1/0” is both an even number and an odd number, because one can be added to “1/0” without changing it. For “log23”, “2^(1/0) = 3^(1/0)”. Per the proposed axiom a ratio “p/q = 0/0” is not possible, therefore, equations “q*2 = p” and “q*log23 = p” apply instead, as “(1/0)*2 = 1/0” and “(1/0)*log23 = 1/0”. Per those two equations, “1/0” is the smallest number multiplied by an irrational number for which the result does not have a contribution after the decimal point. “1/0” is larger than the positive “N0”, “0 < N0 < 1/0”, because if “N0 < 2^N0” then “N0  1/0” and “2^N0  1/0”. Another property of “N0” is that it has no contribution after the decimal point, analogous to an integer, because “N0” is the quantity of members in a set. Given those two properties, neither “(2^N0)*2” nor “(2*N0)*log23” can equal “2^N0”, and that proves irrational numbers are not included in Cantor’s set of real numbers. The proof required the proposed reciprocal of zero axiom so that an irrational number had a quantity “1/0” non-pattern place-value digits after the decimal point. It supposes “1/0 + N0 = 1/0”, “0*N0 = 0”. 







A quantity in the physical world cannot be the idealized “1/0” or the hypothetical positive actual infinity “N0”, therefore, the largest possible quantity in the physical world is the finite potential infinity “L”. A number expressed in base two (only “0”’s and “1”’s as place-value digits) has a maximum quantity of place-value digits before or after the decimal point limited to the finite potential infinity “L”. A number has finite precision because of the finite limitation of place-value digits. For example, if one side of a square is unit length “f” and the adjacent side is unit length “g”, then, idealistically, “g = f = 1”, but when the square is constructed in the physical world “f” and “g” each have finite precision and therefore “f  g”. The imprecision adder is the local-zero “CminC0”. Select a number “Lmax” as the quantity of knowable place-value digits after the decimal point and make all those place-value digits zero. A place-value digit of “b” can be either a “0” or a “1” with that value unknowable until an observation is made, analogous to Schrödinger’s Cat. Each “b” is independent of every other “b”. Assign “d = b - b” and place “d”’s after the zeros. For “Lmax = 3”, “CminC0 = 0.000dddd……”.

xii SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Six dots “……” represents a quantity “1/0”, for example, the quantity of zeros after the decimal point for an integer. To create an integer, we truncate the number at the decimal point to remove those place-value digits in one bulk operation, and that contrasts to the three dots “…” which represent one at a time in a count. Another example of six dots is the quantity of non-pattern place-value digits for an irrational number. Also, for example, “1 = 0.9999……” to contrast with “1  0.9999…” because “0.9999…” has a local-zero added to it. As time increases, Aristotle’s potential infinity “L” increases in value, and that same property applies to “Lmax”. As “Lmax” increases, each “d” in sequence becomes “-1”, “0”, “0”, or “1”. The continuum of the number-line is a geometric physically real construction, and has resolution limited to “2^-Lmax” that gets more precise as time progresses. If each and every “d” of a local-infinity are randomly replaced with one of “-1”, “0”, “0”, or “1”, then, because there is a quantity “1/0” of “d”’s (per the six dots “……”), not all will be zero. Therefore, a division reciprocal of a local-zero exists, called a local-infinity, “CmaxC = 1/CminC0”. A local-infinity “CmaxC” has the property of being both positive and negative. Also, the magnitude of a local-infinity is larger than a number created from base two quantity “Lmax” place-value-digits before the decimal point.

Probability Distribution for “Lmax=0” for a local-zero “CminC0 = 0.dddd……” left and a local-infinity “CmaxC = 1/CminC0” right.

xiii SYNOPSIS – FINITE PRECISION FOR NUMBERS A truncated number is the finite rational number limited to “Lmax” place-value digits before and after the decimal point (in base two). To the truncated number we add either a local-infinity or else a local-zero (with the selection unknowable) to form a local-real number. Local-real numbers are our replacement for what were traditionally the real numbers formed using an actual infinity. 







Mechanical energy-momentum components and time-space location components for an electron are given below, respectively. mB*c*coshM + q*mB*c*sinhM c*tB*coshM + q*c*tB*sinhM “mB” is electron rest mass and “tB” is time on a clock mounted on the electron. “c” is the speed-of-light. Electron speed “vM” relates to hyperbolic angle “M” by “vM = c*tanhM”. The “q” is a 2x2 Pauli Spin Matrix conforming to “q2 = 1” and indicates direction in space. Reference frame “B” is stationary with respect to the particle. Reference frame “M” is moving, for example the interior of a bus in which the electron is moving along the floor toward the front with speed “vM”. Reference frame “S” is the stationary reference frame, for example the roadside, where the observer (you or me) is located. The speed of the bus relative to the roadside is “vS/M = c*tanhS/M”. Because hyperbolic angle “M” is a truncated number, add to it a local-zero or else local-infinity. Use Lorentz Transformation hyperbolic angle “S/M = (1 - q)*” to account for the large and small magnitude imprecision in the hyperbolic angle. Imprecision term “” (xi) is either a local-infinity “CmaxC” or else a local-zero “CminC0”. The “1 - q” factor keeps components finite and applies because rest mass is a truncated number, as is time measured on the electron’s clock. The Lorentz Transformation “S = M + S/M” for the two invariants has a First Case for “” “CmaxC” and a Second Case for “” “CminC0”. The Second Case is trivial because “CminC0” is zero, so that “S/M” is zero such that the electron when observed in “S” is the same as when

xiv SPECIAL ALGEBRA FOR SPECIAL RELATIVITY observed in “M”. In contrast, the First Case (with “C” positive, “+”) has the electron observed as a photon because the energy component “mB*c*exp(M)/2” equals the momentum component “mB*c*exp(M)/2”, and the time component “c*tB*exp(M)/2” equals the space component “c*tB*exp(M)/2”. mB*c*coshS + q*mB*c*sinhS = mB*c*exp(M)/2 + q*mB*c*exp(M)/2

First Case “C” positive

c*tB*coshS + q*c*tB*sinhS = c*tB*exp(M)/2 + q*c*tB*exp(M)/2

First Case “C” positive

“Lmax” increases with time to make the unknowable known in observations of the electron and its projection as emitted photons. Electron dynamics are modeled by the Dirac Equation. The Dirac Equation is developed below with a First Case and a Second Case. Begin with a relationship of momentum components per the Pythagorean Theorem, and split it into two equations to form a matrix equation. (mB*c*(1 + coshS)*PPS)*(mB*c*(-1 + coshS)*-QQS) = (q*mB*c*sinhS*-QQS)*(q*mB*c*sinhS*PPS) (mB*c*(1 + coshS))

(q*mB*c*sinhS)

PPS *

( q*mB*c*sinhS) (mB*c*(-1 + coshS))

0 =

QQS

0

For the First Case, rest mass “mB” terms are zero because the “q” in “S/M = (1 - q)*” creates a zero factor “exp(-CmaxC)” (for “C” “+”). And, because “S/M” also creates an infinity factor “exp(CmaxC)” that cancels that zero factor, “mB*c*coshS = mB*c*exp(M)/2” and “mB*c*sinhS = mB*c*exp(M)/2”. To account for electric charge, the right-side is given a non-zero value “a”. (mB*c*exp(M)/2) (q*mB*c*exp(M)/2)

PPS *

(q*mB*c*exp(M)/2) (mB*c*exp(M)/2)

a =

QQS

q*a

xv SYNOPSIS – FINITE PRECISION FOR NUMBERS Mechanical energy-momentum equals total energy-momentum minus electrical energy-momentum. mB*c*cosh(S) = i*ħ*tS - QB*VtS q*mB*c*sinh(S) = qx*(-i*ħ*xS - QB*VxS) + qy*(-i*ħ*yS - QB*VyS) + qz*(-i*ħ*zS - QB*VzS) “” is the gradient operator, with “tS = /ctS”, “xS = /xS”, “yS = /yS”, and “zS = /zS”. “V” components are from the external voltage invariant. “ħ” is Planck’s constant. For the First Case, a zero electron rest mass “mB” coincides to a zero electron electric charge “QB”. The above operators reduce to: mB*c*exp(M)/2 = i*ħ*tS q*mB*c*exp(M)/2 = qx*(-i*ħ*xS) + qy*(-i*ħ*yS) + qz*(-i*ħ*zS) Substitute the Pauli Spin Matrices for “qx”, “qy”, and “qz”. (“qy” is the negative.) 0 1 qx =>

0 i qy =>

1 0

-i 0

1000 0100 /ct

1

+ -/x

-1

0 1 000 i 0 0 -i 0

+ -/y 0 i00 -i 0 0 0 1_FirstCase 2_FirstCase

1_FirstCase 2_FirstCase

* 1 000 0 -1 0 0

0

0100 1000

001 0 0 0 0 -1

1 0 1 =>

0001 0010

0010 0001

+ -/z

0

qz =>

= 3_FirstCase 4_FirstCase

3_FirstCase 4_FirstCase

“PPS” and “QQS = q*PPS” were replaced by the Dirac Spinor (). Similarly, “a” and “q*a” were replaced by a spinor ().

xvi SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Electric field components are “E”. Magnetic field components are “B”. Electric current density components are “J”, with adjusted measurement units. 1_FirstCase = 3_FirstCase = Ez + i*c*Bz 2_FirstCase = 4_FirstCase = (Ex + i*c*Bx) - i*(Ey + i*c*By)

1_FirstCase = 3_FirstCase = -Jt - Jz 1_FirstCase = 3_FirstCase = -Jx + i*Jy 1000 0100 /ct

0001 0010 + -/x

0010 0001

+ -/y 0100 1000

001 0 0 0 0 -1 + -/z

000 i 0 0 -i 0 0 i00 -i 0 0 0

(Ez + i*c*Bz) (Ex + i*c*Bx) - i*(Ey + i*c*By) *

1 000 0 -1 0 0

(-Jt - Jz) (-Jx + i*Jy) =

(Ez + i*c*Bz) (Ex + i*c*Bx) - i*(Ey + i*c*By)

(-Jt - Jz) (-Jx + i*Jy)

Rotate the three Pauli Spin Matrices, each through “qx”, “qy” and “qz”. Second Portion: 1000 0100 /ct

0001 0010 + -/y

0010 0001

+ -/x

+ -/z 0100 1000

001 0 0 0 0 -1

000 i 0 0 -i 0 0 i00 -i 0 0 0

(Ex + i*c*Bx) (Ey + i*c*By) - i*(Ez + i*c*Bz) *

1 000 0 -1 0 0

Third Portion:

(-Jt - Jx) (-Jy + i*Jz) =

(Ex + i*c*Bx) (Ey + i*c*By) - i*(Ez + i*c*Bz)

(-Jt - Jx) (-Jy + i*Jz)

xvii SYNOPSIS – FINITE PRECISION FOR NUMBERS 1000 0100 /ct

0001 0010 + -/z

0010 0001

+ -/y

+ -/x 0100 1000

001 0 0 0 0 -1

000 i 0 0 -i 0 0 i00 -i 0 0 0

(Ey + i*c*By) (Ez + i*c*Bz) - i*(Ex + i*c*Bx) *

1 000 0 -1 0 0

(-Jt - Jy) (-Jz + i*Jx) =

(Ey + i*c*By) (Ez + i*c*Bz) - i*(Ex + i*c*Bx)

(-Jt - Jy) (-Jz + i*Jx)

Second Case Dirac Equation: 1000 0100 m*c*

1000 0100 + (i*ħ*/ct - Q*Vt)*

0 0 -1 0 0 0 0 -1

0001 0010 + (-i*ħ*/x - Q*Vx)*

0010 0001 000 i 0 0 -i 0

+ (-i*ħ*/y - Q*Vy)*

0100 1000

+ (-i*ħ*/z - Q*Vz)* 0 i00 -i 0 0 0

1 2

001 0 0 0 0 -1 * 1 000 0 -1 0 0

0 0 =

3 4

0 0

The Dirac Spinor is the column-vector with the four components “1”, “2”, “3”, “4”. Electric charge of the electron is “Q”. Voltage invariant components are “V”. Electron rest mass is “m”. The First Case and Second Case, together, as one mathematical model, pertain to a combined photon/electron particle. Per the model, the electromagnetic field is developed from “CmaxC” with “C” “+”, and the electron matter-wave Dirac Spinor is developed from “CminC0”. As time progresses, “M” becomes more precise to change observations of the projected photon. The Dirac Spinor is post-processed into a prediction for a measured electric current density invariant. By the same equations, the First Case is post-processed into a prediction for a measured electromagnetic field force density invariant, and it includes the empirically discovered energy density and Poynting Vector. The uniting of separate empirically discovered models of physics means the First Case Dirac Equation form of Maxwell’s Equations appears to be fundamental

xviii SPECIAL ALGEBRA FOR SPECIAL RELATIVITY to electromagnetic field theory. The discovery of something more fundamental justifies a claim that the new number system with finite precision is valid in applied mathematics and is ready for use in more modern theories. For further information, please read Special Algebra for Special Relativity. What didn’t fit into a textbook was placed in Alien Invasion Math Story.

1 CHAPTER 1 - NUMBERS

Chapter 1 – Numbers 1.1 Process from Descartes Formal mathematics in the written record began about twenty-five hundred years ago in ancient Greece, when Pythagoras presented his proof for the Pythagorean Theorem. Along with that proof came the fresh idea that numbers could be separated from items being counted. About four hundred years ago in La Géométrie, Descartes solved geometric problems using an all-number algebra and plotted solutions on a (Des)Cartesian grid. Algebra had become separated from the geometry being analyzed. The three steps that form the “Process from Descartes” are: 1) Mathematically model a physically real phenomenon with a geometric representation 2) Translate the mathematical model into a more abstract allnumber algebra to do the analysis 3) Translate the all-number algebraic finished-calculation back into a geometric representation as a final-result The Process from Descartes, what he called “analytic geometry”, explicitly removed the analysis from the world of measurements and geometry but did so temporarily. The Process from Descartes is the basis of the applied mathematics in this book.

2 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

1.2 Geometric-Vectors Geometric-Unit-Vectors and Components. Newton’s Second Law is our example: Force “F” equals mass “m” times acceleration “a”. “F” and “a” are geometric-vectors, as indicated by bold font. F = m*a ;

Fx*ix + Fy*iy + Fz*iz = m*ax*ix + m*ay*iy + m*az*iz

Fx = m*ax ; Fy = m*ay ; Fz = m*az

(Component equations)

Figure 1. Tabletop Coordinate System, Location P is 3r = x*ix + y*iy + z*iz

Figure 2. Right-hand coordinate system, and Left-hand system

3 CHAPTER 1 - NUMBERS Three perpendicular geometric-unit-vectors “ix”, “iy”, and “iz” identify directions along which length components “x”, “y” and “z” are measured by counting measurement units (inches or centimeters) on measuring tapes that have their zero points at the origin. Per tradition, a right-hand coordinate system for “x, y, z” is always selected: The right-hand fingers curl through positive “ix” and then positive “iy”, and the thumb points to positive “iz”. Push slippery ice of mass “m” with force “Fx”, and the ice moves with acceleration “ax” per “Fx = m*ax”. The push was parallel with the “x” direction and not necessarily along the front edge of the table. Dot-Product Operations: ix•ix = 1 ; ix•iy = 0 ; ix•iz = 0 iy•ix = 0 ; iy•iy = 1 ; iy•iz = 0 iz•ix = 0 ; iz•iy = 0 ; iz•iz = 1

Cross-Product Operations: ixxix = 0 ; ixxiy = iz ; ixxiz = -iy iyxix = -iz ; iyxiy = 0 ; iyxiz = ix izxix = iy ; izxiy = -ix ; izxiz = 0

Time Geometric-Unit-Vector. The time component in time-space is “c*t”. Time “t” is measured by a clock in seconds relative to a time origin by counting a repeated ticking internal to the clock mechanism. “c  300,000,000” meters/second is the speed-of-light and is a measurement unit conversion factor from time units to length units. “it” is the time geometric-unit-vector. Location “P” in time-space: 4r

= c*t*it + x*ix + y*iy + z*iz

The “4” pre-subscript indicates the quantity of terms in the expression. Each term consists of a component factor (“c*t”, “x”, “y”, or “z”) multiplied by a geometric direction factor (“it”, “ix”, “iy”, or “iz”). We could speculate “ixxit” should include an “i” factor (“i2 = -1”) because, perhaps, “it•it = -1”, but we don’t go there because geometricvectors are only used to set up the problem. Engineering-Calculation-Algebra. The lack of division as an inverse to the dot-product and/or cross-product illustrates that physical properties of geometric vectors are a priority over mathematical properties. Giving priority to physical properties is appropriate in engineering because an

4 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY engineer applies known theory in a simple and efficient calculation. In the time of Maxwell (late 1800’s) and Minkowski (early 1900’s) (slightly over a hundred years ago) geometric vectors had just been introduced by Heaviside and was the best algebra available.

1.3 Quaternions An all-number algebra using quaternions fits Special Relativity so well it appears quaternions are inherently the correct algebra. An all-number algebra has only mathematical properties, no physical properties. It might be counter-intuitive, but because the algebra is separated from the physical, geometric world, more of the subtle symmetries and features of the physics can be expressed algebraically. Theory-Development-Algebra is the name for the new intention of the all-number algebra, which is to expand known theories into new theories, as opposed to using known theories in engineering. The difference between engineering-calculation-algebra and theory-development-algebra evolved (in the writing of this book) from the difference in Pythagorean mathematics between logistica (routine calculations) and arithmetica (theory). Quaternions and Label-Numbers. There are three quaternions “jx”, “jy”, and “jz” (with the number “1” as the fourth) in Hamilton’s paper dated 1843. Each of the three quaternions squares to negative one. Quaternions “jx”, “jy”, and “jz”, the complex number factor “i”, the number one “1”, and unit magnitude constructions of these special numbers, are given the categorical name (in this book) “label-numbers”. Translation to all-number. Label-numbers are placed in front of the components. “4r = c*t*it + x*ix + y*iy + z*iz” becomes i*4r = i*c*t + jx*x + jy*y + jz*z qx = jx/i ; qy = jy/i ; qz = jz/i ;

or

4r

= 1*c*t + qx*x + qy*y + qz*z

qx2 = qy2 = qz2 = +1

5 CHAPTER 1 - NUMBERS Matrix Isomorphs of Quaternions. An “isomorph” is a functional equal with respect to an algebra. Products of label-numbers are calculated using matrix isomorphs because matrix isomorphs provide internal mathematical structure for the calculation. A traditional association of matrix isomorphs to quaternion label-numbers is given below. 0 i jx =>

0 -1

i

jy => i 0

0 i qy =>

1 0

i

0

0

i

i =>

1 0

0 1 qx =>

0

jz => 0

-i

1

0

qz => -i 0

1 0 1 =>

0

-1

0 1

(“qy” is the negative of its corresponding Pauli Spin Matrix.) The “1” and “i” terms in the matrices also have isomorphs. 1

0

0 -1

1 =>

i => 0 1

1 0

The above ordering of “jx”, “jy”, “jz” can be rotated to “jy”, “jz”, “jx” or to “jz”, “jx”, “jy”, but with the matrices not being rotated. Matrix multiplication is per the below equation. It applies only if the matrix terms (“a, b, c, d, e, f, g, h”) commute. a

b

c

d

e

f

g

h

*

a*e + b*g

a*f + b*h

c*e + d*g

c*f + c*h

=

Label-numbers have the following product equations. jx*jy = -jy*jx = jz jy*jz = -jz*jy = jx jz*jx = -jx*jz = jy

; ; ;

i*jx = jx*i i*jy = jy*i i*jz = jz*i

jx2 = jy2 = jz2 = i2 = -1

;

qx2 = qy2 = qz2 = 12 = +1

6 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY qx = jx/i ; qy = jy/i ; qz = jz/i qx*jy = -jy*qx = -qy*jx = jx*qy = qz ; qy*jz = -jz*qy = -qz*jy = jy*qz = qx ; qz*jx = -jx*qz = -qx*jz = jz*qx = qy ;

qx*qy = -qy*qx = -i*qz = -jz qy*qz = -qz*qy = -i*qx = -jx qz*qx = -qx*qz = -i*qy = -jy

Quaternion Division Operation. To divide one matrix by another, alter the numerator to create a matrix divided by itself. All possible factorings and regroupings end with the same result. Alternatively, use a reciprocal. qy/jz = (jz*qx)/jz = -(qx*jz)/jz = -qx*(jz/jz) = -qx ; qy/jz = qy*(-jz) = -qx Comparing Quaternions to Geometric-Unit-Vectors. Geometric-unitvectors had no “i” factor in a cross-product result (“ixxiy = iz”). In contrast, quaternions did have an “i” factor (“qx*qy = -i*qz”). An engineer does not want an “i” factor in a calculation that predicts a measurement because every measurement is a real number. In contrast, the “i” is wanted for theory development. Exponential Function with Quaternions. Argument “s” can have a labelnumber included as a factor, for example “s = qx*α” (alpha). In contrast, a geometric-unit-vector cannot be in the argument of an exponential function because there is no algebra by which to calculate. exp(s) = 1 + s + s2/2 + s3/6 + s4/24 + s5/120 + s6/6! + … exp(qx*α) = 1 + qx*α + (qx*α)2/2 + (qx*α)3/6 + (qx*α)4/24 + … = 1 + qx*α + α2/2 + qx*α3/6 + α4/24 + … = (1 + α2/2 + α4/24 + …) + qx*(α + α3/6 + …) = cosh(α) + qx*sinh(α) Complex-conjugate “*i” swaps “i” for “-i” and reverses factors. i*i = -i

1*i = 1

7 CHAPTER 1 - NUMBERS Quaternion hypercomplex-conjugate “*j” swaps “jx”, “jy”, “jz” for “-jx”, “-jy”, “-jz”, respectively, and reverses factors. i*j = i

; 1*j = 1

jx*j = -jx ; jy*j = -jy ; jz*j = -jz ;

qx*j = -qx ; qy*j = -qy ; qz*j = -qz

-qx = (qx)*j = (jy*qz)*j = (qz)*j*(jy)*j = (-qz)*(-jy) = qz*jy = -qx Both an all-number expression and its conjugate relate to the same geometric-vector expression. The “*j” symbol is removed after the “4r*j” is used as a factor in a multiplication operation, and not before. 4r

*j

= (1*c*t + qx*x + qy*y + qz*z)*j = c*t*1*j + x*qx*j + y*qy*j + z*qz*j

A reciprocal can substitute for the negative. i*i = -i = 1/i ;

jx*j = -jx = 1/jx ; jy*j = -jy = 1/jy ; jz*j = -jz = 1/jz

An Algebra that Includes Quaternions. The product equations above must be organized into a mathematical group. The group’s criteria are used in a proof so that the proof is not ambiguous or illogical. Criteria for a group: • • • • •

Closure Identity element Commutative property Associative property Inverse operation

Two groups can be combined to form an algebra field with: •

Distributive property

8 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Number Sets. Every natural number, integer, and rational number is finite. Natural numbers “N” used for counting start with one and have one added repeatedly. Include zero and negatives to form integers “Z”. N = {1, 2, 3, …}

;

Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

Rational numbers “Q” are each a ratio of two integers: “q  Q” if “q = m/n” with “m, n  Z”, “n ≠ 0”. A rational number “q” has multiple selections of “m” and “n”, for example “1.5 = 3/2 = -6/-4”. Integers are a subset of rational numbers: “Z  Q” because the denominator “n” may equal “1”. An integer lacks a decimal point. Descartes assigned the name “real” to real numbers to distinguish them from complex or imaginary numbers. Today, real numbers “R” are rational numbers united with numbers proven to be irrational. “Q  R” Algebra Field for Real Numbers. Integers “Z”, rational numbers “Q”, and real numbers “R” each form a group with respect to addition. The example group “{R, +}” uses real numbers “R”. Criteria are: •

.1. Addition Closure Property. Each element “a” formed by addition is an element of “R”: “a = b + c”. “a, b, c  R”

•

.2. Addition Identity Property. An addition identity element “e” is an element of the set “R”, for which “a = a + e”. “a, e R” (“e” is the integer zero “0”)

•

.3. Addition Commutative Property. The order of elements in an addition operation has no effect on the result of the operation, so that “a + b = b + a”. “a, b  R”

•

.4. Addition Associative Property. The order of addition operations has no effect on the result of the operations, so that “(a + b) + c = a + (b + c)”. “a, b, c  R”

•

.5. Addition Inverse Property. Each element “a” formed by the addition inverse operation, “a = -b” (such that “a + b = e”), is an element of “R”. “a, b, e  R”

9 CHAPTER 1 - NUMBERS The rational numbers “Q” and the real numbers “R” each form a group with respect to multiplication. “{R, *}” Criteria: •

.6. Multiplication Closure Property. Each element “a” formed by multiplication “a = b*c” is an element of “R”. “a, b, c  R”

•

.7. Multiplication Identity Property. A multiplication identity element “f” is an element of the set “R”, for which “a = a*f”. “a, f  R” (“f” is the integer one “1”)

•

.8. Multiplication Commutative Property. The order of elements in a multiplication operation has no effect on the result of the operation, so that “a*b = b*a”. “a, b  R”

•

.9. Multiplication Associative Property. The order of multiplication operations has no effect on the result of the operations, so that “(a*b)*c = a*(b*c)”. “a, b, c  R”

•

.10. Multiplication Inverse Property. Each element “a” formed by the multiplication inverse operation, “a = 1/b” (such that “a*b = f”), is an element of “R”. “a, b, f  R” with the exception “b  0”

The two groups above are brought together to form an algebra field “{R, +, *}” (or “{Q, +, *}”) by use of the distributive property. •

.11. Distributive Property of Multiplication over Addition. “a*(b + c) = a*b + a*c” and “(a + b)*c = a*c + b*c”. “a, b, c  R” Use algebra field criteria to prove properties.

•

.12. Property of One Addition Identity Element “e”: If “e1 + a = a” and if also there is “e2 + a = a” then from “a = a” we have “e1 + a = e2 + a”, and after subtracting “a” we have “e = e1 = e2”.

•

.13. Property of One Multiplication Identity Element “f”: If “f1*a = a” and if “f2*a = a”, then from “a = a” we have “f1*a = f2*a”, and, after dividing out “a” (for “a  0”), we have “f = f1 = f2”.

10 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •

.14. Property of “0 = -0”: The addition identity element “0” is its own addition inverse, so that “0 = -0”.

•

.15. Property of “1 = 1/1” and “-1 = 1/(-1)”: The multiplication identity element “1” is its own multiplication inverse, “1 = 1/1”. And, the addition inverse of the multiplication identity element is its own multiplication inverse, “-1 = 1/(-1)”.

•

.16. Property of “b*0 = 0”: Any number “b” multiplied by the additive identity element zero “0” equals the additive identity element zero “0” as the product. If “a = a + 0”, then “0 = a - a” and “b*0 = b*a - b*a = 0”.

•

.17. Inverse Anti-Commutative Properties: The addition inverse anti-commutative property is “-(a - b) = b - a”. The multiplication inverse anti-commutative property is “1/(a/b) = b/a”.

Exponent operation properties for rational numbers and integers are listed below. The exponent operation “^” is repeated multiplication (for example, “4^5 = 4*4*4*4*4”). Similarly, the multiplication operation “*” is repeated addition “+” (for example, “4*5 = 4 + 4 + 4 + 4 + 4”). •

.18. Exponent Closure Property: Each element “a” formed by the exponent operation “a = b^c” is an element of “Q”. “a, b  Q”, “c  Z”, and “b ≠ 0” if “c < 0”.

•

.19. Exponent Identity Property: An exponent identity element “g” is an element of the set “Z” for which “a = a^g”. “a  Q”, “g  Z” (“g” is the integer one “1”)

•

.20. No Exponent Inverse Property: Element “c” formed by the root inverse exponent operation, “c = a^(1/b)”, is not an element of “Q” if “b ≠ 1” or “b ≠ -1”. “a  Q”, “b  Z”

•

.21. Base Inverse Property: Element “b” formed by the logarithm inverse exponent operation, “logac = b”, is an element of “Z” only if “c = a^b”. “a, c  Q”

11 CHAPTER 1 - NUMBERS •

.22. No Exponent Associative Property: The order of exponent operations cannot be altered, so that “(a^b)^c” does not necessarily equal “a^(b^c)”. “a  Q”, “b, c  Z”

•

.23. No Exponent Commutative Property: The order of elements in an exponent operation has an effect on the result of the operation so that “a^b ≠ b^a”. “a, b  Z”, “a ≠ b”

•

.24. Distributive Property for Exponent Operation. “a^(b + c) = (a^b)*(a^c)”. “a  Q”, “b, c  Z”

Complex Numbers. “{1, i}” in a group with multiplication, “{{1, i}, *}”: •

Closure: No holes in the multiplication table

•

Identity: Identity element is the positive number one.

•

Commutative Property: Applies without exception

•

Associative Property: Applies without exception

•

Inverse: The ratio of any two numbers is in the set of numbers 1/1 = 1 ; i/1 = i ; 1/i = 1*(-i) = -i ; i/i = 1

“{1, i}” combines with real numbers “R” to form “complex numbers”, “C”, that conform to the criteria of an algebra with the same properties as for the real numbers “R”, “{C, +, *}”. Conjugate operation “*i” does not affect “{C, +, *}”. Through an algebraic manipulation “i” is placed in the numerator. 1/(5 + i*7) = (5 - i*7)/((5 - i*7)*(5 + i*7)) = (5 - i*7)/(25 + 49) = 5/74 - i*7/74 “a + i*b” with “a, b  R” is called the “summation-form”. 2x2 matrix isomorphs apply.

12 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1 0 1 =>

0 -1 ;

0

1

i =>

a -b ;

a + i*b =>

1 0

b a

*

+1

-1

+i

-i

+1

(+1)*(+1) = +1

(+1)*(-1) = -1

(+1)*(+i) = +i

(+1)*(-i) = -i

-1

(-1)*(+1) = -1

(-1)*(-1) = +1

(-1)*(+i) = -i

(-1)*(-i) = +i

-i

(-i)*(+1) = -i

(-i)*(-1) = +i

(-i)*(+i) = +1

(-i)*(-i) = -1

+i

(+i)*(+1) = +i

(+i)*(-1) = -i

(+i)*(+i) = -1

(+i)*(-i) = +1

Table 1. Multiplication Table for the Complex Number Factor. “+1” is along the major diagonal by using a conjugate in the left column (the first factor).

Quaternion-Hypercomplex Numbers. “{{1, jx, jy, jz}, *}” criteria: •

Closure: No holes in the multiplication table

•

Identity: Identity element is the positive number one

•

Commutative Property: Applies with “1” or “-1” as a factor or as the product. The anti-commutative property applies when two different (non-“1”) quaternions are factors.

•

Associative Property: Applies without exception

•

Inverse: The ratio of any two numbers is in the set of numbers

jx/jy = (jy*jz)/jy = -(jz*jy)/jy = -jz*(jy/jy) = -jz*1 = -jz ;

jx/jy = jx*(-jy) = -jz

“{1, jx, jy, jz}” with “R” forms “quaternion-hypercomplex numbers”, “QH”, “{QH, +, *(anti-commute)}”. Conjugate operation “*j” does not affect the algebra field.

13 CHAPTER 1 - NUMBERS *

+1

-1

+jx

-jx

+1

(+1)*(+1) = +1

(+1)*(-1) = -1

(+1)*(+jx) = +jx

(+1)*(-jx) = -jx

-1

(-1)*(+1) = -1

(-1)*(-1) = +1

(-1)*(+jx) = -jx

(-1)*(-jx) = +jx

-jx

(-jx)*(+1) = -jx

(-jx)*(-1) = +jx

(-jx)*(+jx) = +1

(-jx)*(-jx) = -1

+jx

(+jx)*(+1) = +jx

(+jx)*(-1) = -jx

(+jx)*(+jx) = -1

(+jx)*(-jx) = +1

-jy

(-jy)*(+1) = -jy

(-jy)*(-1) = +jy

(-jy)*(+jx) = +jz

(-jy)*(-jx) = -jz

+jy

(+jy)*(+1) = +jy

(+jy)*(-1) = -jy

(+jy)*(+jx) = -jz

(+jy)*(-jx) = +jz

-jz

(-jz)*(+1) = -jz

(-jz)*(-1) = +jz

(-jz)*(+jx) = -jy

(-jz)*(-jx) = +jy

+jz

(+jz)*(+1) = +jz

(+jz)*(-1) = -jz

(+jz)*(+jx) = +jy

(+jz)*(-jx) = -jy

*

+jy

-jy

+jz

-jz

+1

(+1)*(+jy) = +jy

(+1)*(-jy) = -jy

(+1)*(+jz) = +jz

(+1)*(-jz) = -jz

-1

(-1)*(+jy) = -jy

(-1)*(-jy) = +jy

(-1)*(+jz) = -jz

(-1)*(-jz) = +jz

-jx

(-jx)*(+jy) = -jz

(-jx)*(-jy) = +jz

(-jx)*(+jz) = +jy

(-jx)*(-jz) = -jy

+jx

(+jx)*(+jy) = +jz

(+jx)*(-jy) = -jz

(+jx)*(+jz) = -jy

(+jx)*(-jz) = +jy

-jy

(-jy)*(+jy) = +1

(-jy)*(-jy) = -1

(-jy)*(+jz) = -jx

(-jy)*(-jz) = +jx

+jy

(+jy)*(+jy) = -1

(+jy)*(-jy) = +1

(+jy)*(+jz) = +jx

(+jy)*(-jz) = -jx

-jz

(-jz)*(+jy) = +jx

(-jz)*(-jy) = -jx

(-jz)*(+jz) = +1

(-jz)*(-jz) = -1

+jz

(+jz)*(+jy) = -jx

(+jz)*(-jy) = +jx

(+jz)*(+jz) = -1

(+jz)*(-jz) = +1

Table 2. Multiplication Table for Quaternions. “+1” is along the major diagonal using the conjugate in the left column.

Quaternions can be placed exclusively in the numerator. 1/(3 + jx*5 + jy*7) = (3 - jx*5 - jy*7)/((3 - jx*5 - jy*7)*(3 + jx*5 + jy*7)) = (3 - jx*5 - jy*7)/(9 + 25 + 49 - 35*(jx*jy + jy*jx)) = (3 - jx*5 - jy*7)/83 = 3/83 - jx*5/83 - jy*7/83 Any quaternion-hypercomplex number can be written in the summation-form “a + jx*b + jy*c + jz*d” with “a, b, c, d  R”. Each of three rotation options for 2x2 matrix isomorph substitutions apply.

14 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY (a + i*d) -(c + i*b)*i a + jx*b + jy*c + jz*d =>

(c + i*b) (a + i*d)*i

Quaternion-Complex-Hypercomplex Numbers. “{{1, i, jx, qx}, *}” criteria: •

Closure: No holes in the multiplication table

•

Identity: The identity element is the positive number one

•

Commutative Property: Applies without exception

•

Associative Property: Applies without exception

•

Inverse: The ratio of any two numbers is in the set of numbers

“{1, i, jx, qx}” combines with “{1, jx, jy, jz}” to form a 16x16 multiplication table for “{1, i, jx, jy, jz, qx, qy, qz}”. Inverse example: qx/jy = (jx/i)/jy = -i*(jx/jy) = -i*(-jz) = -jz/i = -qz “{1, i, jx, jy, jz, qx, qy, qz}” combines with “R” to form “quaternion-complexhypercomplex numbers”, “{QCH, +, *(anti-commute)}”. Conjugates apply. Label-numbers can be placed exclusively in the numerator. 1/(3 + i*5 + qy*7) = (3 + i*5 - qy*7)/((3 + i*5 - qy*7)*(3 + i*5 + qy*7)) = (3 + i*5 - qy*7)/(-16 + i*30 - 49) = (3 + i*5 - qy*7)/(-65 + i*30) = (3 + i*5 - qy*7)*(-65 - i*30)/((-65 - i*30)*(-65 + i*30)) = (-45 - i*415 + qy*455 + jy*210)/5125 Summation-form “ar + i*ai + qx*br + jx*bi + qy*cr + jy*ci + qz*dr + jz*di” with “ar, ai, br, bi, cr, ci, dr, di  R” has three 2x2 matrix isomorphs. ((ar + dr) + i*(ai + di))

-((ci - br) + i*(cr + bi))*i

((ci + br) + i*(-cr + bi)) ((ar - dr) + i*(-ai + di))*i

15 CHAPTER 1 - NUMBERS *

+1

-1

+i

-i

+1

(+1)*(+1) = +1

(+1)*(-1) = -1

(+1)*(+i) = +i

(+1)*(-i) = -i

-1

(-1)*(+1) = -1

(-1)*(-1) = +1

(-1)*(+i) = -i

(-1)*(-i) = +i

-i

(-i)*(+1) = -i

(-i)*(-1) = +i

(-i)*(+i) = +1

(-i)*(-i) = -1

+i

(+i)*(+1) = +i

(+i)*(-1) = -i

(+i)*(+i) = -1

(+i)*(-i) = +1

-jx

(-jx)*(+1) = -jx

(-jx)*(-1) = +jx

(-jx)*(+i) = +qx

(-jx)*(-i) = -qx

+jx

(+jx)*(+1) = +jx

(+jx)*(-1) = -jx

(+jx)*(+i) = -qx

(+jx)*(-i) = +qx

+qx

(+qx)*(1) = +qx

(+qx)*(-1) = -qx

(+qx)*(+i) = +jx

(+qx)*(-i) = -jx

-qx

(-qx)*(+1) = -qx

(-qx)*(-1) = +qx

(-qx)*(+i) = -jx

(-qx)*(-i) = +jx

*

+jx

-jx

+qx

-qx

+1

(+1)*(+jx) = +jx

(+1)*(-jx) = -jx

(1)*(+qx) = +qx

(+1)*(-qx) = -qx

-1

(-1)*(+jx) = -jx

(-1)*(-jx) = +jx

(-1)*(+qx) = -qx

(-1)*(-qx) = +qx

-i

(-i)*(+jx) = +qx

(-i)*(-jx) = -qx

(-i)*(+qx) = -jx

(-i)*(-qx) = +jx

+i

(+i)*(+jx) = -qx

(+i)*(-jx) = +qx

(+i)*(+qx) = +jx

(+i)*(-qx) = -jx

-jx

(-jx)*(+jx) = +1

(-jx)*(-jx) = -1

(-jx)*(+qx) = -i

(-jx)*(-qx) = +i

+jx

(+jx)*(+jx) = -1

(+jx)*(-jx) = +1

(+jx)*(+qx) = +i

(+jx)*(-qx) = -i

+qx

(+qx)*(+jx) = +i

(+qx)*(-jx) = -i

(+qx)*(+qx) = 1

(+qx)*(-qx) = -1

-qx

(-qx)*(+jx) = -i

(-qx)*(-jx) = +i

(-qx)*(+qx) = -1

(-qx)*(-qx) = +1

Table 3. Multiplication Table for a Quaternion and the Complex Number Factor. Conjugates for the first factor place “+1” on the major diagonal.

Dot-Product and Cross-Product. Analogous to geometric vectors, the allnumber expressions, for example “jz*qz = i” and “jy*qz = qx”, can be separated using the dot-product “•” and the cross-product “x”. jz*qz = jz•qz + jzxqz = i + 0 = i 1•1 = 1 jx•1 = 0 jy•1 = 0 jz•1 = 0

1•jx = 0 jx•jx = -1 jy•jx = 0 jz•jx = 0

1•jy = 0 jx•jy = 0 jy•jy = -1 jz•jy = 0

;

jy*qz = jy•qz + jyxqz = 0 + qx = qx 1•jz = 0 jx•jz = 0 jy•jz = 0 jz•jz = -1

16 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1x1 = 0 jxx1 = jx jyx1 = jy jzx1 = jz

1xjx = jx jxxjx = 0 jyxjx = -jz jzxjx = +jy

1xjy = jy jxxjy = +jz jyxjy = 0 jzxjy = -jx

1xjz = jz jxxjz = -jy jyxjz = +jx jzxjz = 0

“4rC” and “4rD” are factors in “6A”. “3A” is “area”. 4rC

= 1rC + 3rC ;

1rC 1rC

= 1*c*tC

*j

;

3rC

= qx*xC + qy*yC + qz*zC

= c*tC*1*j ;

3rC

*j

= xC*qx*j + yC*qy*j + zC*qz*j

*4rD = 4rC*j•4rD + 4rC*jx4rD

4rC

*j

1rC

*j

3rC

*j

4rC

*j

•1rD = c*tC*(1*j*1)*c*tD = c*tC*c*tD •3rD = xC*qx*j*qx*xD + yC*qy*j*qy*yD + zC*qz*j*qz*zD = -(xC*xD + yC*yD + zC*zD) •4rD = 1rC*j•1rD + 3rC*j•3rD = c*tC*c*tD - (xC*xD + yC*yD + zC*zD) 6A

= 4rC*jx4rD = 1rC*j*3rD + 3rC*j*1rD + 3rC*jx3rD

3B

= 1rC*j*3rD + 3rC*j*1rD = (c*tC*qx*xD + c*tC*qy*yD + c*tC*qz*zD) + (qx*j*xC*c*tD + qy*j*yC*c*tD + qz*j*zC*c*tD)

= (c*tC*xD - c*tD*xC)*qx + (c*tC*yD - c*tD*yC)*qy + (c*tC*zD - c*tD*zC)*qz = Bx*qx + By*qy + Bz*qz 3A

= 3rC*jx3rD = xC*qx*j*qy*yD + xC*qx*j*qz*zD + yC*qy*j*qx*xD + yC*qy*j*qz*zD + zC*qz*j*qx*xD + zC*qz*j*qy*yD = (yC*zD - zC*yD)*jx + (zC*xD - xC*zD)*jy + (xC*yD - yC*xD)*jz = Ax*jx + Ay*jy + Az*jz

17 CHAPTER 1 - NUMBERS 6A

= 3B + 3A = Bx*qx + By*qy + Bz*qz + Ax*jx + Ay*jy + Az*jz = (Bx+ i*Ax)*qx + (By+ i*Ay)*qy + (Bz+ i*Az)*qz

The division operation as an inverse applies only to a complete multiplication operation “*” and not to a dot-product or cross-product (or triple-vector-product or remnant-product) alone. Triple-Vector-Product and Remnant-Product. A four-term summationform number “4rB” multiplied by a six-term summation-form number “6A” may be split between the triple-vector-product “■” and the remnantproduct “♦”. The triple-vector-product “■” has a mathematically imaginary result. The remnant-product “♦” has a mathematically real result (as given by the “r” (real) and “i” (imaginary) subscripts in the summation-form in the quaternion-complex-hypercomplex algebra). 4rB*6A

= 4rB*(4rC*jx4rD) = 4rB■(4rC*jx4rD) + 4rB♦(4rC*jx4rD)

Triple-Vector-Product. 4rB■6A

= 1rB*3A + 3rBx3B + 3rB•3A

4rB■(4rC

*j

x4rD) = 1rB*(3rC*jx3rD) + 3rBx(1rC*j*3rD) + 3rBx(3rC*j*1rD) + 3rB•(3rC*jx3rD)

= c*tB*((yC*zD - zC*yD)*jx + (zC*xD - xC*zD)*jy + (xC*yD - yC*xD)*jz) - ((yB*Bz - zB*By)*jx + (zB*Bx - xB*Bz)*jy + (xB*By - yB*Bx)*jz) + xB*(yC*zD - zC*yD)*qx*jx + yB*(zC*xD - xC*zD)*qy*jy + zB*(xC*yD - yC*xD)*qz*jz = c*tB*yC*zD*jx - c*tB*zC*yD*jx + c*tB*zC*xD*jy - c*tB*xC*zD*jy + c*tB*xC*yD*jz - c*tB*yC*xD*jz - ((yB*c*tC*zD - yB*c*tD*zC - zB*c*tC*yD + zB*c*tD*yC)*jx + (zB*c*tC*xD - zB*c*tD*xC - xB*c*tC*zD + xB*c*tD*zC)*jy + (xB*c*tC*yD - xB*c*tD*yC - yB*c*tC*xD - yB*c*tD*xC)*jz) + xB*yC*zD*i - xB*zC*yD*i + yB*zC*xD*i - yB*xC*zD*i + zB*xC*yD*i - zB*yC*xD*i

18 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = i*(xB*yC*zD + yB*zC*xD + zB*xC*yD - zB*yC*xD - xB*zC*yD - yB*xC*zD) + jx*(c*tB*yC*zD + zB*c*tC*yD + yB*zC*c*tD - c*tB*zC*yD - yB*c*tC*zD - zB*yC*c*tD) + jy*(c*tB*zC*xD + xB*c*tC*zD + zB*xC*c*tD - c*tB*xC*zD - zB*c*tC*xD - xB*zC*c*tD) + jz*(c*tB*xC*yD + yB*c*tC*xD + xB*yC*c*tD - c*tB*yC*xD - xB*c*tC*yD - yB*xC*c*tD) Notice there is no term with two “x” dimension factors, etc. Notice “i*4rB■6A  4rB■(i*6A)”. Contravariant vector “4kA” is perpendicular (orthogonal or normal) to each covariant factor “4rB”, “4rC”, and “4rD”. “World volume” “1w = (4rA*j•(4rB■(4rC*jx4rD)))” is in the denominator so that “4kA” is real and is per length, as opposed to the length cubed measurement units of the imaginary volume, “4VA = 4rB■(4rC*jx4rD)”. 4k A

= 4rB■(4rC*jx4rD)/(4rA*j•(4rB■(4rC*jx4rD))) •4rB = 4kA*j•4rC = 4kA*j•4rD = 0

4k A

*j

4r A

= 4kB■(4kC*jx4kD)/(4kA*j•(4kB■(4kC*jx4kD)))

Remnant-Product is the sum of terms not in the triple-vector-product. 4rB♦(4rC 4rB♦6A

*j

x4rD) = 4rB*(4rC*jx4rD) - 4rB■(4rC*jx4rD)

= 1rB*3B + 3rBx3A + 3rB•3B

4rB♦(3rC

*j

x3rD) = 1rB*(1rC*j*3rD) + 1rB*(3rC*j*1rD) + 3rBx(3rC*jx3rD) + 3rB•(1rC*j*3rD) + 3rB•(3rC*j*1rD)

= c*tB*(c*tC*xD - c*tD*xC)*qx + c*tB*(c*tC*yD - c*tD*yC)*qy + c*tB*(c*tC*zD - c*tD*zC)*qz + (yB*Az - zB*Ay)*qx + (zB*Ax - xB*Az)*qy + (xB*Ay - yB*Ax)*qz + xB*(c*tC*xD - c*tD*xC)*qx*qx + yB*(c*tC*yD - c*tD*yC)*qy*qy + zB*(c*tC*zD - c*tD*zC)*qz*qz

19 CHAPTER 1 - NUMBERS = (c*tB*c*tC*xD - c*tB*c*tD*xC)*qx + (c*tB*c*tC*yD - c*tB*c*tD*yC)*qy + (c*tB*c*tC*zD - c*tB*c*tD*zC)*qz + (yB*(xC*yD - yC*xD) - zB*(zC*xD - xC*zD))*qx + (zB*(yC*zD - zC*yD) - xB*(xC*yD - yC*xD))*qy + (xB*(zC*xD - xC*zD) - yB*(yC*zD - zC*yD))*qz + xB*c*tC*xD - xB*c*tD*xC + yB*c*tC*yD - yB*c*tD*yC + zB*c*tC*zD - zB*c*tD*zC = 1*(xB*c*tC*xD + yB*c*tC*yD + zB*c*tC*zD - xB*c*tD*xC - yB*c*tD*yC - zB*c*tD*zC) + qx*(c*tB*c*tC*xD + yB*xC*yD + zB*xC*zD - c*tB*c*tD*xC - yB*yC*xD - zB*zC*xD) + qy*(c*tB*c*tC*yD + zB*yC*zD + xB*yC*xD - c*tB*c*tD*yC - zB*zC*yD - xB*xC*yD) + qz*(c*tB*c*tC*zD + xB*zC*xD + yB*zC*yD - c*tB*c*tD*zC - xB*xC*zD - yB*yC*zD) The triple-vector-product and the remnant-product each result in a four-term summation-form number, one imaginary and the other real. There is a pattern to the count of terms in the products. The pattern is one (for a real number), four (for time-space location), six (for area), four (for volume), and one (for world volume). This pattern of numbers – 1,4,6,4,1 – is a row of Pascal’s Triangle.

1.4 Translation Back to Geometry The translation for step three has the following equivalences. “1” and “i” become “it” “qx” and “jx” become “ix”

; ;

“qy” and “jy” become “iy” “qz” and “jz” become “iz”

Axial Vectors. “Polar vectors” have “1”, “qx”, “qy”, and/or “qz” with real components. Examples are time-space location, contravariant location, energy-momentum, frequency-wavenumber, and electric field. “Axial vectors” have “jx”, “jy”, and/or “jz” such that their components are imaginary. Examples are area (only the imaginary

20 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY portion), torque, and magnetic field. There are other imaginary allnumber expressions, but we do not consider these to be axial vectors: volume four-term summation-form, and world volume. Axial geometric-vectors are called “pseudo-vectors” because the direction of the geometric-vector changes depending if the coordinate system is left-hand or right-hand. In the all-number algebra, “qx”, “qy”, and “qz” may be thought of as the direction indicated by the thumb and “jx”, “jy”, and “jz” may be thought of as direction of the curl of the fingers in a plane, with no thumb applied. The step three translation forces the thumb to be applied.

1.5 Singular-Label-Numbers 1 = 1 + q1

alpha (includes “i - jx”) beta gamma

β1 = q2 + i*q3 = q2*1 1 = 1 + β1 = 2*1

(1, 2, 3) are substituted by (x, y, z), (-x, z, y), (z, x, y), etc. with handedness retained. Therefore, (x, z, y) and (x, -y, z) are not valid.

x =>

1

1

i

i

-i

-i

; βx => 1

1

; x =>

1+i

1+i

1-i

1-i

The above 2x2 matrix isomorphs are singular matrices because the determinant is zero. For terms that commute, the determinant is calculated by multiplying the upper left by lower right and subtracting from it from upper right by lower left. A singular-label-number multiplied by its quaternion hypercomplex-conjugate equals zero:

1*j*1 = (1 - q1)*(1 + q1) = 0

;

β1*j*β1 = (-q2 - i*q3)*(q2 + i*q3) = 0

1*j*1 = (1*j + β1*j)*(1 + β1) = 1*j*1 + 1*j*β1 + β1*j*1 + β1*j*β1 = 0 + (1 - q1)*((q2)*(1 + q1)) + ((1 - q1)*(-q2))*(1 + q1) + 0 = 0

21 CHAPTER 1 - NUMBERS Squares differ between the three varieties.

12 = (1 + q1)*(1 + q1) ; = 1 + 2*q1 + q1*q1 = 2*1

β12 = (q2 + i*q3)*(q2 + i*q3) = q2*q2 + q2*i*q3 + i*q3*q2 + i*q3*i*q3 =0

12 = (1 + β1)*(1 + β1) = 12 + 1*β1 + β1*1 + β12 = 2*1 + q2*1*j*1 + q2*1*1 + 0 = 2*1 + q2*0 + 2*β1 + 0 = 2*1 A singular-label-number in the denominator is a division by zero and not allowed, and that gives us “singularity theorems”. Examples: ((1 + qx)/2)*((3 - qx)/2) = (1 + qx)/2 ((1  qx)/2)*exp(qx*α) = ((1  qx)/2)*exp(α) An exponential function with a singular label number argument has a result of unit magnitude, for example, “exp((i + jx)*/2) = -qx”. Singular-label-numbers should be thought of as part of a group rather than as individual anomalies.

1.6 Exercises 1) Write geometric-unit-vector representations for time-space “4r” and energy-momentum “4p”: Use “E/c” for the energy component (time component), and “px”, “py”, and “pz” for the three momentum components (three space components). Write allnumber representations for time-space location “4r” and “4p”. Write the hypercomplex-conjugates of “4r” and “4p”. 2) Write matrix multiplication operations for the below. jx2 = jy2 = jz2 = i2 = -1

;

qx2 = qy2 = qz2 = 12 = +1

22 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY jx*jy = -jy*jx = jz jy*jz = -jz*jy = jx jz*jx = -jx*jz = jy

; ; ;

i*jx = jx*i i*jy = jy*i i*jz = jz*i

qx = jx/i qy = jy/i qz = jz/i

qx*jy = -jy*qx = -qy*jx = jx*qy = qz ; qx*qy = -qy*qx = -i*qz = -jz qy*jz = -jz*qy = -qz*jy = jy*qz = qx ; qy*qz = -qz*qy = -i*qx = -jx qz*jx = -jx*qz = -qx*jz = jz*qx = qy ; qz*qx = -qx*qz = -i*qy = -jy

x*j*x = (1 - qx)*(1 + qx) = 0 ; x2 = (1 + qx)*(1 + qx) = 2*x y*j*y = (1 - qy)*(1 + qy) = 0 ; y2 = (1 + qy)*(1 + qy) = 2*y z*j*z = (1 - qz)*(1 + qz) = 0 ; z2 = (1 + qz)*(1 + qz) = 2*z βx*j*βx = (-qy - i*qz)*(qy + i*qz) = 0 ; βx2 = (qy + i*qz)*(qy + i*qz) = 0 βy*j*βy = (-qz - i*qx)*(qz + i*qx) = 0 ; βy2 = (qz + i*qx)*(qz + i*qx) = 0 βz*j*βz = (-qx - i*qy)*(qx + i*qy) = 0 ; βz2 = (qx + i*qy)*(qx + i*qy) = 0 x*j*x = (x*j + βx*j)*(x + βx) ; x2 = (x + βx)*(x + βx) = 2*x y*j*y = (y*j + βy*j)*(y + βy) ; y2 = (y + βy)*(y + βy) = 2*y z*j*z = (z*j + βz*j)*(z + βz) ; z2 = (z + βz)*(z + βz) = 2*z 3) Write the dot product and cross product multiplication tables using “q”’s rather than “j”’s. And, again for “q*j” on the left. 4) Translate the below summation-form all-number expressions into summation-form expressions that have geometric-unit-vectors as factors. Explain in words what the geometric-unit-vector translations represent in our physical world. 4rC 3rC 6A

= c*tC + qx*xC + qy*yC + qz*zC = c*tC + xC*qx*j + yC*qy*j + zC*qz*j

*j

= Bx*qx + By*qy + Bz*qz + Ax*jx + Ay*jy + Az*jz

4VA

= i*VAt + jx*VAx + jy*VAy + jz*VAz

World Volume = i*w

23 CHAPTER 1 - NUMBERS 5) For the below set of time-space location geometric-vectors “4rA”, “4rB”, “4rC” and “4rD” find the corresponding set of contravariant geometric-vectors “4kA”, “4kB”, “4kC” and “4kD”. Check with the dot product. And, then, as another check, repeat the process but use the derived contravariant geometric-vectors “4kA”, “4kB”, “4kC” and “4kD” substituting for the original (covariant) geometric-vectors, to find “4rA”, “4rB”, “4rC” and “4rD”. First Exercise: 4rA = 1*ix 4rB = 1*iy

4rC

= 3*it + 5*iz

4rD

= 2*it + 7*iz

Second Exercise: 4rA = 1*ix 4rB = 1*iy

4rC

= 2*it + 5*iz

4rD

= 3*it + 7*iz

Third Exercise: 4rA = 2*it + 1*ix

4rB

= 3*it + 1*iy

4rC

= 5*iz

4rD

= 7*iz

6) Prove the two time-space location geometric-vectors “4rC” and “4rD” factors are each perpendicular to the cross-product of “4rC” and “4rD”. Use the triple-vector-product set equal to zero. 7) A row in Pascal’s Triangle is 1 3 3 1. Using three-dimensional geometric-unit-vectors, relate the first “1” to a real number, the first “3” to the polar vector for location, the second “3” to the axial vector for area formed by the cross-product, and the second “1” to the volume formed by the dot-product. Next, for the row 1 2 1, model space using complex numbers: For two locations on the complex-plane modeled by “2rA” and “2rB”, take the complexconjugate of one of them, and then multiply them using the dot product, and relate the result to the 1 2 1 model of terms in vector expressions for the complex-plane. 8) Prove “a*b = ((a2 + b2) - (a2 - b2))/4” for rational numbers by stating which criteria are used in each step of the proof. Prove “-(a - b) = b - a” and “1/(a/b) = b/a” for rational numbers, also by referencing criteria. Prove the quadratic equation solution “x = -b/(2*a)  ((b*b - 4*a*c))/(2*a)” is the solution for “a*x2 + b*x + c = 0”. For the quadratic equation solution, what limitations must be placed onto “a”, “b”, “c”, and “x” for integers, rational numbers, real

24 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY numbers, and for complex numbers to apply. Prove the distributive property of the exponential operation over multiplication with reference to an algebra and criteria. Explain why this distributive property does not apply to “exp(qx*x + qy*y)”. 9) Prove the below singularity theorem. Why cannot “1  qx” be divided by both sides to equate “exp(qx*α)” to “exp(α)”? (1  qx)*exp(qx*α) = (1  qx)*exp(α) Select Solutions 1)

4p

4r

= c*t*it + x*ix + y*iy + z*iz ; 4r = 1*c*t + qx*x + qy*y + qz*z *j *j *j *j *j 4r = c*t*1 + x*qx + y*qy + z*qz = c*t*1 - x*qx - y*qy - z*qz

= (E/c)*it + px*ix + py*iy + pz*iz ; 4p = 1*(E/c) + qx*px + qy*py + qz*pz *j *j *j *j *j 4p = (E/c)*1 + px*qx + py*qy + pz*qz = (E/c)*1 - px*qx - py*qy - pz*qz 2) 0 i

jx2 = -1 =>

0

i

* i 0

i 0

0 i

0

-1

1

0

jx*jy = jz =>

* i 0

0*0 + i*i

0*i + i*0

=

-1 0 =

i*0 + 0*i i*i + 0*0

0 -1

0*0 + i*1 0*-1 + i*0 =

i 0 =

i*0 + 0*1 i*-1 + 0*0

0 -i

i*jx = jx*i i 0

0 i *

0 i

= i 0

i*0 + 0*i i*i + 0*0 0*i + i*0 0*0 + i*i 0 i i 0 = = * 0*0 + i*i 0*i + i*0 i*i + 0*0 i*0 + 0*i i 0 0 i

25 CHAPTER 1 - NUMBERS qx = jx/i 0 i

i 0 /

= -(

i 0 0 i

jx/i = -(jx*i)*(i/i) = -(jx*i) = qx

0 i i 0 i 0 i 0 0 i i 0 0*i + i*0 0*0 + i*i 0 1 * )*( / )=- * == i 0 0 i 0 i 0 i i 0 0 i i*i + 0*0 i*0 + 0*i 1 0

0 1

0 1

qx2 = +1 =>

0*0 + 1*1 0*1 + 1*0

*

=

1 0

1 0

0 1

1*0 + 0*1 1*1 + 0*0

0 -1

qx*jy = qz => 0 1

= 1*0 + 0*1 1*-1 + 0*0

0 i

0*0 + 1*-i

* 1 0

1 0

= 1 0

qx*qy = -jz =>

0 1

0*0 + 1*1 0*-1 + 1*0

* 1 0

1 0 =

0*i + 1*0

-i 0

= -i 0

0 -1 =

1*0 + 0*-i

1*i + 0*0

0 i

x*j*x = (1 - qx)*(1 + qx) = 0 1 -1

1 1 *

-1 1

1*1 + -1*1

1*1 + -1*1

= 1 1

0 0 =

-1*1 + 1*1

-1*1 + 1*1

0 0

βx*j*βx = (-qy - i*qz)*(qy + i*qz) = 0 -i*1

-i

i*1 i

-i -i

* --i -i*-1

= -i i*-1

i i *

i i

-i*i + -i*-i -i*i + -i*-i =

-i -i

0 0 =

i*i + i*-i

i*i + i*-i

0 0

x*j*x = (x*j + βx*j)*(x + βx) = 0 1-i -1-i * -1+i 1+i

(12-i2)-(12-i2)

1+i 1+i

(12-i2)-(12-i2)

= 1-i 1-i

0 0 =

-(12-i2)+(12-i2) -(12-i2)+(12-i2)

0 0

26 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY x2 = (1 + qx)*(1 + qx) = 2*x 1 1

1 1 *

1 1

1*1+1*1

1*1+1*1

= 1 1

2 2 =

1*1+1*1

1*1+1*1

2 2

βx2 = (qy + i*qz)*(qy + i*qz) = 0 i i

i 2-i2

i i *

-i -i

i 2-i2

= -i -i

0 0 =

-(i 2-i2) -(i 2-i2)

0 0

x2 = (x + βx)*(x + βx) = 2*x 1+i 1+i 1+i 1+i (12+2*i+i2)+(12-i2) (12+2*i+i2)+(12-i2) 2+2*i 2+2*i * = = 1-i 1-i 1-i 1-i (12-i2)+(12-2*i+i2) (12-i2)+(12-2*i+i2) 2-2*i 2+2*i 3) Solution: 1•1 = 1 qx•1 = 0 qy•1 = 0 qz•1 = 0

1•qx = 0 qx•qx = +1 qy•qx = 0 qz•qx = 0

1•qy = 0 1•qz = 0 qx•qy = 0 qx•qz = 0 qy•qy = +1 qy•qz = 0 qz•qy = 0 qz•qz = +1

1x1 = 0 qxx1 = qx qyx1 = qy qzx1 = qz

1xqx = qx qxxqx = 0 qyxqx = +jz qzxqx = -jy

1xqy = qy qxxqy = -jz qyxqy = 0 qzxqy = +jx

1*j•1 = 1 qx*j•1 = 0 qy*j•1 = 0 qz*j•1 = 0

1*j•qx = 0 qx*j•qx = -1 qy*j•qx = 0 qz*j•qx = 0

1*j•qy = 0 1*j•qz = 0 j qx* •qy = 0 qx*j•qz = 0 qy*j•qy = -1 qy*j•qz = 0 qz*j•qy = 0 qz*j•qz = -1

1*jx1 = 0 qx*jx1 = -qx qy*jx1 = -qy qz*jx1 = -qz

1*jxqx = qx qx*jxqx = 0 qy*jxqx = -jz qz*jxqx = +jy

1xqz = qz qxxqz = +jy qyxqz = -jx qzxqz = 0

1*jxqy = qy qx*jxqy = +jz qy*jxqy = 0 qz*jxqy = -jx

1*jxqz = qz qx*jxqz = -jy qy*jxqz = +jx qz*jxqz = 0

27 CHAPTER 1 - NUMBERS

4) Solutions: 4rC

4VA

= c*tC*it + xC*ix + yC*iy + zC*iz

;

3B

= Bx*ix + By*iy + Bz*iz A = Ax*ix + Ay*iy + Az*iz 3

= VAt*it + VAx*ix + VAy*iy + VAz*iz ; World Volume = w 5) Select Solution, First Exercise: 4rA = 1*ix 4rB = 1*iy 4rC = 3*it + 5*iz 4rA 4rA

= qx*1 *j

= 1*qx*j

4rB

= qy*1

4r B

*j

4rC

= 1*qy*j

= 3 + qz*5

4r C

*j

4 rD 4rD

= 3*1*j + 5*qz*j

= 2*it + 7*iz

= 2 + qz*7 4rD

*j

= 2*1*j + 7*qz*j

x3rA = (7*qz*j)x(qx*1) = 7*qz*j*qx*1 = jy*7 *j *j 1rD *3rA = (2*1 )x(qx*1) = 2*1 *qx*1 = qx*2 *j *j 3rD *1rA = (7*qz )x(0) = 0 *j *j *j *j *j 4rD x4rA = (2*1 + 7*qz )x(qx*1) = 2*1 *qx*1 + 7*qz *qx*1 = qx*2 + jy*7 3rD

*j *j

x3rB = (1*qx*j)x(qy*1) = 1*qx*j*qy = jz*1 1rA *3rB = (0)x(qy*1) = 0 *j *j 3rA *1rB = (1*qx )x(0) = 0 *j *j *j 4rA x4rB = (1*qx )x(qy*1) = 1*qx *qy = jz*1 3rA

*j *j

x3rC = (1*qy*j)x(qz*5) = 1*qy*j*qz*5 = jx*5 1rB *3rC = (0)x(qz*5) = 0 *j *j *j 3rB *1rC = (1*qy )x(3) = 1*qy *3 = qy*(-3) j j * * *j *j 4rB x4rC = (1*qy )x(3 + qz*5) = 1*qy *3 + 1*qy *qz*5 = qy*(-3) + jx*5 3rB

*j *j

x3rD = (5*qz*j)x(qz*7) = 0 *j *j 1rC *3rD = (3*1 )x(qz*7) = 3*1 *qz*7 = qz*21 *j *j *j 3rC *1rD = (5*qz )x(2) = 5*qz *2 = qz*(-10) *j *j *j *j *j 4rC x4rD = (3*1 + 5*qz )x(2 + qz*7) = 3*1 *qz*7 + 5*qz *2 = qz*(21 - 10) = qz*11 3rC

*j *j

28 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY *j

x4rA) = 1rC*(3rD*jx3rA) + 3rCx(1rD*j*3rA) + 3rCx(3rD*j*1rA) + 3rC•(3rD*jx3rA) = (3)*(jy*7) + (qz*5)x(qx*2) + (qz*5)x(0) + (qz*5)•(jy*7) = jy*21 + jy*(-10) + 0 + 0 = jy*11

4rC■(4rD

*j

x4rB) = 1rD*(3rA*jx3rB) + 3rDx(1rA*j*3rB) + 3rDx(3rA*j*1rB) + 3rD•(3rA*jx3rB) = (2)*(jz*1) + (qz*7)x(0) + (qz*7)x(0) + (qz*7)•(jz*1) = jz*2 + i*7

4rD■(4rA

*j

x4rC) = 1rA*(3rB*jx3rC) + 3rAx(1rB*j*3rC) + 3rAx(3rB*j*1rC) + 3rA•(3rB*jx3rC) = (0)*(jx*5) + (qx*1)x(0) + (qx*1)x(qy*(-3)) + (qx*1)•(jx*5) = jz*3 + i*5

4rA■(4rB

*j

x4rD) = 1rB*(3rC*jx3rD) + 3rBx(1rC*j*3rD) + 3rBx(3rC*j*1rD) + 3rB•(3rC*jx3rD) = (0)*(0) + (qy*1)x(qz*21) + (qy*1)x(qz*(-10))) + (qy*1)•(0) = jx*(-21+10) = jx*(-11)

4rB■(4rC

1w

= (4rA*j•(4rB■(4rC*jx4rD))) = (1*qx*j)•(jx*(-11)) = i*11

= 4rC■(4rD*jx4rA)/1w = (jy*11)/(i*11) = qy*1 *j 4kC = 4rD■(4rA x4rB)/1w = (jz*2 + i*7)/(i*11) = (7/11) + qz*(2/11) *j 4kD = 4rA■(4rB x4rC)/1w = (jz*3 + i*5)/(i*11) = (5/11) + qz*(3/11) *j 4kA = 4rB■(4rC x4rD)/1w = (jx*(-11))/(i*11) = qx*(-1) 4k B

•4rA = 0 4kC •4rA = 0 *j 4kD •4rA = 0 *j 4kA •4rB = 0 4k B

*j *j

•4rC = 0 4kC •4rB = 0 *j 4kD •4rB = 0 *j 4kA •4rC = 0 4kB

•4rD = 0 •4rD = (7/11)*2 - (2/11)*7 = 0 *j 4kD •4rC = (5/11)*3 - (3/11)*5 = 0 *j 4kA •4rD = 0

*j

4kB

*j

*j

4kC

*j

6) The triple-vector-product being equal to zero indicates the geometric-vector is perpendicular to the cross-product of that geometric-vector with another geometric-vector. *j

x4rD) = 1rC*(3rC*jx3rD) + 3rCx(1rC*j*3rD) + 3rCx(3rC*j*1rD) + 3rC•(3rC*jx3rD)

4rC■(4rC

29 CHAPTER 1 - NUMBERS = i*(xC*yC*zD + yC*zC*xD + zC*xC*yD - zC*yC*xD - xC*zC*yD - yC*xC*zD) + jx*(c*tC*yC*zD + zC*c*tC*yD + yC*zC*c*tD - c*tC*zC*yD - yC*c*tC*zD - zC*yC*c*tD) + jy*(c*tC*zC*xD + xC*c*tC*zD + zC*xC*c*tD - c*tC*xC*zD - zC*c*tC*xD - xC*zC*c*tD) + jz*(c*tC*xC*yD + yC*c*tC*xD + xC*yC*c*tD - c*tC*yC*xD - xC*c*tC*yD - yC*xC*c*tD) = i*(0) + jx*(0) + jy*(0) + jz*(0) =0 7) Solutions: The first “1” in “1 3 3 1” pertains to the real numbers. The first “3” in “1 3 3 1” pertains to the polar vectors. 3rB

= xB*ix + yB*iy + zB*iz = xC*ix + yC*iy + zD*iz 3rD = xD*ix + yD*iy + zD*iz 3rC

The second “3” in “1 3 3 1” pertains to the axial vectors. 3A = 3rCx3rD = (xC*ix + yC*iy + zC*iz)x(xD*ix + yD*iy + zD*iz) = ((yC*zD* - zC*yD)*ix + (zC*xD* - xC*zD)*iy + (xC*yD* - yC*xD)*iz) = Ax*ix + Ay*iy + Az*iz

The second “1” in “1 3 3 1” pertains to the dot-product of a crossproduct to form a scalar (non-vector) result. V = 3rB•(3rCx3rD) = (xB*ix + yB*iy + zB*iz)•(Ax*ix + Ay*iy + Az*iz) = xB*Ax + yB*Ay + zB*Az The first “1” in “1 2 1” pertains to the real numbers. The “2” in “1 2 1” pertains to a complex number, including one complex number times another. = xA + i*yA ; C = 2rA*j*2rB = (xA + yA*i*i)*(xB + i*yB) *i = (xA*xB + yA*yB*(i*i*i)) + (i*xA*yB + i*i*yA*xB) 2rA = xA + yA*i = (xA*xB + yA*yB) + i*(xA*yB - yA*xB) 2rB = xB + i*yB = A + i*B 2rA

30 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The second “1” in “1 2 1” pertains to a scalar number that is the result of a dot-product operation. 2rA

= xA + i*yA *i r 2 A = xA + yA*i 2rB = xB + i*yB

;

A = 2rA*j•2rB = (xA*xB + yA*yB*(i*i*i)) = (xA*xB + yA*yB)

8) Solution not given. 9) Solution:

(1  qx)*exp(qx*α) = (1  qx)*(cosh(α) + qx*sinh(α)) = (1  qx)*cosh(α) + qx*(1  qx)*sinh(α) = (1  qx)*cosh(α)  qx*(1  qx)*sinh(α) = (1  qx)*cosh(α) + (1  qx)*sinh(α) = (1  qx)*(cosh(α) + sinh(α)) = (1  qx)*exp(α)

Division by “1  qx” is prohibited because a singular-labelnumber cannot be in the denominator, to avoid a division by zero that would lead to the non-sense result of equating “exp(qx*α)” to “exp(α)”.

31 CHAPTER 1 - NUMBERS Further Thought. 1) By what criteria are some theories of pure mathematics segregated away from applied mathematics? 2) Label-numbers “i”, “j”, and “k” in “{1, i, j, k, i*j, j*k, k*i, i*j*k}” square to negative one and commute. With negatives, prove this set forms a division algebra by reviewing each of the five criteria. 3) Write analogous equations to those below, using Pauli Spin Matrices (“PSMx = jx/i”, “PSMy = jy*i” and “PSMz = jz/i”). qx = jx/i ; qy = jy/i ; qz = jz/i qx*jy = -jy*qx = -qy*jx = jx*qy = qz qy*jz = -jz*qy = -qz*jy = jy*qz = qx qz*jx = -jx*qz = -qx*jz = jz*qx = qy

qx2 = qy2 = qz2 = 12 = +1

; ; ; ;

qx*qy = -qy*qx = -i*qz = -jz qy*qz = -qz*qy = -i*qx = -jx qz*qx = -qx*qz = -i*qy = -jy

4) 2x2 matrix isomorphs for the quaternions did not include intermediate combinations of the three 2x2 matrix isomorphs. Assign “a = exp(i*)” and “b = exp(i*)” in the below 2x2 matrix. Select “”, “” and “” to create “1”, “jx”, “jy”, and “jz”. Try to select other values for “”, “” and “” retaining “jx = jy*jz = -jz*jy”, “jy = jz*jx = -jx*jz”, and “jz = jx*jy = -jy*jx”. Try a different format for the 2x2 matrix, too. a*sin()

-b*i*cos()

exp(i*)*sin() -exp(-i*)*cos() =

b*cos()

a*i*sin()

exp(i*)*cos()

exp(-i*)*sin()

5) If organized as a group, singular-label-numbers would have each of the five criteria of a mathematical group addressed. Try to define the algebra for singular-label-numbers. 6) For a better understanding, push math to the extremes, and further. Quaternions are pushed to the extreme in octonions, sedonions, and beyond. Read the Appendix on octonionsedonion algebra.

32 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

33 CHAPTER 2 – PARTICLES

Chapter 2 – Particles Einstein’s Special Theory of Relativity provides the method for transforming the mathematical description of physical entities, called invariants, from one constant speed observer vantage to a different constant speed observer vantage. Two examples of invariants are timespace location and energy-momentum. A person seated in a moving bus has their own speed vantage from which they measure time, spacelocation, energy, and momentum component values for a baseball thrown forward from the back of the bus. Because the bus is moving, a person standing on the roadside has a different vantage, and so measures different component values for the same baseball. The change in component values is quantified through the mathematics of Special Relativity by use of the Lorentz Transformation. The Lorentz Transformation is the addition of time-space hyperbolic angles.

2.1 Hypercomplex-Plane In this chapter, Special Relativity is presented with the “x” component the only space component. Time-space location “2r = 1*c*t + qx*x” is visualized by plotting components “c*t” and “x” on a Cartesian grid called the time-space “hypercomplex-plane”. The hypercomplex-plane is an analogy to the “complex-plane” used to illustrate a complex number.

Figure 3. Complex-Plane and Radial Coordinate System.

A complex-plane is drawn on a sheet of paper using two space dimensions “x” and “y”. “” is theta.

34 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2z

= x + i*y = r*cos + i*r*sin = r*exp(i*)

1 = (cos2 + sin2) ; r = (x2 + y2) Rotate the complex-plane coordinate system by rotating the sheet of paper. For the hypercomplex-plane, substitute “jx*x” for “i*y” and “i*c*t” for “x” and find “i*2r = i*c*t + jx*x”. The right-side of “i*2r = i*c*t + jx*x” has two label-numbers that square to negative one, “i2 = -1” and “jx2 = -1”. That makes it different from the right side of “2z = x + i*y”. Because of that difference, the hypercomplex-plane cannot be rotated by rotating the sheet of paper. Divide “i*2r = i*c*t + jx*x” by “i” to get “2r = c*t + qx*x”, in which 2 “qx = +1”. “rhyperbolic” is the hyperbolic-radius. “α” (alpha) is the timespace hyperbolic-angle. 2r

= 1*c*t + qx*x = 1*rhyperbolic*coshα + qx*rhyperbolic*sinhα = rhyperbolic*exp(qx*α)

1 = (cosh2α - sinh2α) coshα = c*t/rhyperbolic

; rhyperbolic = ((c*t)2 - x2) ;

sinhα = x/rhyperbolic

; tanhα = x/(c*t)

exp(qx*α) = coshα + qx*sinhα To illustrate a time-space hypercomplex-plane on a space-space sheet of paper, replace “qx” with “i” and call it “qx-illustrated”. 2rillustrated

= 1*c*t + i*x = 1*rillustrated*cosαillustrated + i*rillustrated*sinαillustrated = rillustrated*exp(i*αillustrated)

1 = (cos2αillustrated + sin2αillustrated)

; rillustrated = ((c*t)2 + x2)

cosαillustrated = c*t/rillustrated ; sinαillustrated = x/rillustrated ; tanαillustrated = x/(c*t) exp(i*αillustrated) = cosαillustrated + i*sinαillustrated

35 CHAPTER 2 – PARTICLES The difference between the circular trigonometric functions used in the illustration and the hyperbolic trigonometric functions in the timespace location expression causes distortion. Vertical and horizontal displacements are illustrated correctly. Lines at an angle are distorted by appearing longer than in time-space reality (because “rillustrated = ((c*t)2 + x2) > rhyperbolic = ((c*t)2 - x2)”). The diagonal lines (for which “(c*t)2 = x2”) are completely distorted because any point illustrated on the diagonal has a zero hyperbolic-radius in time-space reality.

Figure 4. The Illustrated Hypercomplex-Plane with the Hyperbolic Radial Coordinate System.

A light source at the origin emits one photon that travels along the positive “x” axis at the speed-of-light. The plot of points “c*t” and “x” is called the world-line which, for the photon, is the diagonal from the origin to the upper-right. The world-line of the light source is straight up the “c*t” axis, because “x = 0”, and the hyperbolic-radius equals “c*t”. In contrast to the light source, the hyperbolic radius for the photon remains zero because “c*t = x”. Because “x = rhyperbolic*sinhα”, “α” is large (to the ultimate “1/0” if we allow division by zero).

36 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

2.2 Inertial Reference Frames Selected origins for “c*t” and “x” are valid for both the moving and the stationary inertial (constant speed) reference frames. The visualization is a big rectangular prism bus moving down a road toward the right. Inertial reference frame “M” is the interior of the moving bus. Inertial reference frame “S” is the stationary roadside. A person seated on the moving bus takes measurements of component values. These are written with subscript “M”. A person standing on the roadside also takes measurements. Those component values are written with subscript “S”. “(c*tM, xM) = (c*tS, xS) = (0, 0)”: When “c*tM = c*tS = 0”, the back of the bus (“xM = 0”) and the speed limit sign (“xS = 0”) identify the same point in space. “xM” is measured using measurement tape along the floor of the bus. “xS” is measured using measurement tape along the roadside. “tM” is measured using a clock mounted on the bus wall. “tS” is measured using a clock mounted on the roadside. Define a physical system (of components and label-numbers) inside bus “M”, and then apply a “Lorentz Transformation” using the speed of the bus “S/M” to describe the same physical system observed from roadside “S”. A Lorentz Transformation is a hyperbolic-angle rotation. Rest Frame “B” of the object in motion is a third inertial reference frame. The baseball’s own rest frame “B” has the ball stopped. Therefore, in “B”, there is advancement of time “tB”, but there is no change in space location (“xB = 0”). And, there is rest mass energy per “EB = mB*c2”, but there is no momentum, “pxB = 0”. “vM” for the baseball could be written “vM/B” because “vB = 0”. The “Relative” in “Special Relativity”. “Relativity” is the ancient concept that all inertial frames have the same preference. In the movie Agora watch a re-enactment of Hypatia of Alexandria (who lived around 390 AD / 390 CE) measure locations on the deck of weights dropped from a ship’s mast to verify there is no preferred reference frame. Galileo (around 1600 AD / 1600 CE) described in one

37 CHAPTER 2 – PARTICLES of his books the experiment of dropping weights from a ship’s mast. And, Galileo created the visualization for the lack of a preferred reference frame by describing what a person would observe when isolated inside the interior of a moving ship. Einstein (in year 1905) added to relativity the requirement that the speed-of-light be the same regardless of reference frame and that meant time and space measurements were specific to a reference frame. The change in space is easy to imagine. Place a measuring tape along the floor inside the bus and another measuring tape on the ground along the roadside and see the two coincide only when time equals zero (at a non-relativistic speed). The change in time is not so easy to imagine, and that’s the trick. Geometric Problem Definition. On bus “M” we have measured or have calculated component values for time-space location “2r” and energymomentum “2p” for a moving baseball. 2r

;

= c*tM*itM + xM*ixM

2p

= (EM/c)*itM + pxM*ixM

Baseball “B” of rest mass “mB” was thrown at speed “vM” in the positive “+xM” direction inside the bus at “tM = 0”. “xM” can be calculated from “xM = vM*tM”. Total energy “EM” equals rest mass energy “EB = mB*c2” added to kinetic energy. Mechanical momentum “pxM” equals relativistic mass “EM/c2” times speed “vM”, “pxM = vM*(EM/c2)”. Geometric-unit-vectors “itM” and “ixM” are the same in energymomentum “2p” as they are in time-space “2r”. The bus moves at a speed “vS/M”. Our objective is to use the “M” components and the bus speed “vS/M” to calculate the same four components relative to roadside “S”. 2r

= c*tS*itS + xS*ixS

;

2p

= (ES/c)*itS + pxS*ixS

38 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Figure 5. Moving and Stationary Reference Frames.

2.3 The Unspecified-Speed-Parameter Begin the second step by replacing: itM with exp(-qx*)

;

ixM with qx*exp(-qx*)

If “itM” had been replaced with “1” and “ixM” with “qx”, then the replacement would have implied a preference for the moving coordinate system “M”. To avoid that preference, “exp(-qx*)” uses the “unspecifiedspeed-parameter”, “” (sigma, or, in English, esse). If “” is included in a label-number, then we call that label-number a “compound-label-number”. A compound-label-number contrasts with a “simple-label-number”, such as “1” or “qx”. “” is unknown and unknowable. It is different from independent variable “x” because “x” is intended to be substituted with a selected single valued real number. In other words, although “x” is not known, “x” is knowable. In contrast, “” is unknowable. We are less likely to mistakenly assume a preferred reference frame if continuously reminded about the unspecified-speed-parameter

39 CHAPTER 2 – PARTICLES “” by using compound-label-numbers. (Explicit use of “” ensures we do not violate “gauge invariance”.) 1M = exp(-qx*) = cosh - qx*sinh qxM = qx*1M = qx*exp(-qx*) = -sinh + qx*cosh = exp(-qx*)*qx = 1M*qx

2.4 Compound-Label-Numbers and Components Retain the component values in the translation from a geometric representation to the all-number (more abstract) representation. 2r

= c*tM*itM + xM*ixM

2r

= 1M*c*tM + qxM*xM = exp(-qx*)*(1*c*tM + qx*xM) = exp(-qx*)*(1*c*tB*cosh(αM) + qx*c*tB*sinh(αM)) = exp(-qx*)*c*tB*exp(qx*αM)

2p

= (EM/c)*itM + pxM*ixM

2p

= 1M*(EM/c) + qxM*pxM = exp(-qx*)*(1*EM/c + qx*pxM) = exp(-qx*)*(1*mB*c*cosh(αM) + qx*mB*c*sinh(αM)) = exp(-qx*)*mB*c*exp(qx*αM)

An advantage of all-number expressions is the use of “exp”. c*tB*exp(qx*αM) = c*tB*coshαM + qx*c*tB*sinhαM = c*tM + qx*xM c*tM = c*tB*coshαM

;

xM = c*tB*sinhαM

EM/c = mB*c*coshαM

;

pxM = mB*c*sinhαM

“αM” (alpha) relates to speed “vM” of the baseball on the bus. (vM/c) = tanhαM

αM = atanh(vM/c)

40 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Figure 6. The hyperbolic tangent relationship between the speed “v” and the hyperbolic-angle “α”.

For the above equations, the baseball was thrown forward from the back of the bus at time “tM = 0”, and the baseball has a constant speed “vM” in the positive “xM” direction. Speed-parameter “αM” is the hyperbolic-angle for motion of the baseball inside the bus. “c*tB” and “mB*c” are each the hyperbolic-radius in their respective “c*tM”/”xM” or “EM/c”/”pxM” hypercomplex-planes and are calculated from the Pythagorean Theorem. 12 = cosh2αM - sinh2αM ; (c*tB)2 = (c*tM)2 - xM2 ; (mB*c)2 = (EM/c)2 - pxM2

Figure 7. Energy “E” and momentum “px” approach infinity as speed “v” approaches the speed-of-light “c”.

41 CHAPTER 2 – PARTICLES Per the “cosh” and “sinh” functions, energy “E/c” and momentum “px” increase toward infinity as speed “v” increases toward the speed-oflight “c”.

Figure 8. Time and space in “M” for an object “B” at time “tM”.

Figure 9. Energy and momentum for an object “B” of mass “m” moving in “M”.

“xM = vM*tM” and “pxM = vM*EM/c2” are proven valid. xM = c*tB*sinhαM = c*tB*coshαM*tanhαM = c*tM*tanhαM = c*tM*vM/c = vM*tM pxM = mB*c*sinhαM = mB*c*coshαM*tanhαM = (EM/c)*tanhαM = (EM/c)*vM/c = vM*EM/c2

42 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Non-Relativistic Approximation. The non-relativistic approximation “αM < 0.3” means the “tanh” and “sinh” functions are not noticeable. tanhαM  sinhαM  vM/c  αM

αM kxM” because “cosh(αM) > sinh(αM)”, and that contrasts with “M/c < kxM” for “2k”. Space-like yard-stick “2s” partners with time-like “2r”. 2s

= 1M*qx*sB*exp(qx*αM) = qxM*sB*exp(qx*αM)

49 CHAPTER 2 – PARTICLES

Figure 12. Hypercomplex-plane for frequency as the hyperbolic-radius.

Figure 13. Time-like and space-like.

Figure 14. World lines for the location in “M” of many baseballs of different speeds “vM” at the same time “tB” showing the time-like locations on the hyperbola.

50 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Figure 15. Location in “M” of a yardstick.

Figure 16. Locations of the right ends of many yardsticks, each at a different speed, to illustrate the space-like locations.

Lorentz Transformation from Time-Like to Space-Like. αS/M = (i - jx)*/2 2r

= 1M*(c*tB)*exp(qx*αM)*1/1 = 1M*(c*tB)*exp(qx*αM)*(exp(qx*((i - jx)*/2))/exp(qx*((i - jx)*/2))) = 1M*(c*tB)*exp(qx*αM)*(exp(qx*(i*/2 - jx*/2))/exp(qx*(i*/2 - jx*/2))) = 1M*(c*tB)*exp(qx*αM) *(exp(jx*/2)*exp(-i*/2))/(exp(jx*/2)*exp(-i*/2)) = 1M*(c*tB)*exp(qx*αM)*(jx*(-i))/(jx*(-i))

51 CHAPTER 2 – PARTICLES = 1M*(c*tB)*exp(qx*αM)*(qx/qx) = (1M/qx)*((qx*c*tB)*exp(qx*αM)) = 1S*(qx*c*tB)*exp(qx*αM) 1S = 1M/qx = 1M*qx = qxM

;

qxS = qxM/qx = 1M

c*tS + qx*xS = (qx*c*tB)*exp(qx*αM) c*tS = qx*c*tB*qx*sinh(αM) = xM ; qx*xS = qx*c*tB*cosh(αM) = qx*c*tM 2r

= 1S*c*tS + qxS*xS = (qxM)*(xM) + (1M)*(c*tM) = 1M*c*tM + qxM*xM = 2r

vS/c = tanhαS = tanh(αM + (i - jx)*/2) = tanh(αM + i*/2 - jx*/2) = sinh(αM + i*/2 - jx*/2)/cosh(αM + i*/2 - jx*/2) = (sinh(αM)*cosh(i*/2 - jx*/2) + cosh(αM)*sinh(i*/2 - jx*/2)) /(cosh(αM)*cosh(i*/2 - jx*/2) + sinh(αM)*sinh(i*/2 - jx*/2)) = (sinh(αM)*cosh(i*/2)*cosh(-jx*/2) + sinh(αM)*sinh(i*/2)*sinh(-jx*/2) + cosh(αM)*sinh(i*/2)*cosh(-jx*/2) + cosh(αM)*cosh(i*/2)*sinh(-jx*/2)) /(cosh(αM)*cosh(i*/2)*cosh(-jx*/2) + cosh(αM)*sinh(i*/2)*sinh(-jx*/2) + sinh(αM)*sinh(i*/2)*cosh(-jx*/2) + sinh(αM)*cosh(i*/2)*sinh(-jx*/2)) = (sinh(αM)*cos(/2)*cos(/2) + qx*sinh(αM)*sin(/2)*sin(/2) + i*cosh(αM)*sin(/2)*cos(/2) - jx*cosh(αM)*cos(/2)*sin(/2)) /(cosh(αM)*cos(/2)*cos(/2) + qx*cosh(αM)*sin(/2)*sin(/2) + i*sinh(αM)*sin(/2)*cos(/2) - jx*sinh(αM)*cos(/2)*sin(/2)) = (qx*sinhαM)/(qx*coshαM) = tanhαM = vM/c

52 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY vS/M/c = tanhαS/M = tanh((i - jx)*/2) = tanh(i*/2 - jx*/2) = sinh(i*/2 - jx*/2)/cosh(i*/2 - jx*/2) = (sinh(i*/2)*cosh(-jx*/2) + cosh(i*/2)*sinh(-jx*/2)) /(cosh(i*/2)*cosh(-jx*/2) + sinh(i*/2)*sinh(-jx*/2) = (i*sin(/2)*cos(/2) - jx*cos(/2)*sin(/2)) /(cos(/2)*cos(/2) + qx*sin(/2)*sin(/2)) = 0/qx = 0 The tangent with a sum of angles identity was not used because it is derived using a division by cosine, and cosine of half pi equals zero. Bus “M” moves at speed “vS/M/c = 0”. Observer “S” observes the baseball inside the bus per “c*tS = xM”, “xS = c*tM” and “vS = vM”. We look through the bus window and see the ticks of the clock are nodes along a line. Nodes along a line are ticks of a clock. There are no examples in physics for which time and space are swapped, as if a theory of physics prevents it from happening. “αS/M = (i - jx)*/2” includes a singular-label-number factor “i - jx” so that “αS/M = (i - jx)*/2” is a variation of the number zero. As such, “αS/M = (i - jx)*/2” cannot be in a denominator, which it is not. Also, “exp(qx*((i - jx)*/2)) = qx” must be a variation of the number one, which it is, because “qx” is a variation of the number one. A hypercomplex hyperbolic-angle “αS/M” is called (in this book) an “exotic Lorentz Transformation”.

2.8 Electric Current Density Time-like electric current density is a static electric charge. Space-like electric current density is electric current in a wire. Time-Like Electric Current Density. 2J

= 1M*(dQB/dxB)*exp(qx*αM) = 1M*xB*exp(qx*αM)

53 CHAPTER 2 – PARTICLES “xB = dQB/dxB” (Coulombs per centimeter) refers to rest frame “B” twice: Electric charge “QB” and rest space location “xB”.

Figure 17. Hypercomplex-plane for charge density “JtB = xB = dQB/dxB” as the time-like hyperbolic-radius.

Space-Like Electric Current Density. Inside a wire: •

Stationary positive electric charges “2J+” with “xB+ > 0” (in the atomic nuclei), for which “vM+ = 0”, “αM+ = 0” and “exp(qx*αM+) = 1”

•

Moving negative electric charges “2J-” with “xB- < 0” (the valence electrons), for which “vM-/c = tanh(αM-) > 0” and, for a physically real wire, “vM-/c  αM-”.

The sum “2J = 2J+ + 2J-” time component in “M” must be zero. 2J

= 2J+ + 2J= 1M*xB+*exp(qx*αM+) + 1M*xB-*exp(qx*αM-) = 1M*xB+ + 1M*xB-*exp(qx*αM-) = 1M*xB+ + 1M*xB-*(coshαM- + qx*sinhαM-) = 1M*(xB+ + xB-*coshαM-) + qxM*xB-*sinhαM= qxM*xB-*sinhαM= qxM*(-xB+/coshαM-)*sinhαM= qxM*-xB+*tanhαM= qxM*xB+*(-vM-/c)

54 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The requirement of a zero time component was satisfied by setting “xB+ + xB-*coshαM- = 0”, from which “xB- = -xB+/coshαM-”. Regardless of “vM-/c” being small, current “2J = qxM*xB+*(-vM-/c)” in a wire produces a very noticeable magnetic field around the wire. The wire with the electric current can, as a piece of metal, move in the positive “x” direction with speed “vS”. With respect to “S”, the hyperbolic-radius of the current “qx*(xB+*(-vM-/c))” is space-like. Spacelike current density in “S” is not commonly useful. 2J

= 1S*qx*(xB+*(-vM-/c))*exp(qx*αS/M) = qxS*(xB+*(-vM-/c))*exp(qx*αS/M)

Figure 18. Hypercomplex-plane for a space-like current density.

Space-Like Invariant Complement for Energy-Momentum. “nB” relates to rest mass “mB” as length “sB” relates to time “c*tB”. 2pspace-like

= 1M*qx*nB*c*exp(qx*αM) = qxM*nB*c*exp(qx*αM) = 1M*nB*c*sinh(αM) + qxM*nB*c*cosh(αM) = 1M*(EM/c) + qxM*pxM

“αS/M = (i - jx)*/2” has been used to create “2pspace-like” from “2p”, but although we can create “2pspace-like”, we cannot find a physical example. Can there be momentum without energy?

55 CHAPTER 2 – PARTICLES Angular Momentum Density Invariant. In two-dimensional time-space, the axis of rotation for angular momentum is limited to being in the “x” direction. The visualization is a spinning ring. Use the right-hand-rule: The fingers curl with the direction of rotation and the right-hand thumb points in the direction of angular momentum. If we look to a higher value of “x”, then a clockwise rotation (top to the right) has a positive value of angular momentum. “HxB” is the rest-frame angular momentum of a spinning ring. It is similar to rest-frame electric charge “QB” because its value does not change with respect to the speed of an observer. Like “QB”, “HxB” is an invariant in itself: “1 = cosh(αM)/cosh(αM)” has the numerator from relativistic mass and the denominator from time dilation. If “HxB = dHxB/dxB” is the density of angular momentum along a particle’s length, then the (space-like) invariant is 2H

= 1M*qx*HxB*exp(qx*αM) = 1M*HxB*sinh(αM) + qxM*HxB*cosh(αM) = 1M*HtM + qxM*HxM

Space component “HxM = HxB*cosh(αM)” is positive or negative depending on the spin direction of “HxB”. Time component “HtM = HxB*sinh(αM)” is positive or negative depending on “sinh(αM)” and quantifies angular momentum that passes a point “xM”. Angular momentum density is similar to electric current density, with the difference being space-like vs time-like, and that difference shows in their respective conservation laws: “((2)*jsn)x(2H) = 0” as opposed to “((2)*jsn)•(2J) = 0”.

2.9 Motion Faster than Light Inside bus “M” a baseball moves toward the front at speed “vM < c”, as observed by a person seated on the bus. A person standing on the roadside “S” observes the baseball moving at speed “vS > c”. Bus speed hyperbolic-angle “αS/M = i*/2” is not real, “αS/M  R”. vS/M/c = tanhS/M = tanh(i*/2) = i*tan(/2) = i*(1/0) = 1/0

56 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Bus speed “vS/M/c = 1/0” is instantaneous movement along “x” so that the bus is at all locations “x” simultaneously, but only for an instant. General form for “αS/M = i*/2” and “αM  R”: 2r

= 1M*(c*tB)*exp(qx*αM) = 1M*(c*tB)*exp(qx*αM)*(exp(qx*αS/M)/exp(qx*αS/M)) = 1M*(c*tB)*exp(qx*αM)*(exp(qx*i*/2)/exp(qx*i*/2)) = 1M*(c*tB)*exp(qx*αM)*(exp(jx*/2)/(exp(jx*/2)) = 1M*(c*tB)*exp(qx*αM)*(jx/jx) = (1M/jx)*((jx*c*tB)*exp(qx*αM)) = (-i*1M/qx)*((i*qx*c*tB)*exp(qx*αM))

1S = -i*1M/qx = -i*qxM

;

qxS = qx*i*qxM = -i*1M

Hyperbolic radius “i*qx*c*tB” is space-like and imaginary for “tB” real. c*tS + qx*xS = (i*qx*c*tB)*exp(qx*αM) c*tS = i*qx*c*tB*qx*sinh(αM) = i*xM

; c*tS = c*tM*coshαS/M + xM*sinhαS/M = c*tM*cosh(i*/2) + xM*sinh(i*/2) = c*tM*cos(/2) + i*xM*sin(/2) = i*xM

qx*xS = i*qx*c*tB*cosh(αM) = i*qx*c*tM

;

xS = c*tM*sinhαS/M + xM*coshαS/M = c*tM*sinh(i*/2) + xM*cosh(i*/2) = i*c*tM*sin(/2) + xM*cos(/2) = i*c*tM

“2r” is confirmed to be invariant. 2r

= 1S*c*tS + qxS*xS = (-i*qxM)*(i*xM) + (-i*1M)*(i*c*tM) = qxM*xM + 1M*c*tM = 1M*c*tM + qxM*xM = 2r

“c*tB” is real: c*tB = ((c*tS)2 - xS2) = ((i*xM)2 - (i*c*tM)2) = ((c*tM)2 - xM2) = c*tB

57 CHAPTER 2 – PARTICLES Time and space (in “M”) is swapped for space and time (in “S”), respectively, and that means “M” is drawn on the “S” hypercomplexplane up from the “xS” axis toward the “c*tS” axis, to illustrate “αS = αM + i*/2” drawn down from the “c*tS” axis. αS = αM + αS/M = αM + i*/2 vS/c = tanhαS = tanh(αM + i*/2) = sinh(αM + i*/2)/cosh(αM + i*/2) = (sinh(αM)*cosh(i*/2) + cosh(αM)*sinh(i*/2))/ (cosh(αM)*cosh(i*/2) + sinh(αM)*sinh(i*/2)) = (sinh(αM)*cos(/2) + cosh(αM)*i*sin(/2))/ (cosh(αM)*cos(/2) + sinh(αM)*i*sin(/2)) = (i*coshαM)/(i*sinhαM) = cothαM For “αM  R” “vS/c = cothαM > 1” so that “vS > c”. “c*tS = i*xM” and “xS = i*c*tM” mean “c*tS” and “xS” are imaginary for “c*tM” and “xM” real. The “S” hypercomplex-plane requires “c*tS” and “xS” be real. To force “c*tS” and “xS” to be real make the hyperbolicradius (in “B”) imaginary. tB = -i*B mB = -i*B

(tau) ; B  R

;

(nu) ; B  R ;

i*qx*c*tB = i*qx*c*(-i*B) = qx*c*B i*qx*mB*c = i*qx*(-i*B)*c = qx*B*c

The imaginary hyperbolic-radius causes “c*tM” and “xM” to be imaginary and not plotted on the “M” hypercomplex plane. Colliding Rods Visualization. Two parallel rods move perpendicular to their length to collide and bounce. All the points along the contacting surface coincide everywhere in “xB” at one instant in time “tB”. Because we are observing a rod collision along “xB” at one instant “tB” and not one location “xB” for all time “tB”, we have “αB = i*/2” and not “αB = 0”.

58 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The two rods are on the floor of a bus, “αM = 0”. Because “vS/M, αS/M > 0”, the front of the bus is at an earlier time “tM” compared to the back, for each time “tS”, and that means the rod collision observed from “S” starts in the rear of bus and moves forward at speed “vS = c2/vS/M > c”. Spooky Action at a Distance. Einstein gave the name “spooky action at a distance” to the polarization of a pair of entangled photons determined simultaneously in two observations a macroscopic distance away. The information of the direction of polarization had to travel from one particle to its entangled partner particle instantaneously over the macroscopic distance. Perhaps this is another example of “α = i*/2”. Violation of Cause-and-Effect. A hyper-light-speed signal can arrive before it was emitted. The signal can be two rods colliding. Set “M > 0” and “αM = M + i*/2” so that “vM = c*coth(M)”. (If “M < 0” then the adder is “- i*/2”.) Bus “M” moves backward relative to roadside “S” with a negative hyperbolic-angle speed-parameter “αS/M < 0”. M  R ; M > 0 ; αM = M + i*/2 ; αS = αM + αS/M = i*/2 + (M + αS/M ) 2r

αS/M  R ; αS/M < 0

= 1S*(c*tB)*exp(qx*αS) = 1S*(-i*c*B)*exp(qx*αS) = 1S*(-i*c*B)*exp(qx*(αM + αS/M)) = 1S*(-i*c*B)*exp(qx*(i*/2 + M1 + αS/M)) = 1S*(-i*c*B)*exp(qx*i*/2)*exp(qx*(M + αS/M)) = 1S*(-i*c*B)*jx*exp(qx*(M + αS/M)) = qxS*(i*-i*c*B)*exp(qx*(M + αS/M)) = qxS*(c*B)*exp(qx*(M + αS/M))

c*tS = c*B*sinh(M + αS/M)

xS = c*B*cosh(M + αS/M)

The space-like equations for “c*tS” and “xS” depend on angle “M + αS/M”, which is positive for “αS/M = 0” and negative for “αS/M < -M”.

59 CHAPTER 2 – PARTICLES •

For “αS/M = 0”, the colliding rods move forward inside the bus at speed “vM-rods = c*tanh(M) < c” so that their contact point moves at speed “vS = c*coth(M) > c” and “c*tS > 0” and “xS > 0”.

•

In contrast, for “αS/M < -M” the rods move backward at speed “-c < vS-rods = c*tanh(M + αS/M) < 0” so that their contact point moves backward at speed “vS = c*coth(M + αS/M) < -c”. “c*tS < 0” with “xS > 0” means increasing values of “xS” occur for decreasing values of “c*tS” and that properly corresponds to negative direction movement because in “S” time is only seen as moving positively.

The contact point was seen moving forward on the bus but moving backward from the roadside so that past locations of the contact point on the bus are future locations of the contact point from the roadside. Time goes backward.

Figure 19. Illustration of the violation of cause-and-effect for motion faster than the speed-of-light. The observer sees time as moving up vertically. The arrow on the far-right figure points right, but the “S” observer sees the activity as moving to the left, toward the source of the emission.

Visualization of the Violation of Cause-and-Effect. A solar system has two small planets in counter-orbits. Reference frame “M” is Planet M. “S” is Planet S. At “c*tM = 0” and “c*tS = 0” the two planets are a few days passed each other with Planet M on the left and moving to the left. Therefore, “vS/M/c = tanhαS/M” with “αS/M < 0” and “vS/M < 0” (and “vM/S > 0”).

60 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Planet M sends a hyper-light-speed signal to Planet S at speed “vM/c = cothM” with “M > 0” and “vM > 0” / “vM > c”. Planet S received it at “c*tM = 7” and “xM = tM*vM = c*tM*coth(M)” (“xM > 0”). 2r

= 1M*(-i*c*B)*exp(qx*αM) = 1M*(-i*c*B)*exp(qx*(M + i*/2)) = 1M*(-i*c*B)*(jx)*exp(qx*M) = 1M*(c*B)*(qx)*exp(qx*M)

c*tM = c*B*sinh(M)

;

xM = c*B*cosh(M) = (c*tM/sinh(M))*cosh(M) = c*tM*coth(M) = tM*vM

Note that “xM > 0” and “xM > c*tM” because “M  R”. Relative to Planet S, the hyper-light-speed signal time-space location components “c*tS” and “xS” are found using the below component equations, as given previously. c*tS = c*B*sinh(M + αS/M)

;

xS = c*B*cosh(M + αS/M)

If “M = 0”, such that the signal arrived instantly (as would happen for colliding rods), then “c*tS < 0”, and the signal was received on Planet S prior to being emitted, per “c*tS = c*B*sinh(M + αS/M)”.

Figure 20. Instant signal sent from Planet M to Planet S and then another instant signal is sent back to Planet M. The second signal arrives prior to the first signal being emitted because “M = 0” and “S(second) = 0” with the two planets moving apart.

61 CHAPTER 2 – PARTICLES

Figure 21. Hyper light speed signal from Planet M to Planet S and then another signal back to Planet M. The second signal arrives prior to the first signal being emitted because “M - S(second) < -αS/M” with the two planets moving apart.

But, the people on Planet S were unaware of the emission time of the signal because it occurred on Planet M, far away from them, and so are unaware of the signal moving backward in time. To make everyone aware of the reverse passage of time, Planet S sent a second hyper-light-speed signal to Planet M the instant they received the signal from Planet M. For this second signal, “S(second) < 0” because the signal traveled from Planet S to the left towards Planet M. The “-” in “- i*/2” is because the second signal is on the left side of the “c*tS” axis on the hypercomplex-plane. αS(second) = S(second) - i*/2 2r(second)

;

S(second) < 0

= 1S*(-i*c*B(second))*exp(qx*αS(second)) = 1S*(-i*c*B(second))*exp(qx*(S(second) - i*/2)) = 1S*(-i*c*B(second))*(-jx)*exp(qx*S(second)) = 1S*(c*B(second))*(-qx)*exp(qx*S(second))

c*tS(second) = -c*B(second)*sinh(S(second)) ; xS(second) = -c*B(second)*cosh(S(second)) “S(second) < 0” in “c*tS(second)” and “xS(second)” component equations means “c*tS(second) > 0” for “c*B(second) > 0”. And “xS(second) < 0” for left motion. Relative to Planet S, the total (first signal plus second signal) time elapsed, and the total distance traveled, are given below.

62 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY c*tS(total) = c*tS + c*tS(second) = c*B*sinh(M + αS/M) - c*B(second)*sinh(S(second)) xS(total) = xS + xS(second) = c*B*cosh(M + αS/M) - c*B(second)*cosh(S(second)) The general form of the Lorentz Transformation is now applied to find the arrival time of this second signal on Planet M. 2r(second)

= 1S*(c*B(second))*(-qx)*exp(qx*S(second)))*(exp(qx*αM/S)/exp(qx*αM/S)) c*tM(second) = -c*B(second)*sinh(S(second) + αM/S) xM(second) = -c*B(second)*cosh(S(second) + αM/S) c*tM(total) = c*tM + c*tM(second) = c*B*sinh(M) - c*B(second)*sinh(S(second) + αM/S) = c*B*sinh(M) - c*B(second)*sinh(S(second) - αS/M) xM(total) = xM + xM(second) = c*B*cosh(M) - c*B(second)*cosh(S(second) - αS/M) “xM(total) = 0” when “c*B*cosh(M) = c*B(second)*cosh(S(second) - αS/M)”. c*B(second) = c*B*cosh(M)/cosh(S(second) + αM/S) c*tM(total) = c*B*sinh(M) - c*B(second)*sinh(S(second) - αS/M) = c*B*(sinh(M) - cosh(M)*tanh(S(second) - αS/M)) For this second signal to be received by Planet M prior to the first signal being emitted, “c*tM(total) < 0”, and that requires “sinh(M) < cosh(M)*tanh(S(second) - αS/M)”, or “tanh(M) < tanh(S(second) - αS/M)” and that means “M < S(second) - αS/M”. It was specified that “M > 0”, “αS/M < 0” and “S(second) < 0”, therefore “M - S(second) > 0”. If both “M” and “S(second)” are small enough in magnitude (so that the signal speeds are very fast), then “M - S(second) < -αS/M” to the result “c*tM(total) < 0”. If the first signal is a weapon, then the weapon is countered by a second weapon counterattack that goes backward in time to destroy the enemy prior to their initial attack. But then the attack is not initiated,

63 CHAPTER 2 – PARTICLES and, therefore, the counterattack is not initiated. To avoid such a strange condition, a rule of nature prevents controlled or prescribed information from being transmitted faster than the speed-of-light, to not violate cause-and-effect. The hypothetical violation of cause-and-effect for hyper-light-speed signals is a classic feature of Special Relativity. Matter-waves. An electron’s matter-wave moves at the phase speed “vp”. vpM/c = M/kxM = (ħ*M)/(ħ*kxM) = (EM)/(pxM) = coshαM/sinhαM = c/vM Per the above equation (which assumes no potential energy for the electron) the phase speed “vpM/c” is the reciprocal of the group speed of the electron, “c/vM”. If group speed “vM = 0” (for a stopped electron), the phase speed “vpM” equals the reciprocal of zero. If all of space cycled as a wave per “T = exp(i*M*tM)”, then the information of the value of “T” can be thought of as being transmitted in both directions of “x” instantly because it goes to the extremes of “x” and back instantly. The wave carries no controlled information and so is not a signal. The controlled information is formed from the interference pattern of the waves. The interference pattern moves at the group speed. The group forms the probability function of the particle location, and the group speed is slower than the speed-of-light.

Figure 22. Wavelength node points for a matter-wave, with “vpM > c”.

64 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

2.10 Anti-Matter Anti-matter was first proposed because of the alternative/opposite electron identified in the solution of the Dirac Equation. A few years later, the anti-matter electron, called the positron, was discovered experimentally. Many years later, Feynman proposed that anti-matter is matter that moves backward in time. (The below proposed use of Special Relativity as the basis of a theory for anti-matter might be new with this book. A search didn’t find it anywhere.)

Figure 23. Back and front are swapped for anti-matter. But more than that, anti-matter is turned inside-out, like a glove, such that the left glove appears to be a right glove. Momentum for matter is on the left figure and is to the right. Momentum for anti-matter is on the right figure and is to the left. An observer made of matter feels the push of the momentum of anti-matter as if it moves to the right, because the observed momentum is a reaction from the push that drove the anti-matter backward in time and space.

An observer sits inside bus “M” and observes a baseball with speed “vM” pass their seat toward the front of the bus, “αM, vM > 0”. Bus speed is represented by “αS/M”. αS/M = (N + 1/2)*i*2* ;

NZ

“αS/M = (N + 1/2)*i*2*” is written simplified to “αS/M = i*”.

65 CHAPTER 2 – PARTICLES vS/M/c = tanhαS/M = tanh(i*) = i*tan() = i*0 = 0 “αS/M = i*” has no effect on speed. Total speed “vS” of the baseball inside the bus is found by adding hyperbolic-angles. αS = αM + αS/M = αM  i* vS/c = tanhαS = tanh(αM  i*) = sinh(αM  i*)/cosh(αM  i*) = (sinh(αM)*cosh(i*) + cosh(αM)*sinh(i*)) /(cosh(αM)*cosh(i*) + sinh(αM)*sinh(i*)) = (sinh(αM)*cos()  cosh(αM)*i*sin()) /(cosh(αM)*cos()  sinh(αM)*i*sin()) = (-sinhαM)/(-coshαM) = tanhαM “vS = vM = c*tanhαM” even though “αS  αM”. General form: 2r

= 1M*(c*tB)*exp(qx*αM) = 1M*(c*tB)*exp(qx*αM)*(exp(qx*i*)/exp(qx*i*)) = 1M*(c*tB)*exp(qx*αM)*(exp(jx*)/exp(jx*)) = 1M*(c*tB)*exp(qx*αM)*(-1/-1) = (-1M)*((-c*tB)*exp(qx*αM))

c*tS = c*tM*coshαS/M + xM*sinhαS/M = c*tM*cosh(i*) + xM*sinh(i*) = c*tM*cos()  i*xM*sin() = c*tM*(-1)  i*xM*0 = -c*tM c*tS = c*tB*coshαS = c*tB*cosh(αM + αS/M) = c*tB*cosh(αM + i*) = c*tB*(cosh(αM)*cosh(i*) + sinh(αM)*sinh(i*)) = c*tB*(cosh(αM)*cos() + i*sinh(αM)*sin()) = -c*tB*cosh(αM) = -c*tM

66 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY xS = c*tM*sinhαS/M + xM*coshαS/M = c*tM*sinh(i*) + xM*cosh(i*) = i*c*tM*sin() + xM*cos() = i*c*tM*0 + xM*(-1) = -xM xS = c*tB*sinhαS = c*tB*sinh(αM + αS/M) = c*tB*sinh(αM + i*) = c*tB*(sinh(αM)*cosh(i*) + cosh(αM)*sinh(i*)) = c*tB*(sinh(αM)*cos() + cosh(αM)*sin()) = -c*tB*sinh(αM) = -xM 1S = 1M*exp(-qx*αS/M) = 1M*exp(∓qx*i*) = 1M*exp(∓jx*) = 1M*(-1) = -1M qxS = qxM*exp(-qx*αS/M) = -qxM 2r

= 1S*c*tS + qxS*xS = (-1M)*(-c*tM) + (-qxM)*(-xM) = 1M*c*tM + qxM*xM = 2r

Observer “S” sees time “tS” go forward and, due to “c*tS = -c*tM”, sees time going backward for what observer “M” sees going forward. To visualize this: Observer “S” watches time pass “tB-matter” on his clock and through the bus window sees a clock for “tB-antimatter” made of anti-matter and recognizes “tB-antimatter = -tB-matter”. “B-matter” is the rest frame (“αB-matter = 0”) for matter. In contrast, “αB-antimatter = i*”creates the negative. (It is one-in-the-same with “αS/M = i*” for the bus, with the difference being how reference frames are defined.) For anti-matter insert a negative. c*tB-matter = +((c*tS)2 - xS2) c*tB-antimatter = -((c*tS)2 - xS2) tB-antimatter*cosh(αB-antimatter) = -tB-matter*-cosh(αB-matter) = tB-matter*cosh(αB-matter)

67 CHAPTER 2 – PARTICLES “c*tS = -c*tM” is complemented by “xS = -xM”. The measuring tape on the floor of the bus has increasing numbers that are to the negative of the tape on the roadside, and therefore the bus points backward. The clock held by the anti-matter person “M” is observed by person “S” to be moving to smaller numbers and those numbers are reversed left-to-right so that the clock hand moves clockwise, just like the matter clock. Imagine the baseball is rolling on the floor toward the front of the bus (to the right) inside a little toy car frame. Person “M” seated in the bus sees head-lights in front and tail-lights in back. Person “S” standing on the roadside sees the little car frame moving to the right, too, per “vS = vM”, but with tail-lights leading and the headlights following. Person “S” looks at the whole bus and sees the back faces positive “xS” (to the right) and the front faces negative “xS” and concludes “vS = vM” because of the double negative, time and space. Frequency. Person “M” seated in the bus hears clock ticks at frequency “M = B”. Person “S” on the roadside hears tick frequency “S = -M” because “coshS = -coshM”. The hyperbolic-radius of the frequency invariant for anti-matter is negative. B-matter = -B-antimatter

same as

c*tB-matter = -c*tB-antimatter

S = B-antimatter*coshS = B-antimatter*cosh(M + i*) = B-antimatter*cosh(i*) = B-antimatter*-1 = -B-antimatter = B-matter = B-matter*1 = B-matter*cosh(0) = B-matter*coshM Our observation of frequency “S” is positive, both for observing the anti-matter baseball’s clock ticks and for observing the matter baseball’s clock ticks. Therefore “M = B-antimatter” is negative. Energy and Momentum. “ES/c = -EM/c” and “pxS = -pxM”. Person “S” is impacted by an anti-matter baseball and it feels the same as if it were matter. The difference is that the matter baseball was received by person “S”, and the anti-matter baseball was launched by person “S”. Person “S” felt the reaction force from launching it backward in time and space.

68 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY To help visualize matter and anti-matter feeling the same, imagine the baseball re-composes itself into an electric field. The moving electric field induces a magnetic field and therefore has inertia same as mass. Anti-matter has the electric field reversed because anti-matter is what is formed when matter is subtracted from a vacuum. Regardless of the direction of the electric field and its induced magnetic field, the momentum will feel the same when it impacts person “S”. ES(of antimatter)/c = +ES(of matter)/c pxS(of antimatter) = +pxS(of matter) Two particles move down the road and pass person “S”. The matter particle has “S(of matter)  R”, “S(of matter) = M”, and “mB-matter > 0”. ES(of matter)/c = mB-matter*c*cosh(S(of matter)) = mB-matter*c*cosh(M) pxS(of matter) = mB-matter*c*sinh(S(of matter)) = mB-matter*c*sinh(M) Anti-matter particle has “S(of antimatter) = M + i*”, and “mB-antimatter < 0”. ES(of antimatter)/c = mB-antimatter*c*cosh(S(of antimatter)) = mB-antimatter*c*cosh(M + i*) = -mB-antimatter*c*cosh(M) = mB-matter*c*cosh(M) = ES(of matter)/c pxS(of antimatter) = mB-antimatter*c*sinh(S(of antimatter)) = mB-antimatter*c*sinh(M + i*) = -mB-antimatter*c*sinh(M) = mB-matter*c*sinh(M) = pxS(of matter) Newtonian Mechanics with Anti-Matter. “mB-antimatter = -mB-matter” should not be put into the context of mass times acceleration equals force, but rather into the context of force equals the time derivative of momentum, as Newton originally presented his second law. Per “pxS(of antimatter) = +pxS(of matter)” the negative mass of anti-matter is inconsequential. Negative rest mass also affects Newton’s Law of Gravity, but General Relativity, which uses energy and not mass, supersedes it, and per “ES(of antimatter)/c = +ES(of matter)/c” the negative mass of anti-matter is

69 CHAPTER 2 – PARTICLES inconsequential. Note that anti-matter has not been produced in a quantity large enough to measure the force due to gravity. Reverse Parity of Anti-Matter. The bus in reverse parity is perhaps best visualized using a right-hand glove. The fingers point to positive “x”. Pull the fingers through the open end to turn it inside out. The fingers now point to negative “x” and the glove looks like a left-hand glove. That same operation is not possible with a bus or particle. Rather, imagine the bus is an illustration on a sheet of paper as one page in a book sitting flat on a table. The page is turned by lifting it up and placing it upside down on the other side of the book to create a mirror image. If ever anti-matter is made from its matter counterpart, then it will have been rotated through another dimension, just as the page had to be lifted out of the plane of the table. This page-turning visualization is a classic interpretation of the reverse parity of anti-matter. Anti-matter Electric Current Density. Electric charge “QB” is in a space derivative (with respect to “xB” as a second reference to the reference frame “B”) to form electric charge density “B-matter = dQB/dxB” as the hyperbolic-radius of the current density invariant “2J”. B-matter = dQB-matter/dxB-matter

;

2J

= 1M*B-matter*exp(qx*M)

The ratio of anti-matter electric charge to location, formed as a derivative, has a double negative, and, therefore, no negative. QB-antimatter = -QB-matter

;

dQB-antimatter = -dQB-matter

xB-antimatter = -xB-matter

;

dxB-antimatter = -dxB-matter

dQB-antimatter/dxB-antimatter = -dQB-matter/-dxB-matter = dQB-matter/dxB-matter B-antimatter = +B-matter The Lorentz Transformation from matter observed in bus “M” to anti-matter observed from the roadside “S” uses “S/M = i*”.

70 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2J

= 1M*JtM + qxM*JxM = 1M*B-matter*exp(qx*M) = 1M*B-matter*exp(qx*M)*exp(qx*i*)/exp(qx*i*)) = 1M*B-matter*exp(qx*M)*(-1)/(-1) = (1M*/(-1))*(-B-matter)*exp(qx*αM) = 1S*(-B-matter)*coshαM + qxS*(-B-matter)*sinhαM = 1S*(-B-matter)*coshαM + qxS*(-B-matter)*sinhαM = 1S*(-JtM) + qxS*(-JxM) = 1S*JtS + qxS*JxS

“JtS = -JtM” and “JxS = -JxM” state the electric charge density and the electric current density become negative. Because “B-antimatter = B-matter”: JtS(of antimatter) = -JtS(of matter)

JxS(of antimatter) = -JxS(of matter)

(The analogous equations for energy-momentum did not have the negative.) Two particles pass person “S”. Matter particle: “S(of matter)  R”, “S(of matter) = M”, and “B-matter > 0”. JtS(of matter) = B-matter*c*cosh(S(of matter)) = B-matter*c*cosh(M) JxS(of matter) = B-matter*c*sinh(S(of matter)) = B-matter*c*sinh(M) For the anti-matter particle “S(of antimatter) = M + i*”, “B-antimatter > 0”. JtS(of antimatter) = B-antimatter*c*cosh(S(of antimatter)) = B-antimatter*c*cosh(M + i*) = B-antimatter*c*-cosh(M) = B-matter*c*-cosh(M) = -JtS(of matter) JxS(of antimatter) = B-antimatter*c*sinh(S(of antimatter)) = B-antimatter*c*sinh(M + i*) = B-antimatter*c*-sinh(M) = B-matter*c*-sinh(M) = -JxS(of matter) “JtS(of antimatter) = -JtS(of matter)” says anti-matter electric charge density is the negative of matter. Specifically, for the electron, the anti-electron (called the positron) is observed in “S” having a positive electric charge (compared to negative charge of the electron).

71 CHAPTER 2 – PARTICLES “JxS(of antimatter) = -JxS(of matter)” states anti-matter electric current density is the negative of matter. A flow of positively charged particles is the negative of a flow of negatively charged particles.

Figure 24. Anti-matter moving from future to past contrasted with matter moving from past to future, for “S(matter) = M” and “S(anti-matter) = M  i*”.

Matter-waves of Anti-matter. A factory makes anti-matter one particle at a time and assembles an anti-matter bus, an android, and a rubber bounce ball. The (anti-matter) rubber superball bounces off the walls of the (anti-matter) bus with smaller and smaller bounces losing energy to friction and eventually stops. Time appeared to go forward, just like for us in the factory made of matter. To explain this, we can propose that the direction of time is determined by the particle that makes the observation (which is us made of matter), and not by the particle being observed (which is the bouncing ball). This would make sense in terms of quantum mechanics because the “collapse of the wave function” in which particle properties get specified, is an observation. What would the android made of anti-matter observe of us? How would it view the two slit experiment? Can time move in both directions in our cosmological model of the universe? What experiments are needed?

72 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

2.11 Distributed Material Theory A distributed material is a field of point particles, each point with an infinitesimal quantity. Examples are a fluid, a solid, an electromagnetic field, and a distributed electric charge. Each is spread through space as a continuum and varies in time. Differential calculus applies. Space-Negative. The time-space differential gradient operator invariant “4sn” (del) is the mathematical tool for a distributed material. 4

= tM*itM + xM*ixM + yM*iyM + zM*izM

translates to the all number expression 4

sn

= 1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM

The space-negative operator “sn” has two aspects: Space compound-label-numbers “qxM”, “qyM”, and “qzM” are negative, and the Lorentz Transformation is inverted. The inverted Lorentz Transformation compensates for the lack of a negative on “1M”. The space-negative is necessary because time and space are in the denominator for “4”. A long rod has temperature “1T” along its length. Invariant “1T” can loosely be called a “scalar” field. The compound-label-number associated with a scalar field is the number “1”. It is devoid of a reference to the unspecified-speed-parameter “”, but is “compound”, regardless. 1T(tM,

xM) = C + a*tM + b*xM

The rod is mounted inside bus “M” and moves with the bus. “1T” has slope “b” (measured in degrees centigrade per meter) and increases at rate “a” (measured in degrees centigrade per second) when measured by a person seated on bus “M”. T/ctM = a/c

;

T/xM = b

The two gradients are placed into a time-space gradient invariant.

73 CHAPTER 2 – PARTICLES 2

sn

*1T = 1Msn*T/ctM + qxMsn*T/xM = 1Msn*(a/c) + qxMsn*b

The time-space gradient operator is 2

sn

= 1Msn*/ctM + qxMsn*/xM

The multiplication symbol “*” after the operator “2sn” tells us to think of “2sn” as an invariant. Bus “M” moves at speed “vS/M” relative to roadside “S”. Gradients “T/ctS” and “T/xS” are measured. 2

sn

*1T = 1Ssn*T/ctS + qxSsn*T/xS ;

2

sn

= 1Ssn*/ctS + qxSsn*/xS

Figure 25. The temperature “T” at the point “xS” decreases when the bus moves forward at speed “vS/M > 0”. Therefore, “TS/ctS < 0” for “TM/xM > 0”.

Bus “M” moves forward (“vS/M > 0”) to present colder and colder temperature to a location “xS” if “a = 0” and “b > 0”, so that “T/ctS < 0”. The colder temperature requires a negative. T/ctS = (T/ctM)*(coshS/M) + (T/xM)*(-sinhS/M) = (a/c)*(coshS/M) + (b)*(-sinhS/M) T/xS = (T/ctM)*(-sinhS/M) + (T/xM)*(coshS/M) = (a/c)*(-sinhS/M) + (b)*(coshS/M)

74 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY To better see the negative, consider a non-relativistic speed by replacing “sinhS/M” with “vS/M/c” and “coshS/M” with “1”. T/ctS = -b*vS/M/c

T/xS = b

Using the above “T/ctS” and “T/xS” information, the onecomponent invariant “1T” can be expressed in terms of “c*tS” and “xS”. 1T(tS,

xS) = C + (T/ctS)*(c*tS) + (T/xS)*(xS)

= C + ((T/ctM)*(coshS/M) + (T/xM)*(-sinhS/M))*(c*tS) + ((T/ctM)*(-sinhS/M) + (T/xM)*(coshS/M))*(xS) To prove this is correct, derive “1T(tM, xM)” from “1T(tS, xS)”. 1T(tS,

xS) = C + ((T/ctM)*(coshS/M) + (T/xM)*(-sinhS/M)) *((c*tM)*(coshS/M) + (xM)*(sinhS/M)) + ((T/ctM)*(-sinhS/M) + (T/xM)*(coshS/M)) *((c*tM)*(sinhS/M) + (xM)*(coshS/M)) = C + (T/ctM)*tM + (T/xM)*xM = 1T(tM, xM)

The gradient operators based on those equations are below. /ctS = (/ctM)*(coshS/M) + (/xM)*(-sinhS/M) /xS = (/ctM)*(-sinhS/M) + (/xM)*(coshS/M) tS = tM*coshαS/M - xM*sinhαS/M xS = -tM*sinhαS/M + xM*coshαS/M The critical concept here is the “-” sign in front of “sinhS/M”. This “-” sign makes the Lorentz Transformation for “2sn” the opposite (or inverse) of the Lorentz Transformation for “2r”, given below. c*tS = (c*tM)*(coshS/M) + (xM)*(sinhS/M) xS = (c*tM)*(sinhS/M) + (xM)*(coshS/M)

75 CHAPTER 2 – PARTICLES There is no negative sign in the above “2r” invariant equations. The negatives make the two Lorentz Transformations inverses of each other. A matrix multiplied by its inverse equals one. coshS/M

sinhS/M

coshS/M

-sinhS/M

* sinhS/M

coshS/M

1 0 =

-sinhS/M

coshS/M

0 1

Including “-” with “sinhαS/M” makes the Lorentz Transformation for “2sn” special. To show it is special, it is given the symbol “sn”. Space Negative on Other Invariants. “sn” applies to other invariants. “2k = (M/c)*itM + kxM*ixM” translates to “2k = 1M*(M/c) + qxM*kxM”. S/c = (M/c)*(coshS/M) + (kxM)*(sinhS/M) kxS = (M/c)*(sinhS/M) + (kxM)*(coshS/M) “2k = (M/c)*itM - kxM*ixM” translates to“2ksn = 1Msn*(M/c) + qxMsn*kxM”. S/c = (M/c)*(coshS/M) - (kxM)*(sinhS/M) kxS = -(M/c)*(sinhS/M) + (kxM)*(coshS/M) “2ksn” is abnormal (because of the negative before “ixM”). When we use “2ksn”, we are modelling a wave-number in a special way, for example, for anti-matter, if we choose that anti-matter moves to the left for positive “M” when matter moves to the right for positive “M”, as will be done in the chapter on waves. “2ksn” being abnormal is unlike the time-space gradient operator “2sn”, because the space-negative on the gradient operator “2sn” is normal (because of the positive before “ixM”). Multiplication Operation with the Space Negative. Because of the inverted matrix, the space label-numbers are negative. Also, multiplication needs a hypercomplex-conjugate, for another negative. 2

sn

*1T = 1Msn*T/ctM + qxMsn*T/xM (2sn*1T)*j = T/ctM*1M*jsn + T/xM*qxM*jsn

76 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2r

= 1M*c*tM + qxM*xM xM) - C = ((2*1T)*jsn)•(2r) = (T/ctM*((1M)*jsn) + (T/xM*(qxM)*jsn))•(1M*c*tM + qxM*xM) = (T/ctM*((1M)*jsn)*(1M*c*tM) + (T/xM*(qxM)*jsn)*(qxM*xM) = (T/ctM)*(c*tM)*((1M)*jsn)*(1M) + (T/xM)*(xM)*((qxM)*jsn)*(qxM) = (T/ctM)*(c*tM) + (T/xM)*(xM)

1T(tM,

“(T/ctM)*(c*tM) + (T/xM)*(xM)” has a “+” sign between the two terms. It is different from “((2k)*j)•(2r)”, as given below, because “((2k)*j)•(2r)” has a negative sign “-” between the two terms. ((2k)*j)•(2r) = (M/c)*((1M)*j) + (kxM*(qxM)*j))•(1M*c*tM + qxM*xM) = (M/c)*((1M)*j)*(1M*c*tM) + (kxM*(qxM)*j)*(qxM*xM) = (M/c)*(c*tM)*((1M)*j)*(1M) + (kxM)*(xM)*((qxM)*j)*(qxM) = M*tM - kxM*xM To create the “+” sign between the two terms in “((2 *1T)*j)•(2r)”, a negative was on the space compound-label-numbers introduced by the hypercomplex-conjugate operation, and another negative was introduced by the space-negative operator, for a net result of a positive. (“pxM” and “kxM” will be discussed later.) sn

(1M)sn = 1M ; (qxM)sn = -qxM ; (qyM)sn = -qyM ; (qzM)sn = -qzM (1M)*jsn = 1M ; (qxM)*jsn = -qxM*j; (qyM)*jsn = -qyM*j; (qzM)*jsn = -qzM*j ((1)*jsn)*1M = 1 ; ((qxM)*jsn)*qxM = +1 ((qyM)*jsn)*qyM = +1 ; ((qzM)*jsn)*qzM = +1 ((1M)*jsn)*qxM = ((qxM)*jsn)*1M = pxM ((qyM)*jsn)*qzM = -((qzM)*jsn)*qyM = -i*pxM = -kxM Examples of Gradient Operations with Space-Negative. A conservation law uses the divergence operator, “((4)*jsn)•”. ((4)*jsn)•(4) = (tM/ctM) + (xM/xM) + (yM/yM) + (zM/zM) = 0 ((4)*jsn)•(4J) = (JtM/ctM) + (JxM/xM) + (JyM/yM) + (JzM/zM) = 0 ((4)*jsn)•(4V) = (VtM/ctM) + (VxM/xM) + (VyM/yM) + (VzM/zM) = 0

77 CHAPTER 2 – PARTICLES “4” is a density of particles (for example, a count of gravel particles in the back of a truck). “4J” is electric current density and is a special case of “4”. “4V” is voltage from the next chapter. Curl operator, “((4)*jsn)x”: ((4)*jsn)x(4J) = -6G

((4)*jsn)x(4V) = -6E

“6E” is the electromagnetic field invariant from the next chapter. What is special about the four-dimensional time-space curl operator “((4)*jsn)x” is the lack of a negative between the time term and the space term. This special feature is shown in the two-dimensional time-space simplification given below. ((2)*jsn)x(2V) = = (/ctM*((1M)*jsn) + (/xM*(qxM)*jsn))x(1M*VtM + qxM*VxM) = (/ctM)*((1M)*jsn)*(qxM*VxM) + (/xM)*((qxM)*jsn)*(1M*VtM) = (VxM/ctM)*((1M)*jsn)*(qxM) + (VtM/xM)*((qxM)*jsn)*(1M) = (VxM/ctM + VtM/xM)*(pxM) = -ExM*pxM -ExM = VxM/ctM + VtM/xM ; pxM = ((1M)*jsn)*(qxM) = ((qxM)*jsn)*(1M) d’Alembert operator, what is called the harmonic operator: ((4)sn)*((4)*jsn) = ((4)sn)•((4)*jsn) = tM2 - (xM2 + yM2 + zM2) = (4)2 4

2

*4V = 4J

;

4

2

*6E = 6G

Theory-Development-Algebra Mathematics. “((4)*jsn)•(4) = 0” is the same as fluid mass conservation from engineering class, where it is written “/t + 3•(*3v) = 0” for which “3v = vx*ix + vy*iy + vz*iz” is the velocity. In engineering class, it was derived very pragmatically by equating what goes into and out of a cube to what is inside.

78 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY “/t + 3•(*3v) = 0” appeared strange because “+” is between the time and space terms, and not “-” as in “((2k)*j)•(2r) = *t - kx*x”. The “+” is now explained by use of the much less efficient nomenclature of the space-negative in the all-number theory-development-algebra. Theory-development-algebra is much more symbol intensive and is being used because our purpose for math has changed from engineering calculations to theory development, with the benefit being we see how mass conservation fits into the larger scheme of patterns in our world. Tensor notation calculus of General Relativity is a form of differential geometry and also is an engineering-calculation-algebra. Covariant variables time and space are in the denominator of a derivative with the result being a contravariant vector. A contravariant vector is the tensor notation calculus analogy to the space-negative and special rules state where to insert negatives. The challenge is to create a theorydevelopment-algebra for General Relativity that can replace tensor notation calculus. The challenge looks difficult. Example of Particle Count Conservation. A pile of gravel moves in the positive “x” direction. Particle count starts on the left of “xB” in “dCountB/dxB = A*exp( -(xB*kxB)2 )”.

Figure 26. A moving pile of gravel of height “h”.

79 CHAPTER 2 – PARTICLES Subscript “B” identifies the rest frame of the pile of gravel and is the same as “M”. “dCountB/dxB” is placed into a time-space invariant “2lump” by multiplying it by the compound-label-number “1B”. 2lump

= 1B*( dCountB/dxB ) = 1B*( A*exp(-(xB*kxB)2) ) = 1B*lumptB

Use a Lorentz Transformation to derive “lumptS” and “lumpxS”. 2lump

= 1B*( dCountB/dxB ) = 1S*( dCountB/dxB )*exp(qx*αS/B) = 1S*( A*exp(-(xB*kxB)2) )*exp(qx*αS/B) = 1S*( A*exp( -((xS - vS*tS)*coshαM*kxB)2 ) )*exp(qx*αS/B) = 1S*( A*exp( -((xS - vS*tS)*coshαM*kxB)2 ) )*exp(qx*αS) = 1S*( A*exp( -((xS - vS*tS)*coshαS*kxB)2 ) )*coshαS + qxS*( A*exp( -((xS - vS*tS)*coshαS*kxB)2 ) )*sinhαS = 1S*lumptS + qxS*lumpxS lumptS = ( A*exp( -((xS - vS*tS)*coshαS*kxB)2 ) )*coshαS lumpxS = ( A*exp( -((xS - vS*tS)*coshαS*kxB)2 ) )*sinhαS 1S = 1B/exp(qx*αS/B)

The substitution “xB = (xS - vS*tS)*coshαS” was derived from the Lorentz Transformation from “B” to “S”. c*tS = c*tB*coshαS + xB*sinhαS ; c*tB = c*tS*coshαS - xS*sinhαS xS = c*tB*sinhαS + xB*coshαS ; xB = -c*tS*sinhαS + xS*coshαS xB = -c*tS*sinhαS + xS*coshαS ; c*tB = c*tS*coshαS - xS*sinhαS = xS*coshαS - c*tS*sinhαS = c*tS*coshαS - xS*sinhαS = xS*coshαS - c*tS*tanhαS*coshαS = c*tS*coshαS - xS*tanhαS*coshαS = (xS - tanhαS*c*tS)*coshαS = (c*tS - tanhαS*xS)*coshαS = (xS - vS*tS)*coshαS = c*(tS - vS*xS/c2)*coshαS The conservation law is applied.

80 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2

sn

= 1Ssn*tS + qxSsn*xS ; sn sn = 1S */ctS + qxS */xS

2lump

*jsn

= tS*1S*jsn + xS*qxS*jsn

= 1S*(dCountB/dxB)*coshαS + qxS*(dCountB/dxB)*sinhαS

1S*jsn*1S = 1 ; 2

2

qxS*jsn*qxS = 1

•2lump = (tS*1S*jsn)*(1S*(dCountB/dxB)*coshαS) + (xS*qxS*jsn)*(qxS*(dCountB/dxB)*sinhαS)

*jsn

= tS*((dCountB/dxB)*coshαS) + xS*((dCountB/dxB)*sinhαS) For the example above tS*((dCountB/dxB)*coshαS) = (/ctS)*(( A*exp(- ((xS - vS*tS)*(coshαS)*kxB)2 ) )*coshαS) = (-((xS - vS*tS)2)/ctS)*(kxB*coshαS)2 *(( A*exp(- ((xS - vS*tS)*(kxB*coshαS))2 ) )*coshαS) = -2*((xS - vS*tS)/ctS)*(xS - vS*tS)*(kxB*coshαS)2 *(( A*exp(- ((xS - vS*tS)*(kxB*coshαS))2 ) )*coshαS) = -2*(-vS/c)*(xS - vS*tS)*(kxB2*cosh2αS) *A*exp(-((xS - vS*tS)*(kxB*coshαS))2)*coshαS = 2*(sinhαS/coshαS)*(xS - vS*tS)*(kxB2*cosh2αS) *A*exp(-((xS - vS*tS)*(kxB*coshαS))2)*coshαS = 2*(xS - vS*tS)*(kxB2*cosh2αS) *A*exp(-((xS - vS*tS)*(kxB*coshαS))2)*sinhαS xS*((dCountB/dxB)*sinhαS) = (/xS)*(( A*exp(- ((xS - vS*tS)*(kxB*coshαS))2 ) )*sinhαS) = (-((xS - vS*tS)2)/xS)*(kxB*coshαS)2 *(( A*exp(- ((xS - vS*tS)*(kxB*coshαS))2 ) )*sinhαS)

81 CHAPTER 2 – PARTICLES = -2*((xS - vS*tS)/xS)*(xS - vS*tS)*(kxB*coshαS)2 *(( A*exp(- ((xS - vS*tS)*(kxB*coshαS))2 ) )*sinhαS) = -2*(xS - vS*tS)*(kxB2*cosh2αS) *A*exp(-((xS - vS*tS)*(kxB*coshαS))2)*sinhαS The above analysis showed 2

•2lump = 0

*jsn

Conservation Law with a Space-Negative Invariant. The abnormal spacenegative invariant “2anti-lumpsn” has subscript “anti” to indicate it applies to anti-matter because if matter moves to the right, then the equivalent anti-matter is chosen to move to the left (if observed by the same observer), and that is because the space-negative operator means the negative of space, or, with respect to movement, the opposite direction. The “2anti-lumpsn” and “2ksn” invariants are abnormal because the geometric translation has “+it” and “-ix”, same as the abnormal “2”. This is in contrast to “+ix” of the geometric translation for normal invariants “2lump”, “2k” and “2sn”. For the matter invariant “2lump”, movement in the positive “xM” direction is identified by “xB = xM - vM*tM”. In contrast, the space-negative anti-matter invariant “2anti-lumpsn” includes “xB = xM + vM*tM”. The “+” rather than “-” means movement is in the opposite direction. “xB = xM - vM*tM” was derived from the Lorentz Transformation. In contrast, “xB = xM + vM*tM” for the space-negative is derived from the inverse matrix Lorentz Transformation for “2rsn”, rather than for “2r”. (The inverse matrix has the negatives on the “sinhαS/M”.) Alternatively, “xB = xM + vM*tM” can be derived from “-2ksn*j•2r” rather than from “-2k*j•2r” if speed “vM” is frequency divided by wavenumber. The overall movement of electric charge particles is the same, for matter to the right or for anti-matter to the left: For matter, if “q = 1” is the quantity of electric charge per particle, then the count that passes the point “xS” moving right is proportional to the total electric charge and is positive. For anti-matter the count is negative because “q = -1”. Using the space-negative, those negative particles move left past the point “xS” and so get subtracted, for a double negative.

82 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY “2*jsn•(2anti-lump)sn = 0” is shown below to be valid: dCountB/dxB = -A*exp( -(xB*kxB)2 ) sn 2anti-lump

= 1Ssn*(dCountB/dxB)*coshαS + qxSsn*(dCountB/dxB)*sinhαS = 1Ssn*(dCountB/dxB)*exp(-qx*αS) = 1Ssn*( -A*exp(-(xB*kxB)2)*exp(-qx*αS) = 1Ssn*( -A*exp(-((xS + vS*tS)*kxB*coshαS)2) )*exp(-qx*αS) = 1Ssn*( -A*exp(-((xS + vS*tS)*(kxB*coshαS))2) )*coshαS + qxSsn*( -A*exp(-((xS + vS*tS)*(kxB*coshαS))2) )*sinhαS

c*tS = c*tB*coshαS - xB*sinhαS ; c*tB = c*tS*coshαS + xS*sinhαS xS = -c*tB*sinhαS + xB*coshαS ; xB = c*tS*sinhαS + xS*coshαS xB = c*tS*sinhαS + xS*coshαS = xS*coshαS + c*tS*sinhαS = xS*coshαS + c*tS*tanhαS*coshαS = (xS + tanhαS*c*tS)*coshαS = (xS + vS*tS)*coshαS 2

sn

= 1Ssn*tS + qxSsn*xS = 1Ssn*/ctS + qxSsn*/xS

;

2

*jsn

= tS*1S*jsn + xS*qxS*jsn

(1S*jsn)*(1Ssn) = (1S*j)*(1S) = (1)*(1) = 1 (qxS*jsn)*(qxSsn) = (-qxS*j)*(-qxS) = (-qx)*(qx) = -1 2

•2anti-lumpsn = (tS*1S*jsn)*(1Ssn*(dCount/dxB)*coshαS) + (xS*qxS*jsn)*(qxSsn*(dCountB/dxB)*sinhαS)

*jsn

= tS*((dCountB/dxB)*coshαS) - xS*((dCountB/dxB)*sinhαS) tS*((dCountB/dxB)*coshαS) = (/ctS)*(( -A*exp(-((xS + vS*tS)*(kxB*coshαS))2) )*coshαS) = (-((xS + vS*tS)2)/ctS)*(kxB*coshαS)2 *(( -A*exp(-((xS + vS*tS)*(kxB*coshαS))2) )*coshαS) = -2*((xS + vS*tS)/ctS)*(xS + vS*tS)*(kxB*coshαS)2 *(( -A*exp(-((xS + vS*tS)*(kxB*coshαS))2) )*coshαS)

83 CHAPTER 2 – PARTICLES = -2*(vS/c)*(xS + vS*tS)*(kxB2*cosh2αS) *-A*exp(-((xS + vS*tS)*(kxB*coshαS))2)*coshαS = -2*(sinhαS/coshαS)*(xS + vS*tS)*(kxB2*cosh2αS) *-A*exp(-((xS + vS*tS)*(kxB*coshαS))2)*coshαS = -2*(xS + vS*tS)*(kxB2*cosh2αS) *-A*exp(-((xS + vS*tS)*(kxB*coshαS))2)*sinhαS xS*((dCountB/dxB)*sinhαS) = (/xS)*(( -A*exp(-((xS + vS*tS)*(kxB*coshαS))2) )*sinhαS) = (-((xS + vS*tS)2)/xS)*(kxB*coshαS)2 *(( -A*exp(-((xS + vS*tS)*(kxB*coshαS))2) )*sinhαS) = -2*((xS + vS*tS)/xS)*(xS + vS*tS)*(kxB*coshαS)2 *(( -A*exp(-((xS + vS*tS)*(kxB*coshαS))2) )*sinhαS) = -2*(xS + vS*tS)*(kxB2*cosh2αS) *-A*exp(-((xS + vS*tS)*(kxB*coshαS))2)*sinhαS

2.12 Exercises Check on Text Comprehension. 1) For “c*t = 13” and “x = 12” calculate “rhyperbolic = ((c*t)2 - x2)”, “coshα = c*t/rhyperbolic”, “sinhα = x/rhyperbolic”, and “tanhα = x/(c*t)”. Find “α = atanh(x/(c*t))” and confirm “1 = (cosh2α - sinh2α)”. Write “2r = 1*c*t + qx*x” and “2r = rhyperbolic*exp(qx*α)” using numbers. Plot “(c*t, x) = (13, 12)” on the hypercomplex-plane and draw a straight line at hyperbolic-angle “α = atanh(12/13)” and a hyperbola with hyperbolic-radius “rhyperbolic = (132 - 122)”. 2) For “c*tM = 11”, “αM =5” and “αS/M = 3” find “c*tS” and “xS”. For “EM/c = 7” find “ES/c” and “pxS”. What is “c*tB” and what is “mB*c”?

84 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 3) How fast “vS/M/c” must bus “M” move so that its seats have half the spacing of a stationary bus? At that speed, what is the ratio of time between ticks of the bus’s clock to the roadside’s clock? What is the energy ratio of the moving bus to a stationary bus? 4) A hyper-light-speed signal of speed “vM(first) = c*coth(M(first))” is sent from Person M in the positive x-direction to Person S who walks away from Person M at a speed “vM/S” equal to positive seven feet per second. The instant the signal is received, Person S sends a signal back to the source at the same speed but in the opposite direction, “vS(second) = -vM(first)”. At what speed “vM(first)” must the signal travel so that the second signal arrives at the same instant the first signal is emitted? Draw the two signals on the hypercomplex-plane for Person M is stationary and then, again, on the hypercomplex-plane for Person S. 5) An anti-matter electron moves on an anti-matter bus “M” at a speed “vM = c*tanhαM = 11 meters per second” with “αM  R”. What is its electric charge “QB-antimatter”? What is its rest mass “mB-antimatter”? The bus moves relative to the road “S” with hyperbolic angle “αS/M = i*”. What is the electron’s energy “ES” and momentum “pxS”? 6) A wavy pattern for static electric charge is across the roof of a bus: “dQB/dxB = Qwavy = A*sin(xB/a)”. “dQB/dxB” is the hyperbolicradius “Qwavy” of the current density invariant “2J”. Show electric charge is conserved for any speed of the bus by showing “2*jsn•2J = 0”. Write “2Jsn” (to represent an anti-matter car), by analogy using the example in the text above, and show “2*jsn•2Jsn = 0”. 7) A row of spinning disks has a density “HxB+ = dHBx+/dxB” with “HBx+ > 0”. In parallel, a moving row of spinning disks has a density “HxB- = dHBx-/dxB” with “HBx- < 0”. This second row moves at speed “vM- > 0” (to the right), so that “αM- > 0”. The total angular momentum of both rows together is constrained to be zero. For both rows together, what is the hyperbolic radius of the time-like angular momentum invariant for the rate of angular momentum that passes a point “xM”.

85 CHAPTER 2 – PARTICLES 8) Angular momentum density “2H = 1M*qx*dHBx/dxB*exp(qx*αM)” is used in the conservation law “((2)*jsn)x(2H) = 0” for which no torque is applied to change the angular momentum. For “2H” below, verify the conservation law. 2H

= 1M*qx*(dHBx/dxB)*exp(qx*αM) = 1M*qx*( A*exp( -((xB)/a)2 ) )*exp(qx*αM)

Answers to Select Exercises. 1) rhyperbolic = (132 - 122) = 5 ; α = atanh(12/13)) = 1.60943… coshα = 13/5 = 2.6 ; (2.6)2 - (2.4)2 = 6.76 - 5.76 = 1 sinhα = 12/5 = 2.4 ; 2r = 1*13 + qx*12 tanhα = 12/13 = 0.092307… ; 2r = 5*exp(qx*1.60943…) 2) c*tB = c*tM/cosh(αM) = 11/cosh(5) = 0.14822… xM = c*tB*sinh(αM) = 0.14822…*sinh(5) = 10.99900… c*tS = c*tB*cosh(αM + αS/M) = 0.14822…*cosh(5 + 3) = 220.9309… xS = c*tB*sinh(αM + αS/M) = 0.14822…*sinh(5 + 3) = 220.9308… c*tS = c*tM*cosh(αS/M) + xM*sinh(αS/M) = 11*cosh(3) + 10.99900…*sinh(3) = 220.9309… xS = c*tM*sinh(αS/M) + xM*cosh(αS/M) = 11*sinh(3) + 10.99900…*cosh(3) = 220.9308… mB*c = (EM/c)/cosh(αM) = 7/cosh(5) = 0.09432… pxM = mB*c*sinh(αM) = 0.09432…*sinh(5) = 6.99930… ES/c = mB*c*cosh(αM + αS/M) = 0.09432…*cosh(8) = 140.59239… pxS = mB*c*sinh(αM + αS/M) = 0.09432…*sinh(8) = 140.59235… ES/c = (EM/c)*cosh(αS/M) + pxM*sinh(αS/M) = 7*cosh(3) + 6.99930…*sinh(3) = 140.59239… pxS = (EM/c)*sinh(αS/M) + pxM*cosh(αS/M) = 7*sinh(3) + 6.99930…*cosh(3) = 140.59235…

86 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 3) kxS/kxM = 2 ; αS/M = acosh(kxS/kxM) = acosh(2) = 1.31695… vS/M/c = tanh(αS/M) = 3/2 ; tS/tM = ES/EM = coshαS/M = 2 4) M(first) = αM/S/2 = atanh(vM/S/c)/2 5) and 6) solution not given 7)

2H

= 2H+ + 2H= 1M*qx*HxB+*exp(qx*αM+) + 1M*qx*HxB-*exp(qx*αM-) = 1M*qx*HxB+ + 1M*qx*HxB-*exp(qx*αM-) = 1M*qx*HxB+ + 1M*qx*HxB-*(coshαM- + qx*sinhαM-) = 1M*qx*(HxB+ + HxB-*coshαM-) + 1M*HxB-*sinhαM= 1M*HxB-*sinhαM- = 1M*(-HxB+/coshαM-)*sinhαM= 1M*-HxB+*tanhαM- = 1M*HxB+*(-vM-/c)

For both rows together, the hyperbolic radius of the time-like angular momentum invariant for the rate of angular momentum that passes a point “xM” is “HxB+*(-vM-/c)”. 8)

2H

= 1M*qx*(dHBx/dxB)*coshαM + qxM*qx*(dHBx/dxB)*sinhαM = 1M*qx*(dHBx/dxB)*exp(qx*αM) = 1M*qx*( A*exp( -((xB)/a)2 ) )*exp(qx*αM) = 1M*qx*( A*exp( -(((xM - vM*tM)*coshαM)/a)2 ) )*exp(qx*αM) = 1M*qx*( A*exp( -((xM - vM*tM)*(coshαM/a))2 ) )*coshαM + qxM*qx*( A*exp( -((xM - vM*tM)*(coshαM/a))2 ) )*sinhαM

xB = -c*tM*sinhαM + xM*coshαM = xM*coshαM - c*tM*sinhαM = xM*coshαM - c*tM*tanhαM*coshαM = (xM - tanhαM*c*tM)*coshαM = (xM - vM*tM)*coshαM 2

sn

2

*jsn

= 1Msn*tM + qxMsn*xM ; = 1Msn*/ctM + qxMsn*/xM

2

*jsn

= tM*1M*jsn + xM*qxM*jsn

x2H = (tM*1M*jsn)*(1M*qx*(dHxB/dxB)*coshαM) + (xM*qxM*jsn)*(qxM*qx*(dHxB/dxB)*sinhαM)

= (tM*((dHxB/dxB)*coshαM) + xM*((dHxB/dxB)*sinhαM))*qx

87 CHAPTER 2 – PARTICLES tM*((dHxB/dxB)*coshαM) = (/ctM)*(( A*exp(- ((xM - vM*tM)*(coshαM/a))2 ) )*coshαM) = (-((xM - vM*tM)2)/ctM)*(coshαM/a)2 *(( A*exp(- ((xM - vM*tM)*(coshαM/a))2 ) )*coshαM) = -2*((xM - vM*tM)/ctM)*(xM - vM*tM)*(coshαM/a)2 *(( A*exp(- ((xM - vM*tM)*(coshαM/a))2 ) )*coshαM) = -2*(-vM/c)*(xM - vM*tM)*(cosh2αM/a2) *A*exp(-((xM - vM*tM)*(coshαM/a))2)*coshαM = 2*(sinhαM/coshαM)*(xM - vM*tM)*(cosh2αM/a2) *A*exp(-((xM - vM*tM)*(coshαM/a))2)*coshαM = 2*(xM - vM*tM)*(cosh2αM/a2) *A*exp(-((xM - vM*tM)*(coshαM/a))2)*sinhαM xM*((dHxB/dxB)*sinhαM) = (/xM)*(( A*exp(- ((xM - vM*tM)*(coshαM/a))2 ) )*sinhαM) = (-((xM - vM*tM)2)/xM)*(coshαM/a)2 *(( A*exp(- ((xM - vM*tM)*(coshαM/a))2 ) )*sinhαM) = -2*((xM - vM*tM)/xM)*(xM - vM*tM)*(coshαM/a)2 *(( A*exp(- ((xM - vM*tM)*(coshαM/a))2 ) )*sinhαM) = -2*(xM - vM*tM)*(cosh2αM/a2) *A*exp(-((xM - vM*tM)*(coshαM/a))2)*sinhαM 2

x2H = 0

*jsn

Questions for Further Thought.

88 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1) If the unspecified-speed-parameter “” is a rational number, then it cannot be infinite, has an equal probability of being positive or negative, and has an equal probability of magnitude less than one or greater than one. A probability function “P(s)” is given in Chapter 5. What alternative assumption might we make for the structure of “”? Numerator only, for equal probability of being any number on the number-line? 2) Assume “αS/M = i*” and find “c*tS”, “xS”, “vS/M/c” and “vS/c”. If “ ≠ z*/2” for “z  Z”, what problem is there? Should we make a rule (as a theory of physics), so that we mimic nature? 3) Is an invariant with hyperbolic-radius “dmB/dxB” or “QB” useful? 4) What does “N” in “αS/M = i**(2*N + 1)” represent physically? 5) What might an anti-matter person observe of us with respect to entropy increase, cause and effect, and successive collapses of matter-wave functions? Does the anti-matter person see us moving backward relative to their sense of time? Propose a cosmological model of the universe. 6) Rewrite the section on anti-matter electric current but substitute angular momentum for electric current. 7) Explain why “/tM + 3•(*3v) = 0” has the two terms added, in contrast to the subtraction in “(2r*j)•(2r) = (c*tM)2 - (xM2 + yM2 + zM2)”. 8) Assume the information transfer between detectors in the EPR experiment (see an Appendix) occurs instantly. Try to design an experiment in which the detectors move away from each other so that the detectors prove time can step backward. 9) Is the angular momentum density invariant the missing spacelike energy-momentum invariant as measurement units imply it is? If time-like energy-momentum is formed from induced electromagnetic fields (as explained in the next chapter), what fields could produce the space-like energy-momentum invariant?

89 CHAPTER 3 – FIELDS

Chapter 3 – Fields Using his equations, Maxwell derived that the speed for light was independent of the observer’s speed. To explain it, Einstein proposed Special Relativity. Imagine light traveling between two people side-byside on a bus. The path is longer when observed by a person on the roadside because the path has a component in the direction of bus travel. To keep the speed for light constant, a clock-tower shows faster passage of time compared to a clock mounted inside the moving bus.

3.1 Geometric-Vector Notation The four materials in Maxwell’s Equations are physical entities in our geometric world. Bold indicates a geometric-vector. Non-bold a scalar. Electric field 3E Magnetic field 3B

; ;

Current density 3j Charge density 

Each exists as a distribution in time and space, as does voltage (see Panofsky, Wolfgang and Philips, Melba: Classical Electricity and Magnetism, Addison-Wesley Publishing Company, Inc.; 1955). Vector voltage 3A

;

Scalar voltage 

A ground for voltage is analogous to an origin for location and to an inertial reference frame for speed or momentum. Maxwell evolved his equations and settled on component equations. A little before year 1900, Heaviside and others applied geometric-unit-vectors “ix”, “iy”, and “iz”, the dot-product “•”, the crossproduct “x”, and the gradient operator “3”. Geometric-unit-vectors explicitly placed components of electromagnetism into physical space. Three constants for measurement unit conversion:  = 4π*10-7 tesla*meter/amp  1.256637*10-6 tesla*meter/amp ǝ  8.854188*10-12 coulomb2/(newton*meter2) c  2.99792458*108 meters/second

90 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The three constants were selected so that “c2*ǝ* = 1”. The coulomb is the electric charge from a specific finite quantity of protons, or that same quantity of electrons times negative one. 1 coulomb = -6.24150965(16)*1018 electron charges QB-electron = -1.6021176462*10-19 coulomb Alternate units of measure are found from: 1 tesla*meter/amp = 1 newton/amp2 = 1 volt*second/(amp*meter) 1 coulomb2/(newton*meter2) = 1 farad/meter = 1 coulomb/(volt*meter) Maxwell’s Equations in geometric-vector notation: 3E

= Ex*ix + Ey*iy + Ez*iz ; 3j = jx*ix + jy*iy + jz*iz = Bx*ix + By*iy + Bz*iz 3 = x*ix + y*iy + z*iz = /x*ix + /y*iy + /z*iz 3B

3•3E

= /ǝ 2 x B 3 3 = *3j + ((3E)/t)/c

3•3B

; ;

Ex/x + Ey/y + Ez/z = /ǝ Bz/y - By/z = (Ex/t)/c2 + *jx Bx/z - Bz/x = (Ey/t)/c2 + *jy By/x - Bx/y = (Ez/t)/c2 + *jz

=0 x E 3 3 = -(3B)/t

; ; ; ;

Bx/x + By/y + Bz/z = 0 Ez/y - Ey/z = -Bx/t Ex/z - Ez/x = -By/t Ey/x - Ex/y = -Bz/t

Electric Charge Conservation in geometric-vector notation: /t + 3•3j = 0

/t + jx/x + jy/y + jz/z = 0

;

Voltage equations in geometric-vector notation: 3A

= Ax*ix + Ay*iy + Az*iz

3B

= 3x3A

;

3E

= -3 - (3A)/t

91 CHAPTER 3 – FIELDS Az/y - Ay/z = Bx Ax/z - Az/x = By Ay/x - Ax/y = Bz 3•3A

; ; ;

-/x - Ax/t = Ex -/y - Ay/t = Ey -/z - Az/t = Ez

+ (/t)/c2 = 0 ; Ax/x + Ay/y + Az/z + (/t)/c2 = 0

(3•3) - 2()/t2/c2 = -/ǝ ; (3•3)3A - 2(3A)/t2/c2 = -*3j 2/x2 + 2/y2 + 2/z2 - (2/t2)/c2 = -/ǝ 2Ax/x2 + 2Ax/y2 + 2Ax/z2 - (2Ax/t2)/c2 = -*jx 2Ay/x2 + 2Ay/y2 + 2Ay/z2 - (2Ay/t2)/c2 = -*jy 2Az/x2 + 2Az/y2 + 2Az/z2 - (2Az/t2)/c2 = -*jz Gauss’s Law for Electricity. “3•3E = /ǝ” states the divergence of the electric field is proportional to electric charge density. In other words, electric charge is the source of an electric field.

Figure 27. Gauss’s Law for Electricity. Electric field times area of a closed surface (a sphere) equals the electric charge inside. The electric field at the radius of the sphere is calculated from “Eradial*(r2**4/3) = q/ǝ”.

Gauss’s Law for Magnetism. “3•3B = 0” states there are no magnetic field charges (in contrast to the existence of electric charges). Faraday’s Law of Electric Field Induction. “3x3E = -(3B)/t” states a time varying magnetic field is a source of vorticity for an induced electric field.

92 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Figure 28. Gauss’s Law of Magnetism. Magnetic field times the area of a closed surface equals zero. The ends and the curved surface of the cylinder cut in half have “B•dA = 0”. The “B•dA” of the upper rectangle equals the negative of the “B•dA” of the lower rectangle, so that all the magnetic flux out of the identified volume equals all the flux into the volume, for a net total of zero.

Figure 29. Faraday’s Law of Induction. Electric field times length along a closed curve (a circle) equals the rate of change of the magnetic field times the area enclosed by the closed curve. Switch “s” closes to initiate the flow of current “i”, to create a magnet in the iron inside the wire coils. The gap of cross-section area “A” in the iron has an increasing magnitude magnetic field “dBx/dt”, which is negative. The changing magnetic field induces an electric field per “Ecircumferential*(2**r) = -(dBx/dt)*A”, for “r > (A/)” (for “r” outside the gap).

Ampere’s Law of Magnetic Field Induction. “3x3B = *3j + ((3E)/t)/c2” states the vorticity of an induced magnetic field is created by a time varying electric field and/or by electric current density.

93 CHAPTER 3 – FIELDS

Figure 30. Ampere’s Law of Induction. Magnetic field times length along a closed curve (a circle) equals the rate of change of the electric field times the area enclosed by the closed curve, plus the electric current through the area. Switch “s” closes to initiate a current that induces a magnetic field around the wire of strength “Bcircumferential = 0*i/(2**r)”. The magnetic field continues along the wire so that it also exists around the capacitor of area “A”, per “Bcircumferential*(2**r) = (dEx/dt)*A/c2” for the radius outside the area of the capacitor.

Figure 31. Electric Charge Conservation. The flow of electric charge out a closed surface (a cube), calculated as electric current density times area of the surface, equals the negative change in electric charge inside. In other words, electric charge cannot be created or destroyed. The box on the right contains one of the two capacitor plates. The change in electric charge on the capacitor plate inside the box is “dq/dt < 0” as noticed by the increase in the electric field “Ex”. The box is penetrated by the wire out the right for which the current “i” equals the current density “Jx” times the cross-section area of the wire. The conservation equation reduces to “i + dq/dt = 0”.

94 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Electric Charge Conservation. “/t + 3•3j = 0”, rewritten “-/t = 3•3j”, states the divergence of an electric current density equals the negative of the change in electric charge density.

3.2 All-Number Notation New component symbols apply to the second step’s translation of geometry into all-number algebra. Vt =  Vx = Ax*c Vy = Ay*c Vz = Az*c

; ; ;

; ; ; ;

Kx = Bx*c Ky = By*c Kz = Bz*c

Jt = /ǝ = *(*c2) Jx = jx*(*c) = jx/(ǝ*c) Jy = jy*(*c) = jy/(ǝ*c) Jz = jz*(*c) = jz/(ǝ*c)

“*c” and “ǝ*c” are near a numerical value of one. Simple-Label-Numbers. Using the components specified above, Maxwell’s Equations, the electric charge conservation equation and the voltage equation may be written using label-numbers. jx*jy = -jy*jx = jz jy*jz = -jz*jy = jx jz*jx = -jx*jz = jy

; ; ;

i*jx = jx*i i*jy = jy*i i*jz = jz*i

jx2 = jy2 = jz2 = i2 = -1

;

qx2 = qy2 = qz2 = 12 = +1

qx = jx/i ; qy = jy/i ; qz = jz/i qx*jy = -jy*qx = -qy*jx = jx*qy = qz ; qy*jz = -jz*qy = -qz*jy = jy*qz = qx ; qz*jx = -jx*qz = -qx*jz = jz*qx = qy ;

qx*qy = -qy*qx = -i*qz = -jz qy*qz = -qz*qy = -i*qx = -jx qz*qx = -qx*qz = -i*qy = -jy

95 CHAPTER 3 – FIELDS Unspecified-Label-Number. There is an unspecified-speed-parameter “” and an unspecified-label-number “” (kappa) restricted to one of “qx”, “qy” or “qz”. “” is unknown and unknowable. “” and “” are placed into two exponential functions: One for each side, left and right, of a simple-label-number “1”, “qx”, “qy” or “qz”. Each is a square root by dividing the argument by two. 1M = exp(-*/2)*1*exp(-*/2) = exp(-*) qxM = exp(-*/2)*qx*exp(-*/2) qyM = exp(-*/2)*qy*exp(-*/2) qzM = exp(-*/2)*qz*exp(-*/2) Hypothetically: “ = qx” for “1M = exp(-qx*)”, “qxM = qx*exp(-qx*)”, “qyM = qy” and “qzM = qz” from the previous chapter. The concept of using a square root factor on both sides, to take advantage of the anticommute law “qx*qy = -qy*qx”, is a classic concept in Special Relativity. 1M = exp(-*/2)*1*exp(-*/2) = exp(-qx*/2)*1*exp(-qx*/2) “ = qx” = exp(-qx*/2)*exp(-qx*/2) = exp(-qx*) qxM = exp(-*/2)*qx*exp(-*/2) = exp(-qx*/2)*(qx*exp(-qx*/2)) “ = qx” = exp(-qx*/2)*(qx*(cosh(/2) - qx*sinh(/2))) = exp(-qx*/2)*((cosh(/2) - qx*sinh(/2))*qx) = exp(-qx*/2)*(exp(-qx*/2)*qx) = exp(-qx*)*qx qyM = exp(-*/2)*qy*exp(-*/2) = exp(-qx*/2)*(qy*exp(-qx*/2)) “ = qx” = exp(-qx*/2)*(qy*(cosh(/2) - qx*sinh(/2))) = exp(-qx*/2)*((cosh(/2) + qx*sinh(/2))*qy) = exp(-qx*/2)*(exp(qx*/2)*qy) = qy qzM = exp(-*/2)*qz*exp(-*/2) = exp(-qx*/2)*(qz*exp(-qx*/2)) “ = qx” = exp(-qx*/2)*(qz*(cosh(/2) - qx*sinh(/2)))

96 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = exp(-qx*/2)*((cosh(/2) + qx*sinh(/2))*qz) = exp(-qx*/2)*(exp(qx*/2)*qz) = qz Invariants. Compound-label-numbers combine with components to form invariants. “4sn” is a space-negative. 4V

= VtM*it + c*AxM*ix + c*AyM*iy + c*AzM*iz V = 1M*VtM + qxM*VxM + qyM*VyM + qzM*VzM 4 = *(*c2)*it + jx*(*c)*ix + jy*(*c)*iy + jz*(*c)*iz 4J = 1M*JtM + qxM*JxM + qyM*JyM + qzM*JzM 4J

3E

= ExM*ix + EyM*iy + EzM*iz c*3B = c*BxM*ix + c*ByM*iy + c*BzM*iz 6E

= pxM*ExM + pyM*EyM + pzM*EzM + kxM*KxM + kyM*KyM + kzM*KzM

4 4

= tM*it + xM*ix + yM*iy + zM*iz = 1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM = 1Msn*(/ctM) + qxMsn*/xM + qyMsn*/yM + qzMsn*/zM

sn

E-M-Compound-Label-Numbers. Two new sets of compound-labelnumbers (“pxM”, “pyM” and “pzM” and “kxM”, “kyM” and “kzM”) have been introduced for the electromagnetic field “6E”. pxM = exp(*/2)*qx*exp(-*/2) = 1M*jsn*qxM = -qxM*jsn*1M = -qyM*jsn*qzM/i = qzM*jsn*qyM/i = kxM/i pyM = exp(*/2)*qy*exp(-*/2) = 1M*jsn*qyM = -qyM*jsn*1M = -qzM*jsn*qxM/i = qxM*jsn*qzM/i = kyM/i pzM = exp(*/2)*qz*exp(-*/2) = 1M*jsn*qzM = -qzM*jsn*1M = -qxM*jsn*qyM/i = qyM*jsn*qxM/i = kzM/i kxM = exp(*/2)*jx*exp(-*/2) = i*pxM = 1M*jsn*jxM = jxM*jsn*1M = -qyM*jsn*qzM = qzM*jsn*qyM = jyM*jsn*jzM = -jzM*jsn*jyM

97 CHAPTER 3 – FIELDS

= 1M

kyM = exp(*/2)*jy*exp(-*/2) = i*pyM *jyM = jyM*jsn*1M = -qzM*jsn*qxM = qxM*jsn*qzM = jzM*jsn*jxM = -jxM*jsn*jzM

*jsn

= 1M

= exp(*/2)*jz*exp(-*/2) = i*pzM = -jyM*jsn*jxM *jzM = jzM*jsn*1M = -qxM*jsn*qyM = qyM*jsn*qxM = jxM*jsn*jyM

kzM *jsn

1M = qxM*pxM = qyM*kzM*pxM = qyM*pyM = qyM*pzM*kxM = 1M*1 = 1M*pxM*pxM = 1M*1 = -1M*kxM*kxM = qxM*pxM qzM = 1M*pzM = 1M*kxM*pyM = 1M*pzM = 1M*pxM*kyM = jxM*pyM = -qyM*pzM*pyM = qxM*kyM = qyM*kzM*kyM = 1M*pzM = qzM*pzM*pzM = 1M*i*kzM = qzM*kzM*kzM 1M*i = qyM*kyM = qyM*kzM*kxM = jyM*pxM = -qyM*pzM*pxM = 1M*pxM*kxM = 1M*kxM*pxM = qxM*kxM jzM = jxM*kyM = 1M*kxM*kyM = 1M*kzM = -1M*pxM*pyM = -qxM*pyM = -qyM*kzM*pyM = jxM*kyM = -qyM*pzM*kyM = 1M*kzM = qzM*kzM*pzM = 1M*kyM = qzM*pzM*kyM 1M = exp(-*/2)*1*exp(-*/2) = exp(-*) qxM = exp(-*/2)*qx*exp(-*/2) qyM = exp(-*/2)*qy*exp(-*/2) qzM = exp(-*/2)*qz*exp(-*/2) pxM = exp(*/2)*qx*exp(-*/2) pyM = exp(*/2)*qy*exp(-*/2) pzM = exp(*/2)*qz*exp(-*/2) kxM = exp(*/2)*jx*exp(-*/2) = i*pxM kyM = exp(*/2)*jy*exp(-*/2) = i*pyM kzM = exp(*/2)*jz*exp(-*/2) = i*pzM = jxM*jsn*jyM The quaternion hypercomplex-conjugate operation “*j” reverses the sign of each “j”, and, therefore, also reverses the sign of each “q”, and reverses the order of the factors.

98 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY i *j = i

jx*j = -jx ; jy*j = -jy ; jz*j = -jz qx*j = -qx ; qy*j = -qy ; qz*j = -qz

; 1*j = 1

jxM*j = exp(*/2)*(-jx)*exp(*/2) jyM*j = exp(*/2)*(-jy)*exp(*/2) jzM*j = exp(*/2)*(-jz)*exp(*/2) 1M*j = exp(*/2)*(1)*exp(*/2) qxM*j = exp(*/2)*(-qx)*exp(*/2) qyM*j = exp(*/2)*(-qy)*exp(*/2) qzM*j = exp(*/2)*(-qz)*exp(*/2)

kxM*j = -kxM kyM*j = -kyM kzM*j = -kzM

; ; ;

; ; ;

pxM*j = -pxM pyM*j = -pyM pzM*j = -pzM

Conjugate Form of Invariants. *j 4V 4J

*j

*j 6E

4

= VtM*1M*j + VxM*qxM*j + VyM*qyM*j + VzM*qzM*j

= JtM*1M*j + JxM*qxM*j + JyM*qyM*j + JzM*qzM*j = ExM*pxM*j + EyM*pyM*j + EzM*pzM*j + KxM*kxM*j + KyM*kyM*j + KzM*kzM*j

*jsn

= tM*1M*jsn + xM*qxM*jsn + yM*qyM*jsn + zM*qzM*jsn

Governing Equations Voltage Equation

4

Lorenz Condition

4

Maxwell’s Equations

4

*jsn

x(4V) = -6E

•(4V) = 0

*jsn

*4*jsn*(4V) = 4sn*(-6E) sn 4 *(-6E) = 4J sn

Electric Charge Conservation Equation 4

•(4sn*4*jsn*(4V)) = 4*jsn•(4sn*(-6E)) = 4*jsn•(4J) = 0

*jsn

99 CHAPTER 3 – FIELDS Decomposition. Invariants in the above equations separate into pieces per the nomenclature given below. Pieces alone are not invariants. Therefore, pieces are specific to an inertial reference frame even though the “M” or “S” subscript may be dropped to simplify what is written. 4V 4J

= 1V + 3V = 1J + 3J

; ;

1V 1J

= 1M*VtM = 1M*JtM

4

sn

3E

= pxM*ExM + pyM*EyM + pzM*EzM

; ;

3V 3J

= qxM*VxM + qyM*VyM + qzM*VzM = qxM*JxM + qyM*JyM + qzM*JzM

= 1sn + 3sn ; 1sn = 1Msn*tM sn sn sn sn 3 = qxM *xM + qyM *yM + qzM *zM 6E = 3E + 3K ;

3K

= kxM*KxM + kyM*KyM + kzM*KzM

Multiplication operation “*” decomposes into the dot-product “•” and cross-product “x”. qx*qy = qx•qy + qxxqy = qxxqy

;

qx*qx = qx•qx + qxxqx = qx•qx

qx•qy = 0

;

qxxqx = 0

Lorenz Condition “4*jsn•(4V) = 0” applies. 4

*jsn

*(4V) = 4*jsn•(4V) + 4*jsnx(4V)

4

*jsn

4

*jsn

•(4V) = 1*jsn*(1V) + 3*jsn•(3V) = 0 •(4V) = tM*VtM*1M*jsn*1M + xM*VxM*qxM*jsn*qxM + yM*VyM*qyM*jsn*qyM + zM*VzM*qzM*jsn*qzM = tM*VtM + xM*VxM + yM*VyM + zM*VzM = 0

Electromagnetic Field Voltage Equation. The cross-product “4*jsnx(4V)” results in the negative of the electromagnetic field. 4

*jsn

x(4V) = 1*jsn*(3V) + 3*jsn*(1V) + 3*jsnx(3V) = -6E

1

*jsn

*(3V) + 3*jsn*(1V) = -3E

;

3

*jsn

x(3V) = -3K

100 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Electric Field Voltage Equation. 1

*(3V) + 3*jsn*(1V) = -3E

*jsn

tM*VxM*1M*jsn*qxM + xM*VtM*qxM*jsn*1M = tM*VxM*pxM + xM*VtM*pxM = -pxM*ExM -tM*VxM - xM*VtM = ExM -tM*VyM - yM*VtM = EyM -tM*VzM - zM*VtM = EzM Magnetic Field Voltage Equation. 3

*jsn

x(3V) = -3K

yM*VzM*qyM*jsn*qzM + zM*VyM*qzM*jsn*qyM = -yM*VzM*kxM + zM*VyM*kxM = -kxM*KxM qyM*jsn*qzM = -qyM*j*qzM = jyM*j*jzM = -kxM -yM*VzM + zM*VyM = -KxM -zM*VxM + xM*VzM = -KyM -xM*VyM + yM*VxM = -KzM Triple-Vector-Product and Remnant-Product. Dot-product and/or crossproduct do not apply to six-component invariants. Use: The triplevector-product “■” and the remnant-product “♦”. 4

sn

*(4*jsn*(4V)) = 4sn*(4*jsn•(4V)) + 4sn*(4*jsnx(4V))

4

sn

*(4*jsnx(4V)) = 4sn■(4*jsnx(4V)) + 4sn♦(4*jsnx(4V)) = 4sn■(-6E) + 4sn♦(-6E)

101 CHAPTER 3 – FIELDS Triple-Vector-Product Gradient Identities. 4

sn

■(4*jsnx(4V)) = 1sn*(3*jsnx(3V)) + 3snx(1*jsn*(3V)) + 3snx(3*jsn*(1V)) + 3sn•(3*jsnx(3V))

4

sn

■(-6E) = 1sn*(-3K) + 3snx(-3E) + 3sn•(-3K)

“4sn■(4*jsnx(4V))” includes identities. 1

*(3*jsnx(3V)) + 3snx(1*jsn*(3V))  0 *jsn *(1V))  0 3 x(3 sn *jsn x(3V))  0 ; 4sn■(4*jsnx(4V))  0 3 •(3 sn sn

Electromagnetic components substitute into the sum of the first two identities to result in the first of four of Maxwell’s Equations. 1

sn

*(3*jsnx(3V)) + (3snx(1*jsn*(3V)) + 3snx(3*jsn*(1V)))  0

1

sn

*(-3K) + 3snx(-3E) = 0

-tM*KxM*1Msn*kxM - yM*EzM*qyMsn*pzM - zM*EyM*qzMsn*pyM = 0 -tM*KxM*jxM - yM*EzM*jxM + zM*EyM*jxM = 0 -tM*KxM - yM*EzM + zM*EyM = 0 Electromagnetic components substitute into “3sn•(3*jsnx(3V))  0” to result in the second of four of Maxwell’s Equations. 3

•(3*jsnx(3V))  0

sn

;

3

•(-3K) = 0

sn

-xM*KxM*qxMsn*kxM - yM*KyM*qyMsn*kyM - zM*KzM*qzMsn*kzM = 0 xM*KxM + yM*KyM + zM*KzM = 0 The first with the second of the four Maxwell’s Equations are given below. 4

■(4*jsnx(4V)) = 4sn■(-6E) = 0

sn

102 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Remnant-Product Results. 4

♦(4*jsnx(4V)) = 4sn*(4*jsnx(4V)) - 4sn■(4*jsnx(4V))

sn

= 1sn*(1*jsn*(3V)) + 1sn*(3*jsn*(1V)) + 3snx(3*jsnx(3V)) + 3sn•(1*jsn*(3V)) + 3sn•(3*jsn*(1V)) Third of four of Maxwell’s Equations: 1

*(1*jsn*(3V)) + 1sn*(3*jsn*(1V)) + 3snx(3*jsnx(3V)) = 1sn*(-3E) + 3snx(-3K) = 3J

sn

-tM*ExM*1Msn*pxM - yM*KzM*qyMsn*kzM - zM*KyM*qzMsn*kyM = qxM*JxM -tM*ExM*qxM + yM*KzM*qxM - zM*KyM*qxM = qxM*JxM -tM*ExM + yM*KzM - zM*KyM = JxM Fourth of four of Maxwell’s Equations: 3

•(1*jsn*(3V)) + 3sn•(3*jsn*(1V)) = 3sn•(-3E) = 1J

sn

-xM*ExM*qxMsn*pxM - yM*EyM*qyMsn*pyM - zM*EzM*qzMsn*pzM = 1M*JtM xM*ExM + yM*EyM + zM*EzM = JtM The third with the fourth of Maxwell’s Equations: 4

♦(4*jsnx(4V)) = 4sn♦(-6E) = 4J

sn

Maxwell’s Equations.

4

*(4*jsnx(4V)) = 4sn*(-6E) = 4J + 0

sn

-tM*KxM - yM*EzM + zM*EyM = 0 ; -tM*ExM + yM*KzM - zM*KyM = JxM -tM*KyM - zM*ExM + xM*EzM = 0 ; -tM*EyM + zM*KxM - xM*KzM = JyM -tM*KzM - xM*EyM + yM*ExM = 0 ; -tM*EzM + xM*KyM - yM*KxM = JzM xM*KxM + yM*KyM + zM*KzM = 0 ; xM*ExM + yM*EyM + zM*EzM = JtM

103 CHAPTER 3 – FIELDS Electric Charge Conservation. *jsn *4sn*4*jsn*(4V) = 4*jsn*(4sn*(4*jsn*(4V))) 4 = 4*jsn*(4sn*(4*jsn•(4V))) + 4*jsn*(4sn*(4*jsnx(4V))) Lorenz Condition “4*jsn•(4V) = 0” and other identities apply. 4

4

*4sn*4*jsn*(4V) = 4*jsn•(4sn♦(4*jsnx(4V))) + 4*jsnx(4sn♦(4*jsnx(4V)))

*jsn

•(4sn♦(4*jsnx(4V))) = 3*jsn•(1sn*(1*jsn*(3V) + 3*jsn*(1V))) + 3*jsn•(3snx(3*jsnx(3V))) + 1*jsn*(3sn•(1*jsn*(3V) + 3*jsn*(1V)))

*jsn

3

•(1sn*(1*jsn*(3V))) + 1*jsn*(3sn•(1*jsn*(3V)))  0 •(1sn*(3*jsn*(1V))) + 1*jsn*(3sn•(3*jsn*(1V)))  0 3 *jsn •(3snx(3*jsnx(3V)))  0 3 *jsn *jsn

4

*jsn

•(4sn♦(4*jsnx(4V)))  0

4

*jsn

•(4J) = 0

tM*JtM*1M*jsn*1M + xM*JxM*qxM*jsn*qxM + yM*JyM*qyM*jsn*qyM + zM*JzM*qzM*jsn*qzM = 0 tM*JtM + xM*JxM + yM*JyM + zM*JzM = 0 Summary of Component Equations. Lorenz Condition:

4

•(4V) = 0

*jsn

tM*VtM + xM*VxM + yM*VyM + zM*VzM = 0 Electric Field Voltage Equation: 1

*(3V) + 3*jsn*(1V) = -3E

*jsn

Magnetic Field Voltage Equation: 3

*jsn

x(3V) = -3K

;

-tM*VxM - xM*VtM = ExM -tM*VyM - yM*VtM = EyM -tM*VzM - zM*VtM = EzM -yM*VzM + zM*VyM = -KxM -zM*VxM + xM*VzM = -KyM -xM*VyM + yM*VxM = -KzM

104 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Maxwell’s Induced Electric Field Equation: 1

*(-3K) + 3snx(-3E) = 0

sn

-tM*KxM - yM*EzM + zM*EyM = 0 -tM*KyM - zM*ExM + xM*EzM = 0 -tM*KzM - xM*EyM + yM*ExM = 0

Maxwell’s Zero Magnetic Charge Equation:

sn 3 •(-3K)

=0

xM*KxM + yM*KyM + zM*KzM = 0 Maxwell’s Induced Magnetic Field Equation: 1

*(-3E) + 3snx(-3K) = 3J

sn

-tM*ExM + yM*KzM - zM*KyM = JxM -tM*EyM + zM*KxM - xM*KzM = JyM -tM*EzM + xM*KyM - yM*KxM = JzM

Maxwell’s Electric Charge Equation:

3

•(-3E) = 1J

sn

xM*ExM + yM*EyM + zM*EzM = JtM Electric Charge Conservation Equation:

4

•(4J) = 0

*jsn

tM*JtM + xM*JxM + yM*JyM + zM*JzM = 0

105 CHAPTER 3 – FIELDS

3.3 Gauges and Super-Potentials Super-Potentials. Voltage “4V” is called “potential”. By analogy, “4J” is the first “sub-potential”. 4

•(4V) = 0 *jsn  •(4J) = 0 4 *jsn

Lorenz Condition Electric Charge Conservation Equation

“4J” relates to “4V” by the square of the gradient operator. By analogy, voltage is related to the first super-potential “4U”, “4*jsn•(4U) = 0”. (4J) = (4sn•4*jsn)*(4V) = (tM2 - xM2 - yM2 - zM2)*(4V) (4V) = (4sn•4*jsn)*(4U) = (tM2 - xM2 - yM2 - zM2)*(4U) Super-potentials and sub-potentials extend indefinitely. “(4sn•4*jsn) = (tM2 - xM2 - yM2 - zM2)” is called the “harmonic operator” because, if “4J = 0”, then “4V” is a summation of sine waves. The physical example is light waves. Fields. Between each pair of adjacent potentials is a six-component field. “6E” is between “4J” and “4V”. 4

*jsn

*(4V) = -6E ;

sn 4 *(-6E)

= 4J

The first sub-field “6G” has “4J” as its potential. 4

*jsn

*(4J) = 4*jsnx(4J) = -6G

4

*jsn

x(4sn♦(4*jsnx(4V))) = 4*jsnx(4sn♦(-6E)) = 4*jsnx(4J) = -6G

106 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Time-Component Field Gauge. A potential is not unique to the field. An electromagnetic field has a potential “4V’” (“’” is “prime”) from which alternative “4V” are found using a mathematically complex field “2P”. 4V 3

= 4V’ + 4sn*2P x(3sn*2P)  0 *(3sn*2P) + 3*jsn*(1sn*2P)  0

*jsn

1

*jsn

4

*jsn

x(4sn*2P)  0

-6E = 4*jsnx(4V) = 4*jsnx(4V’) + 4*jsnx(4sn*2P) = 4*jsnx(4V’) “6E” is not affected by “2P”. “2P” affects the Lorenz Condition because “2P” adds a harmonic oscillator term “(4*jsn•4sn)*2P”. To keep “4V” real, “2P” must be real. 0 = 4*jsn•(4V) = 4*jsn•(4V’) + 4*jsn•(4sn*2P) = 4*jsn•(4V’) + (4*jsn•4sn)*2P The complex number field “2P” is called the field gauge. The simplest version of “4sn*2P” is a constant, and a change to that constant is a change to the ground of the voltage time component. Space-Component Field Gauge does not affect the Lorenz Condition, but, rather, adds a term to the electromagnetic field “6E”. 3

*jsn

x(3snx6Q)  (3*jsn•3sn)*6Q - 3*jsn*(3sn•6Q)

4

*jsn

4V

•(4sn*(-6Q)) = 4*jsn•(4sn■(-6Q)) + 4*jsn•(4sn♦(-6Q)) = 1*jsn*(3sn•(-3Qi)) + 3*jsn•(1sn♦(-3Qi) + 3snx(-3Qr)) + 1*jsn*(3sn•(-3Qr)) + 3*jsn•(1sn♦(-3Qr) + 3snx(-3Qi)) 0

= 4V’ + 4sn*(-6Q)

107 CHAPTER 3 – FIELDS 4

•(4V) = 4*jsn•(4V’) + 4*jsn•(4sn*(-6Q)) = 4*jsn•(4V’) = 0

*jsn

-6E = 4*jsnx4V = 4*jsnx4V’ + 4*jsnx(4sn*(-6Q)) = 4*jsnx4V’ + 4*jsnx(4■(-6Q)) + 4*jsnx(4♦(-6Q)) = 4*jsnx4V’ + 4*jsnx(3sn•(-3Qi) + 1sn*(-3Qi) + 3snx(-3Qr)) + 4*jsnx(3sn•(-3Qr) + 1sn*(-3Qr) + 3snx(-3Qi)) = 4*jsnx4V’ + 3*jsn*(3•(-3Qi)) + 1*jsn*(1sn*(-3Qi) + 3snx(-3Qr)) + 3*jsnx(1sn*(-3Qi) + 3snx(-3Qr)) + 3*jsn*(3sn•(-3Qr)) + 1*jsn*(1sn*(-3Qr) + 3snx(-3Qi)) + 3*jsnx(1sn*(-3Qr) + 3snx(-3Qi)) = 4*jsnx4V’ + 3*jsn*(3sn•(-3Qi)) + 1*jsn*(1sn*(-3Qi)) + 3*jsnx(3snx(-3Qr)) + 3*jsn*(3sn•(-3Qr)) + 1*jsn*(1sn*(-3Qr)) + 3*jsnx(3snx(-3Qi)) = 4*jsnx4V’ + 3*jsn*(3sn•(-6Q)) + 1*jsn*(1sn*(-6Q)) + 3*jsnx(3snx(-6Q)) = 4*jsnx4V’ + (3*jsn•3sn)*(-6Q) + (1*jsn*1sn)*(-6Q) = 4*jsnx4V’ + (4*jsn•4sn)*(-6Q) “(-6E) = 4*jsnx4V’ + (4*jsn•4sn)*(-6Q)” and “4*jsn•(4V) = 4*jsn•(4V’) + 4*jsn•(4sn*(-6Q)) = 4*jsn•(4V’) = 0” define different electromagnetic fields for one Lorentz Condition. “2P” is a classic feature in electromagnetic theory. “6Q” was not mentioned in reference material available to the author. The electromagnetic field harmonic term “(4*jsn•4sn)*(-6Q)” is similar in structure to the super-field equation given below. (-6E) = (4*jsn•4sn)*(-6H) = (tM2 - xM2 - yM2 - zM2)*(-6H) Concluding Statement. Super-potentials and gauge fields place Maxwell’s Equations into a larger (and mathematically beautiful) structure. The identities are easily identified in an all-number algebra but not in the geometric algebra used when Maxwell’s Equations were first discovered.

108 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

3.4 Lorentz Transformation The Lorentz Transformation transforms components and compoundlabel-numbers from “M” (moving because seated on the bus) to “S” (stationary because standing on the roadside). The general form has two “1” factors, left and right. The technique is checked: An invariant formed as a product of two other invariants must have the same Lorentz Transformation result if the invariant is transformed directly or if the invariant is formed by multiplication of two transformed invariants. Four-Component Vector Lorentz Transformation. General form: 4V = 1M*VtM + qxM*VxM + qyM*VyM + qzM*VzM = exp(-*/2)*(1*VtM + qx*VxM + qy*VyM + qz*VzM)*exp(-*/2) = exp(-*/2)*1*(1*VtM + qx*VxM + qy*VyM + qz*VzM)*1*exp(-*/2)

= exp(-*/2)*exp(-qx*αS/M/2) *exp(qx*αS/M/2)*(1M*VtM + qxM*VxM + qyM*VyM + qzM*VzM)*exp(qx*αS/M/2) *exp(-qx*αS/M/2)*exp(-*/2) Components: 1*VtS + qx*VxS + qy*VyS + qz*VzS = exp(qx*αS/M/2)*(1*VtM + qx*VxM + qy*VyM + qz*VzM)*exp(qx*αS/M/2) VtS = VtM*coshαS/M + VxM*sinhαS/M VxS = VtM*sinhαS/M + VxM*coshαS/M VyS = VyM ; VzS = VzM Same mathematics applies to the other four-component invariants. JtS = JtM*coshαS/M + JxM*sinhαS/M JxS = JtM*sinhαS/M + JxM*coshαS/M JyS = JyM ; JzS = JzM Matrix equation form:

109 CHAPTER 3 – FIELDS VtS

coshαS/M

sinhαS/M

= VxS

VtM *

sinhαS/M

coshαS/M

VxM

Lorentz Transformation of the compound-label-numbers: 1S = exp(-*/2)*exp(-qx*αS/M/2)*exp(-qx*αS/M/2)*exp(-*/2) = exp(-*/2)*exp(-qx*αS/M)*exp(-*/2) qxS = exp(-*/2)*exp(-qx*αS/M/2)*qx*exp(-qx*αS/M/2)*exp(-*/2) = exp(-*/2)*qx*exp(-qx*αS/M)*exp(-*/2) qyS = exp(-*/2)*exp(-qx*αS/M/2)*qy*exp(-qx*αS/M/2)*exp(-*/2) = exp(-*/2)*exp(-qx*αS/M/2)*exp(qx*αS/M/2)*qy*exp(-*/2) = exp(-*/2)*qy*exp(-*/2) = qyM qzS = exp(-*/2)*exp(-qx*αS/M/2)*qz*exp(-qx*αS/M/2)*exp(-*/2) = exp(-*/2)*exp(-qx*αS/M/2)*exp(qx*αS/M/2)*qz*exp(-*/2) = exp(-*/2)*qz*exp(-*/2) = qzM 1S*j = (exp(-*/2)*exp(-qx*αS/M)*exp(-*/2))*j = (exp(*/2)*exp(qx*αS/M)*exp(*/2)) qxS*j = (exp(-*/2)*qx*exp(-qx*αS/M)*exp(-*/2))*j = exp(*/2)*exp(qx*αS/M)*-qx*exp(*/2) qyS*j = qyM*j ; qzS*j = qzM*j “4V” in “S” equals “4V” in “M”, proven using mathematics analogous for the same activity in the two-dimensional time-space. 4V

= 1S*VtS + qxS*VxS + qyS*VyS + qzS*VzS = 1M*VtM + qxM*VxM + qyM*VyM + qzM*VzM = 4V

The dot-product of two four-component invariants is an invariant. *j 4k •4r

= ((S/c)*1S*j + kxS*qxS*j + kyS*qyS*j + kzS*qzS*j) •(1S*c*tS + qxS*xS + qyS*yS + qzS*zS)

110 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = ((S/c)*1M*j*1S*c*tS + kxS*qxS*j*qxS*xS + kyS*qyS*j*qyS*yS + kzS*qzS*j*qzS*zS) = (S/c)*c*tS - kxS*xS - kyS*yS - kzS*zS = ((M/c)*coshαS/M + kxM*sinhαS/M)*(c*tM*coshαS/M + xM*sinhαS/M) - ((M/c)*sinhαS/M + kxM*coshαS/M)*(c*tM*sinhαS/M + xM*coshαS/M) - kyM*yM - kzM*zM = ((M/c)*c*tM*(cosh2αS/M - sinh2αS/M) + (kxM*xM*(sinh2αS/M - cosh2αS/M) + (M/c)*xM*(-sinhαS/M*coshαS/M + coshαS/M*sinhαS/M) + (kxM*c*tM*(-coshαS/M*sinhαS/M + sinhαS/M*coshαS/M) - kyM*yM - kzM*zM = (M/c)*c*tM - kxM*xM - kyM*yM - kzM*zM = 4k*j•4r Space-Negative Lorentz Transformation. The inverted matrix for the space-negative Lorentz Transformation is complemented by an inverted matrix for the space-negative compound-label-numbers. tS = tM*coshαS/M - xM*sinhαS/M xS = -tM*sinhαS/M + xM*coshαS/M yS = yM ; zS = zM 1Ssn = (1M*coshαS/M - qxM*sinhαS/M)sn = 1Msn*coshαS/M + qxMsn*sinhαS/M qxSsn = (-1M*sinhαS/M + qxM*coshαS/M)sn = 1Msn*sinhαS/M + qxMsn*coshαS/M 4

sn

= 1Ssn*tS + qxSsn*xS + qySsn*yS + qzSsn*zS

= (1M*coshαS/M + qxM*sinhαS/M)sn*(tM*coshαS/M - xM*sinhαS/M) + (1M*sinhαS/M + qxM*coshαS/M)sn*(-tM*sinhαS/M + xM*coshαS/M) + qyMsn*yM + qzMsn*zM = (1Msn*coshαS/M + qxMsn*sinhαS/M)*(tM*coshαS/M - xM*sinhαS/M) + (+1Msn*sinhαS/M + qxMsn*coshαS/M)*(-tM*sinhαS/M + xM*coshαS/M) + qyMsn*yM + qzMsn*zM

111 CHAPTER 3 – FIELDS = (1Msn*tM + qxMsn*xM)*(cosh2αS/M - sinh2αS/M) + (qxMsn*tM + 1Msn*xM)*(coshαS/M*sinhαS/M - coshαS/M*sinhαS/M) + qyMsn*yM + qzMsn*zM = 1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM = 4sn “4*jsn•(4V) = 0” and “4*jsn•(4J) = 0” are valid after the Lorentz Transformation. 4

•(4J) = (tS*1S*jsn + xS*qxS*jsn + yS*qyS*jsn + zS*qzS*jsn) •(1S*JtS + qxS*JxS + qyS*JyS + qzS*JzS)

*jsn

= tS*1M*jsn*1S*JtS + xS*qxS*jsn*(qxS)*JxS + yS*qyS*jsn*(qyS)*JyS + zS*qzS*jsn*(qzS)*JzS = tS*JtS + xS*JxS + yS*JyS + zS*JzS = (tM*coshαS/M - xM*sinhαS/M)*(JtM*coshαS/M + JxM*sinhαS/M) + (-tM/c*sinhαS/M + xM*coshαS/M)*(JtM*sinhαS/M + JxM*coshαS/M) + yM*JyM + zM*JzM = tM*JtM*(cosh2αS/M - sinh2αS/M) + xM*JxM*(-sinh2αS/M + cosh2αS/M) + tM*JxM*(sinhαS/M*coshαS/M - coshαS/M*sinhαS/M) + xM*JtM*(-coshαS/M*sinhαS/M + sinhαS/M*coshαS/M) + yM*JyM + zM*JzM = tM*JtM + xM*JxM + yM*JyM + zM*JzM =0

Electromagnetic Field Lorentz Transformation. On the right is “exp(qx*αS/M/2)” from the voltage invariant. On the left is the reciprocal “exp(-qx*αS/M/2)” from the gradient invariant, but with the argument changed to positive by the conjugate operation, and then back to negative by the space-negative. Begin with the general form of the Lorentz Transformation.

112 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 6E

= pxM*ExM + pyM*EyM + pzM*EzM + kxM*KxM + kyM*KyM + kzM*KzM = exp(*/2)*1*(qx*ExM + qy*EyM + qz*EzM + jx*KxM + jy*KyM + jz*KzM)*1*exp(-*/2)

= exp(*/2)*exp(qx*αS/M/2) *exp(-qx*αS/M/2)*(qx*ExM + qy*EyM + qz*EzM + jx*KxM + jy*KyM + jz*KzM) *exp(qx*αS/M/2) *exp(-qx*αS/M/2)*exp(-*/2) The equation to find “S” component values for “6E” is below. qx*(ExS + i*KxS) + qy*(EyS + i*KyS) + qz*(EzS + i*KzS) = exp(-qx*αS/M/2)*( qx*(ExM + i*KxM) + qy*(EyM + i*KyM) + qz*(EzM + i*KzM) )*exp(qx*αS/M/2) Derivations: qy*EyS = cosh(αS/M/2)*( qy*EyM )*cosh(αS/M/2) + -qx*sinh(αS/M/2)*( qy*EyM )*qx*sinh(αS/M/2) + cosh(αS/M/2)*( qz*i*KzM )*qx*sinh(αS/M/2) + -qx*sinh(αS/M/2)*( qz*i*KzM )*cosh(αS/M/2) = cosh(αS/M/2)*cosh(αS/M/2)*( qy*EyM ) + qx2*sinh(αS/M/2)*sinh(αS/M/2)*( qy*EyM ) + -2*qx*sinh(αS/M/2)*cosh(αS/M/2)*( qz*i*KzM ) = coshαS/M*( qy*EyM ) - qx*sinhαS/M*( qz*i*KzM ) = qy*coshαS/M*EyM - qx*qz*i*sinhαS/M*KzM = qy*(coshαS/M*EyM + sinhαS/M*KzM) -qx*qz*i = -( (jx/i)*(jz/i)*i ) = jx*jz*i = -jy*i = qy qy*i*KyS = cosh(αS/M/2)*( qy*i*KyM )*cosh(αS/M/2) + -qx*sinh(αS/M/2)*( qy*i*KyM )*qx*sinh(αS/M/2) + cosh(αS/M/2)*( qz*EzM )*qx*sinh(αS/M/2) + -qx*sinh(αS/M/2)*( qz*EzM )*cosh(αS/M/2) = cosh(αS/M/2)*cosh(αS/M/2)*( qy*i*KyM ) + qx2*sinh(αS/M/2)*sinh(αS/M/2)*( qy*i*KyM ) - 2*qx*sinh(αS/M/2)*cosh(αS/M/2)*( qz*EzM ) = coshαS/M*( qy*i*KyM ) - qx*sinhαS/M*( qz*EzM ) = qy*i*coshαS/M*KyM - qx*qz*sinhαS/M*EzM = qy*i*(coshαS/M*KyM - sinhαS/M*EzM)

113 CHAPTER 3 – FIELDS -qx*qz = -(jx/i)*(jz/i) = -jy = -qy*i qz*EzS = cosh(αS/M/2)*( qz*EzM )*cosh(αS/M/2) + -qx*sinh(αS/M/2)*( qz*EzM )*qx*sinh(αS/M/2) + cosh(αS/M/2)*( qy*i*KyM )*qx*sinh(αS/M/2) + -qx*sinh(αS/M/2)*( qy*i*KyM )*cosh(αS/M/2) = cosh(αS/M/2)*cosh(αS/M/2)*( qz*EzM ) + qx2*sinh(αS/M/2)*sinh(αS/M/2)*( qz*EzM ) + -2*qx*sinh(αS/M/2)*cosh(αS/M/2)*( qy*i*KyM ) = coshαS/M*( qz*EzM ) - qx*sinhαS/M*( qy*i*KyM ) = qz*coshαS/M*EzM - qx*qy*i*sinhαS/M*KyM = qz*(coshαS/M*EzM - sinhαS/M*KyM) -qx*qy*i = -( (jx/i)*(jy/i)*i ) = jx*jy*i = jz*i = -qz qz*i*KzS = cosh(αS/M/2)*( qz*i*KzM )*cosh(αS/M/2) + -qx*sinh(αS/M/2)*( qz*i*KzM )*qx*sinh(αS/M/2) + cosh(αS/M/2)*( qy*EyM )*qx*sinh(αS/M/2) + -qx*sinh(αS/M/2)*( qy*EyM )*cosh(αS/M/2) = cosh(αS/M/2)*cosh(αS/M/2)*( qz*i*KzM ) + qx2*sinh(αS/M/2)*sinh(αS/M/2)*( qz*i*KzM ) + -2*qx*sinh(αS/M/2)*cosh(αS/M/2)*( qy*EyM ) = coshαS/M*( qz*i*KzM ) - qx*sinhαS/M*( qy*EyM ) = qz*i*coshαS/M*KzM - qx*qy*sinhαS/M*EyM = qz*i*(coshαS/M*KzM + sinhαS/M*EyM) -qx*qy = -(jx/i)*(jy/i) = jz = qz*i ExS = ExM KxS = KxM EyS = EyM*coshαS/M + KzM*sinhαS/M KyS = -EzM*sinhαS/M + KyM*coshαS/M EzS = EzM*coshαS/M - KyM*sinhαS/M KzS = EyM*sinhαS/M + KzM*coshαS/M EyS + i*KyS

coshαS/M

-i*sinhαS/M

= EzS + i*KzS

EyM + i*KyM *

i*sinhαS/M

coshαS/M

EzM + i*KzM

114 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 4

*jsn

The equation to find “S” compound-label-numbers for “6E = *(4V)” uses the left and right denominators. pxS = pxM pyS = exp(*/2)*exp(qx*αS/M/2)*qy*exp(-qx*αS/M/2)*exp(-*/2) = exp(*/2)*qy*exp(qx*αS/M)*exp(-*/2) pzS = exp(*/2)*exp(qx*αS/M/2)*qz*exp(-qx*αS/M/2)*exp(-*/2) = exp(*/2)*qz*exp(qx*αS/M)*exp(-*/2) kxS = kxM kyS = exp(*/2)*jy*exp(qx*αS/M)*exp(-*/2) kzS = exp(*/2)*jz*exp(qx*αS/M)*exp(-*/2)

A check ensured “6E = 6E”: To check that the theory is correct, “M” Lorentz Transformation components and label-numbers for “4*jsn” and “4V” substitute into the “S” version of “-6E = 4*jsnx(4V)”, so that the subscripts are all “M” and not “S”. Components and compound-labelnumbers are grouped to “-6E” expressed in “M”. Also, the component transformations are confirmed by reference to Page 71 of Introduction to Modern Physics by Richtmyer, Kennard and Lauritsen, McGraw-Hill Book Company, Inc., 1955 and by reference to Page 210 of Methods of Theoretical Physics Part I by Morse and Feshbach, McGraw-Hill Book Company, Inc., 1953. The third check ensures the current density “4sn♦(-6E) = 4J”. (as calculated in “S” from the components in “M”) matches the current density (as calculated from “4sn” and “6E” in “S”). 1S*JtS + qxS*JxS + qyS*JyS + qzS*JzS = (1Ssn*tS + qxSsn*xS + qySsn*yS + qzSsn*zS) ♦(pxS*ExS + pyS*EyS + pzS*EzS + kxS*KxS + kyS*KyS + kzS*KzS)*(-1) 1S*JtS = -xS*qxSsn*pxS*ExS - yS*qySsn*pyS*EyS - zS*qzSsn*pzS*EzS qxS*JxS = -tS*1Ssn*pxS*ExS - yS*qySsn*kzS*KzS - zS*qzSsn*kyS*KyS qyS*JyS = -tS*1Ssn*pyS*EyS - zS*qzSsn*kxS*KxS - xS*qxSsn*kzS*KzS qzS*JzS = -tS*1Ssn*pzS*EzS - xS*qxSsn*kyS*KyS - yS*qySsn*kxS*KxS

115 CHAPTER 3 – FIELDS The compound-label-numbers multiply as 1S = -qxSsn*pxS = -qySsn*pyS = -qzSsn*pzS qxS = -1Ssn*pxS*(-1) = -qySsn*kzS = -qzSsn*kyS*(-1) qyS = -1Ssn*pyS*(-1) = -qzSsn*kxS = -qxSsn*kzS*(-1) qzS = -1Ssn*pzS*(-1) = -qxSsn*kyS = -qySsn*kxS*(-1) JtS = xS*ExS + yS*EyS + zS*EzS JxS = -tS*ExS + yS*KzS - zS*KyS JyS = -tS*EyS + zS*KxS - xS*KzS JzS = -tS*EzS + xS*KyS - yS*KxS To prove each of the four component equations is correct, “M” component expressions substitute for “S” component expressions. After manipulation of the equations, the previously stated component Lorentz Transformation equations for current density “4J” are found to apply. Time “t” component:

JtS = xS*ExS + yS*EyS + zS*EzS

(JtM*coshαS/M + JxM*sinhαS/M) = (-tM*sinhαS/M + xM*coshαS/M)*(ExS) + (yM)*(EyM*coshαS/M + KzM*sinhαS/M) + (zM)*(EzM*coshαS/M - KyM*sinhαS/M) JtM = xM*ExS + yM*EyM + zM*EzM JxM = -tM*ExS + yM*KzM - zM*KyM “x” component:

JxS = -tS*ExS + yS*KzS - zS*KyS

(JtM*sinhαS/M + JxM*coshαS/M) = -(tM*coshαS/M - xM*sinhαS/M)*(ExS) + (yM)*(EyM*sinhαS/M + KzM*coshαS/M) - (zM)*(-EzM*sinhαS/M + KyM*coshαS/M) JtM = xM*ExS + yM*EyM + zM*EzM JxM = -tM*ExS + yM*KzM - zM*KyM “y” component:

JyS = -tS*EyS + zS*KxS - xS*KzS

JyM = -(tM*coshαS/M - xM*sinhαS/M)*(EyM*coshαS/M + KzM*sinhαS/M) + zM*KxM - (-tM*sinhαS/M + xM*coshαS/M)*(EyM*sinhαS/M + KzM*coshαS/M)

116 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY JyM = (-tM*EyM - xM*KzM)*(cosh2αS/M - sinh2αS/M) + zM*KxM + (-tM*KzM - xM*EyM)*(2*sinhαS/M*coshαS/M - 2*sinhαS/M*coshαS/M) JyM = -tM*EyM - xM*KzM + zM*KxM “z” component:

JzS = -tS*EzS + xS*KyS - yS*KxS

JzM = -(tM*coshαS/M - xM*sinhαS/M)*(EzM*coshαS/M - KyM*sinhαS/M) yM*KxM +(-tM*sinhαS/M + xM*coshαS/M)*(-EzM*sinhαS/M +KyM*coshαS/M) JzM = (-tM*EzM + xM*KyM)*(cosh2αS/M - sinh2αS/M) - yM*KxM + (tM*KyM + xM*EzM)*(2*sinhαS/M*coshαS/M - 2*sinhαS/M*coshαS/M) JzM = -tM*EzM + xM*KyM - yM*KxM

3.5 Biot-Savart Law The Biot-Savart Law for the electromagnetic field of a moving particle is derived from the electric field of a stationary particle. 6E

= (q/(4**ǝ*rM2))*((xM/rM)*pxM + (yM/rM)*pyM + (zM/rM)*pzM)

ExM = (q/(4**ǝ*rM2))*(xM/rM) EyM = (q/(4**ǝ*rM2))*(yM/rM) EzM = (q/(4**ǝ*rM2))*(zM/rM)

; ; ;

KxM = 0 KyM = 0 KzM = 0

ExS = ExM = (q/(4**ǝ*rM3))*xM EyS + i*KyS

coshαS/M =

EzS + i*KzS

(q/(4**ǝ*rM2))*(yM/rM)

-i*sinhαS/M *

i*sinhαS/M

coshαS/M

(q/(4**ǝ*rM2))*(zM/rM)

EyS = (q/(4**ǝ*rM2))*(yM/rM)*(coshαS/M) KyS = -(q/(4**ǝ*rM2))*(zM/rM)*(sinhαS/M) EzS = (q/(4**ǝ*rM2))*(zM/rM)*(coshαS/M) KzS = (q/(4**ǝ*rM2))*(yM/rM)*(sinhαS/M)

117 CHAPTER 3 – FIELDS “rM2” must be independently Lorentz Transformed into “S” from “M”. In “M” there is no time “tM” difference between the electrically charged particle and the point (xM, yM, zM) where the electromagnetic field is being measured. locationM2 = xM2 + yM2 + zM2 - c2*tM2 = rM2 - c2*tM2 Equations in “S” have that same assumption of no time discrepancy, but in “S” not “M”. Those surrounding points in “S” are at different times with respect to “M”. xS2 + yS2 + zS2 - c2*tS2 = xM2 + yM2 + zM2 - c2*tM2 rS2 - c2*tS2 = rM2 - c2*tM2 rS2 = rM2 - c2*tM2 ; rM2 = rS2 + c2*tM2 c*tS

coshαS/M

sinhαS/M

sinhαS/M

coshαS/M

= xS c*tM

* coshαS/M

xM

-sinhαS/M

= xM

c*tM

c*tS *

-sinhαS/M

coshαS/M

xS

c*tM = c*tS*coshαS/M - xS*sinhαS/M c2*tM2 = c2*tS2*cosh2αS/M + xS2*sinh2αS/M - c2*tS*xS*sinh(2*αS/M) = xS2*sinh2αS/M rM2 = rS2 + c2*tM2 = rS2 + xS2*sinh2αS/M xM = xS*coshαS/M - c*tS*sinhαS/M = xS*coshαS/M yS = yM

;

zS = zM

ExS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*xS*coshαS/M ; KxS = 0 EyS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*yS*coshαS/M KyS = -(q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*zS*sinhαS/M EzS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*zS*coshαS/M KzS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*yS*sinhαS/M

118 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 6E

= pxS*(ExS + i*KxS) + pyS*(EyS + i*KyS) + pzS*(EzS + i*KzS)

(The author hasn’t seen the above component equations in a book or paper. That statement is common in this book, but with the expectation there is a book or paper not yet seen.) From Wikipedia (“Biot-Savart Law”) attributed to Oliver Heaviside in year 1888 (before the discovery of Special Relativity in 1905): ExSW = (q/(4**ǝ))*((1 - vS/M2/c2)/(1 - sin2*(vS/M2/c2))3/2)*(xS/rS3) KxSW = 0 EySW = (q/(4**ǝ))*((1 - vS/M2/c2)/(1 - sin2*(vS/M2/c2))3/2)*(yS/rS3) KySW = -(vS/M/c)*EzS EzSW = (q/(4**ǝ))*((1 - vS/M2/c2)/(1 - sin2*(vS/M2/c2))3/2)*(zS/rS3) KzSW = (vS/M/c)*EyS cos = xS/rS ; sin2 = (yS2 + zS2)/rS2 ; rS2 = xS2 + yS2 + zS2 The two sets of electromagnetic component expressions in “S” appear very different. Component values deviate with slightly higher values for Heaviside’s expressions (with the “W” subscript) for relativistic speeds and for “xS” locations far from the origin. For example, at “S/M = 6” and “xS = yS = 1” with “zS = 0” there is “ExS = EyS = 0.0000246” and “ExSW = EySW = 0.0000548”, for “q/(4**ǝ) = 1”. The deviation between “EyS” and “EySW” may possibly be attributed to a non-relativistic approximation in Heaviside’s derivation (for example “sinh2αS/M  vS/M2/c2”). That guess has not been confirmed. Translation from All-Number Algebra to Geometry. We could use measurements to resolve the discrepancy. The all-number algebra would be translated into geometry, per the third step. “ixS”, “iyS” and “izS” substitute for “pxS”, “pyS” and “pzS” and for “kxS”, “kyS” and “kzS”.

119 CHAPTER 3 – FIELDS

3.6 Electric Energy-Momentum of an Electron An energy-momentum invariant “4p” is calculated from an electromagnetic field invariant “6E” using empirically derived relationships. Bus “M” moves at speed “vS/M” relative to roadside “S”. “6E” components are used to find the energy-momentum invariant “4p” components. As a check, the energy-momentum invariant (“4p”) components in “S” found from “6E” must also be found from the energymomentum invariant (“4p”) components in “M”.

Figure 32. On the left: Two parallel capacitor plates. On the right: Two concentric capacitor plates.

Select the simplest geometry possible. •

For the field parallel with the motion, the two capacitor plates are perpendicular to the motion.

•

For the field perpendicular to the motion, the two capacitor plates are cylindrical concentric circular surfaces with the axis parallel to the direction of motion.

Electric charge may be placed onto the capacitor plates using a battery. Alternatively, the plates are close together and then electrically charged, and pulled apart: Energy equals force times distance. Bus mass converts to the electric field energy and back again. Energy and momentum (and therefore speed) remain constant during the mass to electric field energy conversion.

120 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Electromagnetic Field Perpendicular to Motion. EyM = A*y/(y2 + z2) = A*y/r2

;

EzM = A*z/(y2 + z2) = A*z/r2

The outer cylinder has an excess of negatively charged electrons because the electric field points outward from positive to negative. (The electric field is the negative of the gradient of the voltage, and, therefore, the inner cylinder has the higher voltage.) Energy density “tM” (sigma) was discovered empirically through the mathematical modeling of measured energy in experiments. tM = ǝ*(ExM*ExM + EyM*EyM + EzM*EzM + KxM*KxM + KyM*KyM + KzM*KzM)/2 Symbol “⊥” applies to fields perpendicular with motion. ⊥tM = ǝ*(EyM*EyM + EzM*EzM)/2 = ǝ*((A*y/r2)*(A*y/r2) + (A*z/r2)*(A*z/r2))/2 = ǝ*(A2*(y2 + z2))/((r2)2)/2 = ǝ*(A2/r2)/2 EyS + i*KyS

coshαS/M

-i*sinhαS/M

i*sinhαS/M

coshαS/M

= EzS + i*KzS

A*y/r2 *

EyS = cosh(αS/M)*A*y/r2 EzS = cosh(αS/M)*A*z/r2

; ;

A*z/r2

KyS = -sinh(αS/M)*A*z/r2 KzS = sinh(αS/M)*A*y/r2

⊥tS = ǝ*(EyS*EyS + EzS*EzS + KyS*KyS + KzS*KzS)/2 = ǝ*((y2 + z2)*(cosh2(αS/M) + sinh2(αS/M))*A2/((r2)2)/2 = ǝ*(cosh(2*αS/M)*A2/r2)/2 To complement energy density “⊥tS”, there is the energy per area per time Poynting Vector “⊥xS*ixS + ⊥yS*iyS + ⊥zS*izS”, empirically found equal to the cross product of the electric and magnetic fields, “ǝ*(3Ex3B)”. ⊥xS = ǝ*(EyS*KzS - EzS*KyS) ⊥yS = ǝ*(EzS*KxS - ExS*KzS) ⊥zS = ǝ*(ExS*KyS - EyS*KxS)

121 CHAPTER 3 – FIELDS ⊥xS = ǝ*(EyS*KzS - EzS*KyS) = ǝ*(y2 + z2)*cosh(αS/M)*sinh(αS/M)*A2/(r2)2 = ǝ*(sinh(2*αS/M)*A2/(y2 + z2))/2 = ǝ*(sinh(2*αS/M)*A2/r2)/2 Energy density “⊥tS” is positive because of squares. And, energy per area per time “⊥xS” is positive for a positive “αS/M”, so that energy moves in the direction of motion. Because they were empirically derived, “⊥tS” and “⊥xS” are combined without a compound-label-number. ⊥tS + qx*⊥xS = ǝ*((A2/r2)/2)*(cosh(2*αS/M) + qx*sinh(2*αS/M)) = ǝ*((A2/r2)/2)*exp(qx*2*αS/M) = ǝ*((A2/r2)/2)*exp2(qx*αS/M) exp(qx*2*αS/M) = cosh(2*αS/M) + qx*sinh(2*αS/M) = cosh2(αS/M) + sinh2(αS/M) + qx*2*sinh(αS/M)*cosh(αS/M) = cosh2(αS/M) + qx*qx*sinh2(αS/M) + qx*2*sinh(αS/M)*cosh(αS/M) = (cosh(αS/M) + qx*sinh(αS/M))2 = exp2(qx*αS/M) To remove the square operation in “exp2(qx*αS/M)”, “⊥tS + qx*⊥xS” is multiplied by the volume between the plates. The multiplication by volume must be an integral because of the radius “r” dependency in “⊥tS” and “⊥xS”. We continue the empirical analysis by using the spacenegative of the volume, and by making the volume invariant mathematically real rather than imaginary. The differential volume element “d(4Volsn)” consists of an “LxM” factor in the x-direction, a factor in the radial direction, and a factor in the tangential direction, “d(4Volsn) = 1S*(LxM)*(2**r)*dr*exp(-qx*αS/M)”. The integration was from “rinnerplate” to “routerplate”. d4p = (⊥tS + qx*⊥xS)*d(4Volsn)/c = ǝ*((A2/r2)/2)*exp2(qx*αS/M)*(1S*dVolB*exp(-qx*αS/M))/c = ǝ*((A2/r2)/2)*exp2(qx*αS/M)*(1S*(LxM)*(2**r)*dr*exp(-qx*αS/M))/c = 1S*(ǝ/c)*(A2/2)*exp(qx*αS/M)*LxM*(2*)*(dr/r) 4p

= 1S*(ǝ/c)*(A2/2)*LxM*(2*)*ln(routerplate/rinnerplate)*exp(qx*αS/M) = 1S*mB*c*exp(qx*αS/M)

122 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The above expression is the energy-momentum invariant for the electromagnetic field perpendicular to motion and formed from two cylindrical concentric circular capacitor plates. Rest mass “mB*c”: mB*c = (ǝ/c)*(A2/2)*LxM*(2*)*ln(routerplate/rinnerplate) We conclude, in general, any electromagnetic field perpendicular to the direction of motion may be replaced with an equivalent rest mass. An Electromagnetic Field Parallel to Motion. ExS = ExM

;

KxS = KxM

Positive “ExM” points right, in the direction of motion, because of a deficit of electrons on the left plate, and an excess of electrons on the right. (Left plate has the higher voltage.) Per “ExS = ExM”, a positive test charge “qTest” stationary with the roadside “S” experiences the same force “qTest*ExS” regardless of the speed “vS/M” of the bus. But note that energy must be applied to the bus to keep its speed constant because the test charge is repulsed by the left capacitor plate. If the test charge is stationary with respect to the bus, there is force “FxM = qTest*ExM” observed with both time and space components from the roadside because “xM = xS*cosh(αS/M) - c*tS*sinh(αS/M)”. To satisfy energy and momentum conservation, use the same basic form of the energy density equations as given for the case of the perpendicular electric field. (The below equations weren’t found in a book or paper.) ||tS = ǝ*(ExS*ExS + KxS*KxS)*cosh(2*αS/M)/2 = ǝ*(ExM*ExM + KxM*KxM)*cosh(2*αS/M)/2 ||xS = ǝ*(ExS*ExS + KxS*KxS)*sinh(2*αS/M)/2 = ǝ*(ExM*ExM + KxM*KxM)*sinh(2*αS/M)/2 “||tS” is positive because of squares and the “cosh” function. Energy per area per time “||xS” (that passes a location “xS”) is positive for positive “αS/M”, so that energy moves in the direction of motion.

123 CHAPTER 3 – FIELDS ||tS + qx*||xS = ǝ*(ExM2/2)*(cosh(2*αS/M) + qx*sinh(2*αS/M)) = ǝ*(ExM2/2)*exp(qx*2*αS/M) = ǝ*(ExM2/2)*exp2(qx*αS/M) VolM = LxM*LyM*LzM 4Vol

sn

= 1Ssn*VolM*exp(qx*αS/M) = 1S*VolM*exp(-qx*αS/M)

4p

= (||tS + qx*||xS)*4Volsn/c = ǝ*(ExM2/2)*exp2(qx*αS/M)*(1S*VolM*exp(-qx*αS/M))/c = 1S*ǝ*(ExM2/2)*(VolM)*exp(qx*αS/M)/c

4p

= 1S*mB*c*exp(qx*αS/M)

;

mB*c = (ǝ/c)*(ExM2/2)*(VolM)

Electric Field of a Stationary Electron. The radius of an electron is calculated from rest mass and electric charge using a macroscopic model. Assume negative electric charge is evenly distributed on the surface of a spherical electron. Electric field lines of force extend radially inward to the surface. Electric field magnitude decreases inversely with the square of the distance from the center. ExM = A*(xM/rM3) A = q/(4**ǝ)

;

EyM = A*(yM/rM3) ;

;

EzM = A*(zM/rM3)

rM2 = xM2 + yM2 + zM2

tM = ǝ*(ExM*ExM + EyM*EyM + EzM*EzM + KxM*KxM + KyM*KyM + KzM*KzM)/2 = ǝ*(ExM*ExM + EyM*EyM + EzM*EzM)/2 = ǝ*(A2*(xM2 + yM2 + zM2)/rM6)/2 = ǝ*(A2/rM4)/2 Energy density “tM = ǝ*(A2/rM4)/2” is integrated from an infinite radius inward (following the electric field lines) to the assumed classical radius “re” of the electron. “dVol = 4**rM2*drM”. d(EB/c) = (ǝ*(A2/rM4)/2)*dVol/c = ((ǝ/c)*(A2/rM4)/2)*4**rM2*drM = (1/2)*((ǝ/c)*(q/(4**ǝ))2/rM4)*4**rM2*drM = (1/2)*(q2/(4**ǝ*c))*(drM/rM2)

124 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The integrand “(1/2)*(q2/(4**ǝ*c))*(drM/rM2)” is integrated to the below rest energy (divided by the speed-of-light) of the electron. mB*c = (1/2)*(q2/(4**ǝ*c))*(1/re) Substitute in measured rest mass “mB = 9.11*10^-31 kg” and measured electric charge “q = -1.60*10^-19 C” to calculate the classical radius “re”. re = (1/2)*(q2/(4**ǝ*c))*(1/(mB*c)) = 1.409… *10-15 meters If the electron had a radius less than “re”, then more electric field would cause more rest mass than what is measured. The classical radius macroscopic model of the electron fails when extended to the magnetic field caused by rotation, to suggest the classical radius is not a physically real surface. Mu and tau particles differ from the electron by having more rest mass and by being unstable. Rather than calculate a smaller classical radius for these particles, the extra mass can be attributed to whatever field causes the instability. The whatever field behaves like mass inertia, as does the electromagnetic field, and so probably creates an induced field when in motion. A similar statement applies to protons, etc. •

Inside the classical radius is the electric charge. If we count electrons, then we are counting a quantity of electric charge.

•

Outside the classical radius is the electric field. The inertia of the energy-momentum invariant is, per this model, due to electromagnetic field induction.

Energy-Momentum Density for the Biot-Savart Law. A = q/(4**ǝ) ExS = (A/(rS2 + xS2*sinh2αS/M)3/2)*xS*coshαS/M KxS = 0 EyS = (A/(rS2 + xS2*sinh2αS/M)3/2)*yS*coshαS/M KyS = -(A/(rS2 + xS2*sinh2αS/M)3/2)*zS*sinhαS/M EzS = (A/(rS2 + xS2*sinh2αS/M)3/2)*zS*coshαS/M KzS = (A/(rS2 + xS2*sinh2αS/M)3/2)*yS*sinhαS/M

125 CHAPTER 3 – FIELDS ⊥tS = ǝ*(EyS*EyS + EzS*EzS + KyS*KyS + KzS*KzS)/2 = ǝ*A2*(yS2 + zS2)*(cosh2αS/M + sinh2αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 = ǝ*A2*(yS2 + zS2)*cosh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 ⊥xS = ǝ*(EyS*KzS - EzS*KyS) = ǝ*A2*(yS2 + zS2)*(coshαS/M*sinhαS/M)*(1/(rS2 + xS2*sinh2αS/M)3) = ǝ*A2*(yS2 + zS2)*sinh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 ||tS = ǝ*(ExS*ExS + KxS*KxS)*cosh(2*αS/M)/2 = ǝ*A2*(xS2)*cosh2αS/M*cosh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 ||xS = ǝ*(ExS*ExS + KxS*KxS)*sinh(2*αS/M)/2 = ǝ*A2*(xS2)*cosh2αS/M*sinh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 tS = ⊥tS + ||tS = ǝ*A2*(xS2*cosh2αS/M + yS2 + zS2)*cosh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 xS = ⊥xS + ||xS = ǝ*A2*(xS2*cosh2αS/M + yS2 + zS2)*sinh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 tS + qx*xS = ǝ*A2*(xS2*cosh2αS/M + yS2 + zS2)*exp(2*qx*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 = ǝ*A2*(xS2*cosh2αS/M + yS2 + zS2)*exp2(qx*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 = ǝ*A2*(xM2 + yM2 + zM2)*exp2(qx*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 = ǝ*A2*rM2*exp2(qx*αS/M)*(1/(rS2 + (c*tM)2)3)/2 = ǝ*A2*rM2*exp2(qx*αS/M)*(1/(rM2)3)/2 = ǝ*((A2/rM4)/2)*exp2(qx*αS/M) Hyperbolic-radius “ǝ*(A2/rM4)/2” is the same as the energy density of the stationary electron, “tM = ǝ*(A2/rM4)/2”.

126 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

3.7 Maxwell’s Wave Equation Maxwell’s Wave Equation is given in three forms below. 4

2

*(-6E) = 0

;

(4*jsn•4sn)*(-6E) = 0 (tM2 - (xM2 + yM2 + zM2))*(-6E) = 0

The “42” operator is called the “harmonic” operator because it applies to functions that are sine waves. (tM2 - xM2)*sin(kx*(x - c*t)) = 0 (sin(kx*(x - c*t)))/x = kx*cos(kx*(x - c*t)) (kx*cos(kx*(x - c*t)))/x = kx2*sin(kx*(x - c*t)) (sin(kx*(x - c*t)))/ct = kx*cos(kx*(x - c*t)) (kx*cos(kx*(x - c*t)))/ct = kx2*sin(kx*(x - c*t)) An electromagnetic field “6E” that satisfies Maxwell’s Wave Equation “42*(-6E) = 0” must be a summation of sine waves. These include cosine waves (because ninety degrees can be added into the argument) and any motion at the speed of light (because its wave form can be a summation of sine waves per a Fourier transform). Maxwell’s Wave Equation is derived by taking the gradient of both sides of Maxwell’s Equations “4sn*(-6E) = 4J”. 4

*4sn*(-6E) = 4*jsn*(4J)

*jsn

Apply the identity “(4*jsnx4sn)*(-6E)  0” to the left side and apply the electric charge conservation equation “4*jsn•(4J) = 0” to the right side. 4

*4sn*(-6E) = (4*jsn•4sn)*(-6E)

*jsn

;

4

*(4J) = 4*jsnx(4J)

*jsn

Set both sides equal to zero. Typically, “4J = 0”. (4*jsn•4sn)*(-6E) = 0

;

4

*jsn

x(4J) = 0

127 CHAPTER 3 – FIELDS Spiral Waves. Each spiral wave is the sum of two perpendicular waves. First Spiral Wave (-6Efirst) = Eamp*(kyM + pzM)*exp(i*(kxM*(xM + c*tM))) -(EyM + i*KyM)first = i*Eamp*exp(i*(kxM*(xM + c*tM))) -(EzM + i*KzM)first = Eamp*exp(i*(kxM*(xM + c*tM))) EyMfirst = Eamp*sin(kxM*(xM + c*tM)) KyMfirst = -Eamp*cos(kxM*(xM + c*tM)) EzMfirst = -Eamp*cos(kxM*(xM + c*tM)) KzMfirst = -Eamp*sin(kxM*(xM + c*tM)) Second Spiral Wave (-6Esecond) = Eamp*(kyM - pzM)*exp(i*(kxM*(xM - c*tM))) -(EyM + i*KyM)second = i*Eamp*exp(i*(kxM*(xM - c*tM))) -(EzM + i*KzM)second = -Eamp*exp(i*(kxM*(xM - c*tM))) EyMsecond = Eamp*sin(kxM*(xM - c*tM)) KyMsecond = -Eamp*cos(kxM*(xM - c*tM)) EzMsecond = Eamp*cos(kxM*(xM - c*tM)) KzMsecond = Eamp*sin(kxM*(xM - c*tM)) Third Spiral Wave (-6Ethird) = Eamp*(kyM + pzM)*exp(-i*(kxM*(xM + c*tM))) -(EyM + i*KyM)third = i*Eamp*exp(-i*(kxM*(xM + c*tM))) -(EzM + i*KzM)third = Eamp*exp(-i*(kxM*(xM + c*tM))) EyMthird = -Eamp*sin(kxM*(xM + c*tM)) KyMthird = -Eamp*cos(kxM*(xM + c*tM)) EzMthird = -Eamp*cos(kxM*(xM + c*tM)) KzMthird = Eamp*sin(kxM*(xM + c*tM)) Fourth Spiral Wave (-6Efourth) = Eamp*(kyM - pzM)*exp(-i*(kxM*(xM - c*tM))) -(EyM + i*KyM)fourth = i*Eamp*exp(-i*(kxM*(xM - c*tM))) -(EzM + i*KzM)fourth = -Eamp*exp(-i*(kxM*(xM - c*tM))) EyMfourth = -Eamp*sin(kxM*(xM - c*tM)) KyMfourth = -Eamp*cos(kxM*(xM - c*tM)) EzMfourth = Eamp*cos(kxM*(xM - c*tM)) KzMfourth = -Eamp*sin(kxM*(xM - c*tM)) To check that the Fourth Spiral Wave is valid, first check it against Maxwell’s Wave Equation by taking derivatives:

128 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2(-6Efourth)/(ctM)2 = -kxM2*(-6Efourth) 2(-6Efourth)/(xM)2 = -kxM2*(-6Efourth) 2(-6Efourth)/(ctM)2 - 2(-6Efourth)/(xM)2 = 0 The above check only confirmed the Fourth Spiral Wave equation was constructed of sine waves that move at the speed-of-light. A second check confirms the electric and magnetic fields induce each other, to ensure there is not a misplaced negative: “1sn*(-6E) + 3snx(-6E) = 0” 1

sn

3

sn

*(-6Efourth) = (1Msn)*(-6Efourth)/(ctM) = (1M)*(kyM - pzM)*Eamp*(exp(-i*(kxM*(xM - c*tM))))/(ctM) = (i*kxM)*Eamp*(jyM - qzM)*exp(-i*(kxM*(xM - c*tM))) x(-6Efourth) = (qxMsn)*(-6Efourth)/(xM) = Eamp*(qxMsn*kyM - qxMsn*pzM)*(exp(-i*(kxM*(xM - c*tM))))/(xM) = (-i*kxM)*Eamp*(-qzM + jyM)*exp(-i*(kxM*(xM - c*tM))) qxMsn*kyM = -qxM*kyM = -qzM -qxMsn*pzM = qxM*kzM/i = -qzM*kxM/i = -qyM*(-i) = jyM Alternatively, as a quick check on the math, the Poynting Vector gives the direction of energy travel and must point in the direction of motion. The Poynting Vector is the cross product of the electric field by the magnetic field. For example, for the First Spiral Wave there are “EyMfirst = Eamp*sin(kxM*(xM + c*tM))” and “KzMfirst = -Eamp*sin(kxM*(xM + c*tM))” as two components of a plane wave that induce each other. For a slightly positive “xM” at “tM = 0” there is “EyMfirst > 0” and “KzMfirst < 0”. The righthand fingers pass through positive “yM” and then through negative “zM” and the thumb points in negative “xM”, which is correct for the First Spiral Wave. Therefore, the induction is correctly modeled mathematically. Right-Handed and Left-Handed Spiral Waves. Point the thumb in the direction of wave travel (“+x” for “xM - c*tM” and “-x” for “xM + c*tM”). The maximum of the helix of the wave spiral is similar to the threads on a screw. Curl the fingers to ride the maximum as the thumb moves in the direction of wave travel.

129 CHAPTER 3 – FIELDS The coordinate system is a right-hand coordinate system with “x” the front table edge going right, with “y” the left table edge going away, and with “z” up. The example is the Fourth Spiral Wave. 3Efourth

= Eamp*(pzM*cos(kxM*(xM - c*tM)) - pyM*sin(kxM*(xM - c*tM)))

Set “tM = 0”. The fingers pass positive “Ez”, then at “kxM*xM = /2”, negative “Ey” using a right hand. The Fourth Spiral Wave is right-handed. The Third Spiral Wave is left hand because the fingers pass negative “Ez” at “xM = 0”, and then positive “Ey” at “kxM*xM = -/2”. 3Ethird

= Eamp*(-pzM*cos(kxM*(xM + c*tM)) - pyM*sin(kxM*(xM + c*tM))) Second Spiral Wave is left-handed.

3Esecond

= Eamp*(pzM*cos(kxM*(xM - c*tM)) + pyM*sin(kxM*(xM - c*tM)))

First Spiral Wave is right-handed. 3Efirst

= Eamp*(-pzM*cos(kxM*(xM + c*tM)) + pyM*sin(kxM*(xM + c*tM)))

A (non-spinning) source disintegrates into two photons with the same handedness, like a nut that separates from a bolt that depart in opposite directions with the same handedness of spin. Right-Handed Photon Pair (first and fourth): 6E+fourth-first

= -Eamp*(kyM - pzM)*exp(-i*(kxM*(xM - c*tM)))

Left-Handed Photon Pair (third and second): 6E+second-third

= -Eamp*(kyM - pzM)*exp(i*(kxM*(xM - c*tM)))

Quantum Effect on Spin. The four spiral waves each represent an idealized hypothetical photon’s wave. They are idealized because direction is known, as is frequency, and the spin axis is known to be parallel or else anti-parallel to the direction of motion. Actual photon waves, per quantum effects, have the spin in the direction of motion only

130 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY if the measurement of the axis of the spin is in the direction of motion, and when the spin axis is not being measured, it includes a component perpendicular to the direction of motion, just like an electron. Matter and Anti-Matter for Electromagnetic Waves. Because a photon has zero length due to the extreme of length contraction, its back side (taillights) is confounded with its front side (headlights), so there is no distinguishing between matter and anti-matter. Regardless, they are paired with direction of motion: First with third and second with fourth. Lorentz Transformation of a Spiral Wave. The example below has wave fronts that move in the positive “x” direction because of the “xM - c*tM”. (-6Efourth) = Eamp*(kyM - pzM)*exp(-i*(kxM*(xM - c*tM))) -(EyM + i*KyM) = i*Eamp*exp(-i*(kxM*(xM - c*tM))) -(EzM + i*KzM) = -Eamp*exp(-i*(kxM*(xM - c*tM))) -(EyS + i*KyS)

coshαS/M

-i*sinhαS/M

= -(EzS + i*KzS)

-(EyM + i*KyM) *

i*sinhαS/M

coshαS/M

-(EzM + i*KzM)

-(EyS + i*KyS) = Eamp*exp(-i*(kxM*(xM - c*tM)))*(i*coshαS/M - -i*sinhαS/M) = i*Eamp*exp(-i*(kxM*(xM - c*tM)))*exp(αS/M) = i*Eamp*exp(-i*(kxS*(xS - c*tS)))*exp(αS/M) -(EzS + i*KzS) = Eamp*exp(-i*(kxM*(xM - c*tM)))*(i*i*sinhαS/M - coshαS/M) = -Eamp*exp(-i*(kxM*(xM - c*tM)))*exp(αS/M) = -Eamp*exp(-i*(kxS*(xS - c*tS)))*exp(αS/M) Observer “S” sees amplitude increased by factor “exp(αS/M)” compared to “M”, accompanied by an increase in frequency “kxS > kxM”. “kxS*(xS - c*tS)” equals “kxM*(xM - c*tM)” because it is an invariant.

131 CHAPTER 3 – FIELDS Geometric-Vector Notation. Induction equations “3x3E = -(3B)/t” and “3x3B = ((3E)/t)/c2” are satisfied for the fourth wave. 3E

= ExM*ixM + EyM*iyM + EzM*izM ; 3B = BxM*ixM + ByM*iyM + BzM*izM = -Eamp*sin(kxM*(xM - c*tM))*iyM = -(Eamp/c)*cos(kxM*(xM - c*tM))*iyM + Eamp*cos(kxM*(xM - c*tM))*izM + -(Eamp/c)*sin(kxM*(xM - c*tM))*izM 3x3E

= (/xM*ixM)x(Eamp*cos(kxM*(xM - c*tM))*izM) + (/xM*ixM)x(-Eamp*sin(kxM*(xM - c*tM))*iyM) = -kxM*Eamp*sin(kxM*(xM - c*tM))*(-iyM) + -kxM*Eamp*cos(kxM*(xM - c*tM))*(+izM) = kxM*Eamp*sin(kxM*(xM - c*tM))*(iyM) + -kxM*Eamp*cos(kxM*(xM - c*tM))*(izM)

(3B)/t = (/tM)(-(Eamp/c)*cos(kxM*(xM - c*tM))*iyM) + (/tM)(-(Eamp/c)*sin(kxM*(xM - c*tM))*izM) = -kxM*-Eamp*-sin(kxM*(xM - c*tM))*(iyM) + -kxM*-Eamp*cos(kxM*(xM - c*tM))*(izM) = -kxM*Eamp*sin(kxM*(xM - c*tM))*(iyM) + kxM*Eamp*cos(kxM*(xM - c*tM))*(izM) 3x3B

= (/xM*ixM)x((-Eamp/c)*sin(kxM*(xM - c*tM))*izM) + (/xM*ixM)x((-Eamp/c)*cos(kxM*(xM - c*tM))*iyM) = kxM*(-Eamp/c)*cos(kxM*(xM - c*tM))*(-iyM) + kxM*(-Eamp/c)*-sin(kxM*(xM - c*tM))*(+izM) = kxM*(Eamp/c)*cos(kxM*(xM - c*tM))*(iyM) + kxM*(Eamp/c)*sin(kxM*(xM - c*tM))*(izM)

((3E)/t)/c2 = (/tM)((-Eamp)*sin(kxM*(xM - c*tM))*iyM)/c2 + (/tM)((Eamp)*cos(kxM*(xM - c*tM))*izM)/c2 = -kxM*c*-Eamp*cos(kxM*(xM - c*tM))*(iyM)/c2 + -kxM*c*Eamp*-sin(kxM*(xM - c*tM))*(izM)/c2 = kxM*(Eamp/c)*cos(kxM*(xM - c*tM))*(iyM) + kxM*(Eamp/c)*sin(kxM*(xM - c*tM))*(izM)

132 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

3.8 Forces Using Geometric-Vector Notation Evidence the electromagnetic field “6E” and electric charge density “4J” are physically real comes from measurements of force. The force causes a reaction force or else accelerates the test electric charge “q”.

Figure 33a. “Ez” is negative because “Ez = -Vt/z < 0”. “q > 0” imposes a force “Fz = q*Ez < 0” on whatever is holding it stationary. Figure 33b. Magnetic field “Bx < 0” surrounds electric charge “q > 0” that moves at a speed “vy > 0” to create force “Fz = -q*vy*Bx > 0”, which accelerates the particle so that it travels in a circle. More mass “m” means less curvature.

Imaginary Force. There is no measurable force due to a stationary electric charge in a magnetic field, or a moving charge in an electric field. But, “q*3B” and “(q*3v)x3E” do not simply equal zero. 3Fr

= q*3E + (q*3v)x3B

Frx = q*Ex + q*vy*Bz - q*vz*By Fry = q*Ey + q*vz*Bx - q*vx*Bz Frz = q*Ez + q*vx*By - q*vy*Bx 3F

= 3Fr  i*3Fi

; 3Fi = q*c*3B - (q*3v/c)x3E ; Fix = q*c*Bx - ((q*vy/c)*Ez - (q*vz/c)*Ey) ; Fiy = q*c*By - ((q*vz/c)*Ex - (q*vx/c)*Ez) ; Fiz = q*c*Bz - ((q*vx/c)*Ey - (q*vy/c)*Ex)

133 CHAPTER 3 – FIELDS Energy Rate. A current of positive charge flows from a high voltage “+” battery terminal to the “-” at a lower value of “x”, “q*v x < 0”. “Ex < 0” because “Ex = -Vt/x”. Energy transfers out from the electrical system into heat when both “vx” and “Ex” are the same sign (both are negative) because “Ftr = q*(3v/c)•3E > 0”. An imaginary rate of energy change “Fti” that cannot be measured occurs when current flows parallel to a magnetic field. Ft = Ftr  i*Fti = q*(3v/c)•3E  i*q*3v•3B

Figure 34. Energy is lost in a resistor so that “Ftr = q*(3v/c)•3E > 0”.

3.9 Force Density Invariant “3F” and “Ft” are to be expressed using all-number algebra. First guess: 4g

= -(4J)*(-6E) = -(4J)■(-6E) - (4J)♦(-6E) = -((3J)•(-3K) + (3J)x(-3E) + (1J)*(-3K)) - ((3J)•(-3E) + (3J)x(-3K) + (1J)*(-3E)) Two terms of “4g” are expanded out. 1M*gtM = (JxM*KxM*qxM*kxM + JyM*KyM*qyM*kyM + JzM*KzM*qzM*kzM) + (JxM*ExM*qxM*pxM + JyM*EyM*qyM*pyM + JzM*EzM*qzM*pzM) = 1M*((JxM*KxM + JyM*KyM + JzM*KzM)*i + (JxM*ExM + JyM*EyM + JzM*EzM)) qxM*kxM = jxM*kxM*(-i) = (-1M)*(-i) = 1M*i qxM*pxM = jxM*kxM*(-1) = (-1M)*(-1) = 1M

134 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY qxM*gxM = ((JyM*EzM*qyM*pzM + JzM*EyM*qzM*pyM) + JtM*KxM*1M*kxM) + ((JyM*KzM*qyM*kzM + JzM*KyM*qzM*kyM) + JtM*ExM*1M*pxM) = qxM*(-(JyM*EzM - JzM*EyM) + JtM*KxM))*i + ((JyM*KzM - JzM*KyM) + JtM*ExM)) qyM*pzM = -jyM*kzM = -jxM = qxM*(-i) qyM*kzM = -jyM*kzM*i = -jxM*i = qxM “4g” has the correct arrangement of plus and minus signs. gtr = Jx*Ex + Jy*Ey + Jz*Ez ; gti = Jx*Kx + Jy*Ky + Jz*Kz ;

Frt = q*(vx*Ex + vy*By + vz*Ez)/c Fit = q*(vx*Bx + vy*Ey + vz*Bz)

gxr = (Jy*Kz - Jz*Ky) + Jt*Ex ; Frx = (q*vy*Bz - q*vz*By) + q*Ex gxi = -(Jy*Ez - Jz*Ey) + Jt*Kx ; Fix = -(q*(vy/c)*Ez - q*(vz/c)*Ey) + q*c*Bx Applying Maxwell’s Equations. If the field from test charge “4J” were included in the externally applied electromagnetic field “6E”, then the force would be there, regardless. Substitute “4sn*(-6E) = (4J)” into “4g”. 4g

= -(4sn*(-6E))*(-6E) = -(4sn*6E)*6E = -((4sn*6E)♦6E + (4sn*6E)■6E) = -((4sn♦6E)♦6E + (4sn■6E)♦6E + (4sn♦6E)■6E + (4sn■6E)■6E)

= -((3sn•3E)*3E + (1sn*3E + 3snx3K)x3K + (1sn*3E + 3snx3K)•3E + (3sn•3K)*3E + (1sn*3K + 3snx3E)x3K + (1sn*3K + 3snx3E)•3E + (3sn•3E)*3K + (1sn*3E + 3snx3K)x3E + (1sn*3E + 3snx3K)•3K + (3sn•3K)*3K + (1sn*3K + 3snx3E)x3E + (1sn*3K + 3snx3E)•3K) The first and fourth rows have mathematically real components. Second and Third have imaginary components. The first row sums to the real portion of “4g”. The third row sums to the imaginary portion of “4g”. The second and fourth rows each sum to zero but individual terms in the second and fourth rows are not zero, for example, “(1sn*3K)•3K  0”.

135 CHAPTER 3 – FIELDS Time component terms of the first row are given below. Gradient operator “” only applies to the component factor immediately behind it, and not to both field component factors, to simplify what is written. Portion of (1M*gtrM) = (1sn*3E + 3snx3K)•3E = -(tM*ExM*ExM*1Msn*pxM*pxM + tM*EyM*EyM*1Msn*pyM*pyM + tM*EzM*EzM*1Msn*pzM*pzM) + -(yM*KzM*ExM*qyMsn*kzM*pxM + zM*KyM*ExM*qzMsn*kyM*pxM + zM*KxM*EyM*qzMsn*kxM*pyM + xM*KzM*EyM*qxMsn*kzM*pyM + xM*KyM*EzM*qxMsn*kyM*pzM + yM*KxM*EzM*qyMsn*kxM*pzM) = -1M*((tM*ExM*ExM + tM*EyM*EyM + tM*EzM*EzM) + (-yM*KzM*ExM + zM*KyM*ExM) + (-zM*KxM*EyM + xM*KzM*EyM) + (-xM*KyM*EzM + yM*KxM*EzM)) qyMsn*kzM*pxM = jyM*kzM*kxM = jxM*kxM = -1M A Negative is Needed. “gtrM”, like “Frt”, models the loss of energy from the system, per the example of the battery and resistor, and this is properly modelled by the “-(1sn*3E)•3E” term in the first row with the example “-(tM*ExM)*ExM = -tM*(ExM*ExM)/2”. A positive value is a loss of energy, and the term is positive because energy “ExM*ExM” is positive, the gradient is negative because it is a loss of energy, and the “-1” makes “gtM” positive. The math is as expected. Good. “-(1sn*3K)•3K” of the fourth row has “-(tM*KxM)*KxM*i2 = tM*(KxM*KxM)/2”. The “i2” was introduced by the label-numbers because “3K” is mathematically imaginary if “3E” is mathematically real. The extra negative is bad because “3K•3K = -K2” is subtracted from “3E•3E = +E2” in “3E•3E + 3K•3K = E2 - K2”, in contradiction to experiments for which the total energy is “E2 + K2”. To fix this, a negative is needed for “3K”. To help specify how to insert the negative, second row term “3E•3K” plus third row term “3K•3E” should subtract to cancel. And, first row “3Ex3K” needs to subtract from its negative “3Kx3E” of the fourth row for conformance to the experimentally derived Poynting Vector. The conclusion is that the fourth row needs a negative relative to the first row, and that negative is acceptable because the fourth row sums to zero. The second row also sums to zero and it needs a negative

136 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY relative to the third row. Also, because the imaginary force of the third row is not measurable, the second and third rows can be either plus or minus relative to the first row. Place “1” with the first “■” and “-1” with the second “■”. 4f

= -((4sn♦6E)♦6E  (4sn■6E)♦6E  -(4sn♦6E)■6E - (4sn■6E)■6E)

The components of “4f” separate into models of physics with two examples being electromagnetic field energy and the Poynting vector. Note that “4fr = 4gr”. Also, “4f = 4fr  i*4fi”. Mathematically Ugly. Because of the inserted negatives, “4f” cannot be expressed using “*” operator gradients, and that means “4f” is ugly. By analogy, the dot product in the exponential function, “4k*j•4r”, is ugly because the cross-product “4k*jx4r” is not added to it. This ugliness is explained by equating “4k*j•4r” to the product of hyperbolic radii. The ugliness of “4f”, too, goes away. “4f” becomes beautiful using the proposed algebra in this book’s last chapter. Terms of the Force Density Invariant. -4f = ((3sn•3E)*3E + (1sn*3E + 3snx3K)x3K + (1sn*3E + 3snx3K)•3E)  ( (3sn•3K)*3E + (1sn*3K + 3snx3E)x3K + (1sn*3K + 3snx3E)•3E )  -( (3sn•3E)*3K + (1sn*3E + 3snx3K)x3E + (1sn*3E + 3snx3K)•3K ) - ( (3sn•3K)*3K + (1sn*3K + 3snx3E)x3E + (1sn*3K + 3snx3E)•3K ) = ( (-1J)*3E + (-3J)x3K + (-3J)•3E )  ( (10)*3E + (30)x3K + (30)•3E )  -( (-1J)*3K + (-3J)x3E + (-3J)•3K ) - ( (10)*3K + (30)x3E + (30)•3K ) -1f = (-3J)•3E  (30)•3E  -(-3J)•3K - (30)•3K = ( (1sn*3E + 3snx3K)•3E )  ( (1sn*3K + 3snx3E)•3E )  -( (1sn*3E + 3snx3K)•3K ) - ( (1sn*3K + 3snx3E)•3K )

137 CHAPTER 3 – FIELDS -ftr = -(Jx*Ex + Jy*Ey + Jz*Ez) = ( (Ex/ct)*Ex + (Ey/ct)*Ey + (Ez/ct)*Ez ) - ( -(Kx/ct)*Kx - (Ky/ct)*Ky - (Kz/ct)*Kz ) + ( (-Kz/y + Ky/z)*Ex + (-Kx/z + Kz/x)*Ey + (-Ky/x + Kx/y)*Ez )

- ( (Ey/z - Ez/y)*Kx + (Ez/x - Ex/z)*Ky + (Ex/y - Ey/x)*Kz )

y*Kz*Ex*(-qyM)*kzM*pxM = y*Kz*Ex*(-qyM)*pyM = -y*Kz*Ex*1M -z*Ey*Kx*(-qzM)*pyM*kxM = -z*Ey*Kx*(-qzM)*(-pzM) = -z*Ey*Kx*1M t*Ex*Ex*(1M)*pxM*pxM = t*Ex*Ex*(1M)*(1) = t*Ex*Ex*1M -t*Kx*Kx*(1M)*kxM*kxM = -t*Kx*Kx*(1M)*(-1) = t*Kx*Kx*1M -Jx*Ex*qxM*pxM = -Jx*Ex*1M -fti =  (Jx*Kx + Jy*Ky + Jz*Kz) =  ( (Kx/ct)*Ex + (Ky/ct)*Ey + (Kz/ct)*Ez )  ( -(Ex/ct)*Kx - (Ey/ct)*Ky - (Ez/ct)*Kz )  ( (Kz/y - Ky/z)*Kx + (Kx/z - Kz/x)*Ky + (Ky/x - Kx/y)*Kz )  -( (Ey/z - Ez/y)*Ex + (Ez/x - Ex/z)*Ey + (Ex/y - Ey/x)*Ez ) -y*Kz*Kx*(-qyM)*kzM*kxM = y*Kz*Kx*(qyM)*(kyM) = i*y*Kz*Kx*1M z*Ey*Ex*(-qzM)*pyM*pxM = -z*Ey*Ex*(qzM)*(kzM) = -i*z*Ey*Ex*1M t*Kx*Ex*(1M)*kxM*pxM = t*Kx*Ex*(1M)*(i) = i*t*Kx*Ex*1M -t*Ex*Kx*(1M)*pxM*kxM = -t*Kx*Kx*(1M)*(i) = -i*t*Ex*Kx*1M  Jx*Kx*qxM*kxM = i*Jx*Kx*1M -3f = ( (-1J)*3E + (-3J)x3K )  ( (0)*3E + (30)x3K )  -( (-1J)*3K + (-3J)x3E ) - ( (0)*3K + (30)x3E ) = ( (3sn•3E)*3E + (1sn*3E + 3snx3K)x3K )  ( (3sn•3K)*3E + (1sn*3K + 3snx3E)x3K )  -( (3sn•3E)*3K + (1sn*3E + 3snx3K)x3E ) - ( (3sn•3K)*3K + (1sn*3K + 3snx3E)x3E ) -fzr = -Jt*Ez + (Jy*Kx - Jx*Ky) = ( (Ky/ct)*Ex - (Kx/ct)*Ey ) + ( (Ex/ct)*Ky - (Ey/ct)*Kx ) + ( (-Kz/y + Ky/z)*Ky ) - ( (Kz/x - Kx/z)*Kx ) + ( (-Ez/y + Ey/z)*Ey ) - ( (Ez/x - Ex/z)*Ex ) + ( -(Ex/x + Ey/y + Ez/z)*Ez ) - ( (Kx/x + Ky/y + Kz/z)*Kz )

138 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY -t*Ky*Ex*(1M)*kyM*pxM = -t*Ky*Ex*(1M)*(-pzM) = t*Ky*Ex*qzM t*Ex*Ky*(1M)*pxM*kyM = t*Ex*Ky*(1M)*(pzM) = t*Ex*Ky*qzM -y*Ez*Ey*(-qyM)*pzM*pyM = -y*Ez*Ey*(-qyM)*kxM = -y*Ez*Ey*qzM y*Kz*Ky*(-qyM)*kzM*kyM = y*Kz*Ky*(-qyM)*(-kxM) = -y*Kz*Ky*qzM x*Ex*Ez*(-qxM)*pxM*pzM = x*Ex*Ez*(-qxM)*kyM = -x*Ex*Ez*qzM -x*Kx*Kz*(-qxM)*kxM*kzM = -x*Kx*Kz*(-qxM)*(-kyM) = -x*Kx*Kz*qzM -Jx*Ky*qxM*kyM = -Jx*Ky*qzM -Jt*Ez*1M*pzM = -Jt*Ez*qzM -fzi = Jt*Kz  (Jy*Ex - Jx*Ey) = ( -(Ky/ct)*Kx + (Kx/ct)*Ky )  ( (Ex/ct)*Ey - (Ey/ct)*Ex )  (( (-Kz/y + Ky/z)*Ey ) - ( (Kz/x - Kx/z)*Ex ))  (( (Ez/y - Ey/z)*Ky ) + ( (Ez/x - Ex/z)*Kx ))  ( (Ex/x + Ey/y + Ez/z)*Kz )  ( -(Kx/x + Ky/y + Kz/z)*Ez ) t*Ky*Kx*(1M)*kyM*kxM = t*Ky*Kx*(1M)*(-kzM) = -t*Ky*Kx*jzM -t*Ex*Ey*(1M)*pxM*pyM = -t*Ex*Ey*(1M)*(-kzM) = t*Ex*Ey*jzM y*Ez*Ky*(-qyM)*pzM*kyM = y*Ez*Ky*qyM*pxM = y*Ez*Ky*jzM -y*Kz*Ey*(-qyM)*kzM*pyM = -y*Kz*Ey*qyM*pxM = -y*Kz*Ey*jzM -x*Ex*Kz*(-qxM)*pxM*kzM = -x*Ex*Kz*(-1M)*kzM = x*Ex*Kz*jzM x*Kx*Ez*(-qxM)*kxM*pzM = x*Kx*Ez*(-i*1M)*(pzM) = -x*Kx*Ez*jzM Jx*Ey*qxM*pyM = -Jx*Ey*jzM Jt*Kz*1M*kzM = Jt*Kz*jzM As a check on the compound-label-number products, the component force equations have as factors Maxwell’s Equations. Component Maxwell’s Equations: -Kx/ct - Ez/y + Ey/z = 0 -Ky/ct - Ex/z + Ez/x = 0 -Kz/ct - Ey/x + Ex/y = 0 Kx/x + Ky/y + Kz/z = 0

; ; ; ;

-Ex/ct + Kz/y - Ky/z = Jx -Ey/ct + Kx/z - Kz/x = Jy -Ez/ct + Ky/x - Kx/y = Jz Ex/x + Ey/y + Ez/z = Jt

In the component equation for “-fzr” are six of Maxwell’s Equations. • • •

“-Jt*Ez” equates to “-(Ex/x + Ey/y + Ez/z)*Ez” “Jy*Kx” equates to “-(Ey/ct)*Kx - (Kz/x - Kx/z)*Kx” “-Jx*Ky” equates to “(Ex/ct)*Ky + (-Kz/y + Ky/z)*Ky”

139 CHAPTER 3 – FIELDS • • •

“0” equates to “-(Kx/x + Ky/y + Kz/z)*Kz” “0” equates to “(Ky/ct)*Ex - (Ez/x - Ex/z)*Ex” “0” equates to “-(Kx/ct)*Ey + (-Ez/y + Ey/z)*Ey”

In the component equation for “-fzi” are these other six Equations that also conform to Maxwell’s Equations. • • • • • •

“Jt*Kz” equates to “(Ex/x + Ey/y + Ez/z)*Kz” “Jy*Ex” equates to “-(Ey/ct)*Ex  -(Kz/x - Kx/z)*Ex” “ -Jx*Ey” equates to “(Ex/ct)*Ey  (-Kz/y + Ky/z)*Ey” “0” equates to “-(Kx/x + Ky/y + Kz/z)*Ez” “0” equates to “-(Ky/ct)*Kx  (Ez/x - Ex/z)*Kx” “0” equates to “(Kx/ct)*Ky  (Ez/y - Ey/z)*Ky”

Example Use of First Case Force Density. The gradient components of the negative of the Third Spiral Wave Solution are given below. Third Spiral Wave (-6Ethird) = Eamp*(kyM + pzM)*exp(-i*(kxM*(xM + c*tM))) -(EyM + i*KyM)third = i*Eamp*exp(-i*(kxM*(xM + c*tM))) -(EzM + i*KzM)third = Eamp*exp(-i*(kxM*(xM + c*tM))) EyMthird = -Eamp*sin(kxM*(xM + c*tM)) KyMthird = -Eamp*cos(kxM*(xM + c*tM)) EzMthird = -Eamp*cos(kxM*(xM + c*tM)) KzMthird = Eamp*sin(kxM*(xM + c*tM)) 3E

= Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) K = Eamplitude*(kyM*cos(kxM*(xM + c*tM)) - kzM*sin(kxM*(xM + c*tM))) 3 E = 6 3E + 3K 1

sn

3

sn

*3E = 1M*/ct*3E = Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM)) - qzM*sin(kxM*(xM + c*tM))) Using: 1M*pyM = qyM 1M*pzM = qzM x3E = (-qxM*/x - qyM*/y - qzM*/z)x3E = Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM))) Using: -qxM*pyM = jxM*kyM = jzM -qxM*pzM = jxM*kzM = -jyM

140 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 3

sn

•3E = (-qxM*/x - qyM*/y - qzM*/z)•3E = 0 1

sn

3

sn

3

sn

*3K = 1M*/ct*3K = Eamplitude*kxM*(-jyM*sin(kxM*(xM + c*tM)) - jzM*cos(kxM*(xM + c*tM))) Using: 1M*kyM = jyM 1M*-kzM = -jzM x3K = (-qxM*/x - qyM*/y - qzM*/z)x3K = Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM)) - qyM*cos(kxM*(xM + c*tM))) Using: -qxM*kyM = -jxM*kyM/i = -qzM -qxM*-kzM = jxM*kzM/i = -qyM •3K = (-qxM*/x - qyM*/y - qzM*/z)•3K = 0 (1sn*3E)•3E = (Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM)) - qzM*sin(kxM*(xM + c*tM)))) • (Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(1M - 1M)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: qyM*pyM = 1M -qzM*pzM = -1M (1sn*3E)x3E = (Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM)) - qzM*sin(kxM*(xM + c*tM)))) x(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(-jxM)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(-jxM) Using: qyM*pzM = -jyM*kzM = -jxM -qzM*pyM = -jyM*kzM = -jxM (3snx3E)•3E = (Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM)))) •(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*i*1M*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*i*1M Using: jzM*pzM = i*1M jyM*pyM = i*1M

141 CHAPTER 3 – FIELDS (3snx3E)x3E = (Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM)))) x(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*(qxM - qxM)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: jzM*pyM = -jyM*kzM/i = -qxM jyM*pzM = jyM*kzM/i = qxM (1sn*3E)•3K = (Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM)) - qzM*sin(kxM*(xM + c*tM)))) •(Eamplitude*(kyM*cos(kxM*(xM + c*tM)) - kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(i*1M)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(i*1M) Using: qyM*kyM = i*jyM*-kyM = i*1M -qzM*-kzM = i*1M (1sn*3E)x3K = (Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM)) - qzM*sin(kxM*(xM + c*tM)))) x(Eamplitude*(kyM*cos(kxM*(xM + c*tM)) - kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(pxM - pxM)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: qyM*-kzM = -pxM -qzM*kyM = pxM (3snx3E)•3K = (Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM)))) •(Eamplitude*(kyM*cos(kxM*(xM + c*tM)) - kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(1M - 1M)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: jzM*-kzM = 1M jyM*kyM = -1M (3snx3E)x3K = (Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM)))) x(Eamplitude*(kyM*cos(kxM*(xM + c*tM)) - kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*-jxM*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*-jxM Using: jzM*kyM = -jyM*kzM = -jxM jyM*-kzM = -jxM (1sn*3K)•3E = (Eamplitude*kxM*(-jyM*sin(kxM*(xM + c*tM)) - jzM*cos(kxM*(xM + c*tM)))) •(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*(-i*1M)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(-i*1M) Using: -jyM*pyM = -i*1M -jzM*pzM = -i*1M

142 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY (1sn*3K)x3E = (Eamplitude*kxM*(-jyM*sin(kxM*(xM + c*tM)) - jzM*cos(kxM*(xM + c*tM)))) x(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*(qxM - qxM)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: -jyM*pzM = -qxM -jzM*pyM = qxM (3snx3K)•3E = (Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM)) - qyM*cos(kxM*(xM + c*tM)))) •(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*(1M - 1M)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: qzM*pzM = 1M -qyM*pyM = -1M (3snx3K)x3E = (Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM)) - qyM*cos(kxM*(xM + c*tM)))) x(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*jxM*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*jxM Using: qzM*pyM = jyM*kzM = jxM -qyM*pzM = jyM*kzM = jxM (1sn*3K)•3K = (Eamplitude*kxM*(-jyM*sin(kxM*(xM + c*tM)) + -jzM*cos(kxM*(xM + c*tM)))) •(Eamplitude*(kyM*cos(kxM*(xM + c*tM)) + -kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(1M - 1M)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: -jyM*kyM = 1M -jzM*-kzM = -1M (1sn*3K)x3K = (Eamplitude*kxM*(-jyM*sin(kxM*(xM + c*tM)) - jzM*cos(kxM*(xM + c*tM)))) x(Eamplitude*(kyM*cos(kxM*(xM + c*tM)) - kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(jxM)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(jxM) Using: -jyM*-kzM = jxM -jzM*kyM = jxM (3snx3K)•3K = (Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM)) - qyM*cos(kxM*(xM + c*tM)))) •((Eamplitude*(kyM*cos(kxM*(xM + c*tM)) - kzM*sin(kxM*(xM + c*tM))) = Eamplitude2*kxM*(-i*1M)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(-i*1M) Using: qzM*-kzM = -i*1M -qyM*kyM = -i*1M

143 CHAPTER 3 – FIELDS (3snx3K)x3K = (Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM)) - qyM*cos(kxM*(xM + c*tM)))) x(Eamplitude*(kyM*cos(kxM*(xM + c*tM)) - kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(qxM - qxM)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: qzM*kyM = -qyM*kzM = -qxM -qyM*-kzM = qxM -4f = -1f + -3f = ( (1sn*3E + 3snx3K)•3E ) W ( (1sn*3K + 3snx3E)•3E ) W -( (1sn*3E + 3snx3K)•3K ) - ( (1sn*3K + 3snx3E)•3K ) + (( (3sn•3E)*3E + (1sn*3E + 3snx3K)x3K ) W ( (3sn•3K)*3E + (1sn*3K + 3snx3E)x3K ) W -( (3sn•3E)*3K + (1sn*3E + 3snx3K)x3E ) - ( (3sn•3K)*3K + (1sn*3K + 3snx3E)x3E )) = (1sn*3E)•3E + (3snx3K)•3E W (1sn*3K)•3E W (3snx3E)•3E W -(1sn*3E)•3K W -(3snx3K)•3K - (1sn*3K)•3K - (3snx3E)•3K + (3sn•3E)*3E + (1sn*3E)x3K + (3snx3K)x3K W (3sn•3K)*3E W (1sn*3K)x3K W (3snx3E)x3K W -(3sn•3E)*3K W -(1sn*3E)x3E W -(3snx3K)x3E - (3sn•3K)*3K - (1sn*3K)x3E - (3snx3E)x3E = 0 + 0 W Eamplitude2*kxM*-i*1M W Eamplitude2*kxM*i*1M W -Eamplitude2*kxM*i*1M W -Eamplitude2*kxM*-i*1M - 0 - 0 + 0 + 0 + 0 W 0 W Eamplitude2*kxM*jxM W Eamplitude2*kxM*-jxM W -0 W -Eamplitude2*kxM*-jxM W -Eamplitude2*kxM*jxM - 0 - 0 - 0 = 0 “4f = 0” because “4J = 0”.

3.10 Area and Volume Differential Operators • • • • •

Per nothing Gradient (per length) differential operator “4sn” (del) Per area differential operator “6sn” (theta) Per volume differential operator “4sn” (xi) Per world volume differential operator “1” (omega)

144 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The count of terms matches a row of Pascal’s Triangle: 1, 4, 6, 4, 1. The first “1” is for a One-Component Invariant. It is Lorentz Transformed by Lorentz Transforming the components it depends on. Electric current density “4J” uses the per volume differential operator. Finding Wavenumber. “4sn” can be made equivalent to a wavenumber/frequency invariant “4k” with consideration as to what the function “1T” is. The first example is “1T = A*sin(kxM*xM - M*tM)”.

=

sn sn 4 *(A*sin(kxM*xM - M*tM)) = A*(4 *sin(kxM*xM - M*tM)) sn sn sn sn A*(1M */ct + qxM */x + qyM */y+qzM */z)*sin(kxM*xM - M*tM) = A*(1Msn*/ctM + qxMsn*/xM)*sin(kxM*xM - M*tM) = A*(1Msn*(-M/c) + qxMsn*(kxM))*cos(kxM*xM - M*tM)

= A*(1M*(-M/c) + qxM*(-kxM))*cos(kxM*xM - M*tM) = A*(-4k)*cos(kxM*xM - M*tM)

“1T = A*exp(i*(kxM*xM - M*tM))” avoids the change from sine to cosine. 4

sn

*(A*exp(i*(kxM*xM - M*tM))) = A*(4sn*exp(i*(kxM*xM - M*tM))) = A*(1Msn*/ctM + qxMsn*/xM)*exp(i*(kxM*xM - M*tM)) = A*(1Msn*(-i*M/c) + qxMsn*(i*kxM))*exp(i*(kxM*xM - M*tM)) = A*i*(1M*(-M/c) + qxM*(-kxM))*exp(i*(kxM*xM - M*tM)) = i*(-4k)*A*exp(i*(kxM*xM - M*tM))

“4sn = -i*4k” (which has terms “1Msn*/ctM = 1M*(-i*M/c)” and “qxMsn*/xM = qxM*(-i*kxM)”) is an abbreviation of “4sn*(1T) = -i*4k*(1T) and is exclusive to “1T = A*exp(i*(kxM*xM - M*tM))”. “i*4sn = 4k” is used in the next chapter on waves. A Linear Scalar Field’s Gradient. A count (zero, one, two, three, four, and higher) is equally spaced along the “x” axis. The count is an invariant scalar field represented by the below expression. The compound-labelnumber in “1Count” is the number “1”. 1Count

= b*xM

145 CHAPTER 3 – FIELDS Visualize “1Count” as a long rod on which numbers are written: Zero at the back of the bus and increasing forward (“b > 0”).

=

sn sn 4 *(1Count) = 4 *(b*xM) sn sn (1M */ctM + qxM */xM + qyMsn*/yM = (qxMsn*/xM)*(b*xM) = qxMsn*b

+ qzMsn*/zM)*(b*xM)

“4sn*(1Count) = qxMsn*b” states “1Count” varies by gradient “b” in the “qxMsn” direction (that is, positive “ixM” direction). “1Count” is Lorentz Transformed from “1Count = b*xM” to “1Count = b*(xS*cosh(αS/M) - c*tS*sinh(αS/M))”, after which a gradient “4sn” is taken. 4

*(b*xM) = 4sn*(b*(xS*cosh(αS/M) - c*tS*sinh(αS/M))) = b*(-1Msn*/ctM*c*tS*sinh(αS/M) + qxMsn*/xM*xS*cosh(αS/M)) = b*(-1Msn*sinh(αS/M) + qxMsn*cosh(αS/M))

sn

Alternatively, “4sn*(1Count) = qxMsn*b” is Lorentz Transformed to the result “4sn*(1Count) = 1Ssn*b*sinh(αS/M) + qxSsn*b*cosh(αS/M)”, per the matrix equation below. “/ctM*(1Count) = 0”. /ctS*(1Count)

/ctM*(1Count)

cosh(αS/M) -sinh(αS/M) =

/xS*(1Count)

* -sinh(αS/M)

/xM*(1Count)

cosh(αS/M)

cosh(αS/M) -sinh(αS/M) =

0 *

-sinh(αS/M)

cosh(αS/M)

-b*sinh(αS/M) =

b

b*cosh(αS/M)

Either method has the same result. 4

sn

*(1Count) = 1Ssn*b*(-sinh(αS/M)) + qxSsn*b*cosh(αS/M)

Time component “-b*sinh(αS/M)” of “4sn*(1Count)” is negative because “1Count” decreases at rate “-b*sinh(αS/M)” when observed from one location “xS” (for “αS/M > 0”). If we are standing on the roadside “S” looking at the numbers, then the numbers decrease.

146 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Rather than care about the numbers, we can care about the spacing of the numbers and the rate the numbers pass us. The spacing of the numbers, called the wavenumber, is a positive value (if we select it to be positive). The rate at which the numbers pass us, called the frequency, is a positive value (per that selection). We continue to use the gradient operator, but we must transition away from the gradient operator invariant with its space-negative to the wavenumber-frequency invariant that has no space-negative (because now “qxM” translates to “ixM”). Remove the space-negative in “4sn*(1Count) = qxMsn*b” by replacing “qxMsn” with “-qxM” so that “4sn*(1Count) = -qxM*b = -4k”. 4

sn

*(1Count) = -4k

;

4k

= qxM*kxM = qxM*b = 1S*b*sinh(αS/M) + qxS*b*cosh(αS/M) = 1S*S/c + qxS*kxS

Per-Area Differential Operator “6sn” has six terms. sn 6

= pxMsn*(2/(xM*ctM) + i*(2/(yM*zM))) + pyMsn*(2/(yM*ctM) + i*(2/(zM*xM))) + pzMsn*(2/(zM*ctM) + i*(2/(xM*yM))) = pxMsn*(xrM + i*xiM) + pyMsn*(yrM + i*yiM) + pzMsn*(zrM + i*ziM)

On the floor inside bus “M” is a rectangular array of particles. On each particle is a sequential number in the “x” direction and a sequential number in the “y” direction. The first particle counted is in the rear of the bus on the right side, “x = 0” and “y = 0”. The count of particles is “1Count = (bx*xM)*(by*yM)”. (“bx > 0”, “by > 0”) “6sn” operates on “1Count = (bx*xM)*(by*yM)” to quantify the change in count relative to area. sn 6 *(1Count)

= pxMsn*(2/(xM*ctM) + i*(2/(yM*zM)))*1Count + pyMsn*(2/(yM*ctM) + i*(2/(zM*xM)))*1Count + pzMsn*(2/(zM*ctM) + i*(2/(xM*yM)))*1Count

147 CHAPTER 3 – FIELDS

= pxMsn*(2/(xM*ctM) + i*(2/(yM*zM)))*(bx*xM)*(by*yM) + pyMsn*(2/(yM*ctM) + i*(2/(zM*xM)))*(bx*xM)*(by*yM) + pzMsn*(2/(zM*ctM) + i*(2/(xM*yM)))*(bx*xM)*(by*yM) = pzMsn*(i*(2/(xM*yM)))*(bx*xM)*(by*yM) = pzMsn*i*bx*by “6sn*(1Count) = pzMsn*i*bx*by” states the change in count with respect to area equals “bx*by”, as observed by a person seated on the bus “M”. The “i” factor in “pzMsn*i” means the “pzMsn*i = kzMsn” label number translates to the “x / y” plane and not to the “z” direction. The result “6sn*(1Count)” is Lorentz Transformed. (yrS + i*yiS)*(1Count) cosh(αS/M) i*sinh(αS/M) (yrM + i*yiM)*(1Count) = * (zrS + i*ziS)*(1Count) -i*sinh(αS/M) cosh(αS/M) (zrM + i*ziM)*(1Count) The right-side column vector has “(yrM + i*yiM)*(1Count) = 0” and “(zrM + i*ziM)*(1Count) = i*bx*by”. The matrix operator is a space negative. The terms of the left side column vector, “(yrS + i*yiS)*(1Count) = (i*bx*by)*(i*sinh(αS/M))” and “(zrS + i*ziS)*(1Count) = (i*bx*by)*(cosh(αS/M))”, are in the invariant expression below. sn 6 *(1Count)

= pySsn*(yrS+i*yiS)*(1Count) + pzSsn*(zrS+i*ziS)*(1Count) = pySsn*(-bx*by*sinh(αS/M)) + pzSsn*(i*bx*by*cosh(αS/M))

“pzSsn*(i*bx*by*cosh(αS/M))” of “6sn*(1Count)” states there are more particles for the same amount of x*y area if we use “xS” rather than “xM” for that x*y area, by a factor of “cosh(αS/M)”. The term “pySsn*(-bx*by*sinh(αS/M))” of “6sn*(1Count)” is a measure of per-length and per-time, because it is real and is not imaginary. The per-length is in the “y” direction, perpendicular to the direction of motion. The per-time is associated with the negative. The count as written on each of the particles decreases relative to a person standing on the roadside “S”.

148 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The complementary invariant to the per-area differential operator “6 ” is the wave-number-area invariant “6kk”. There is no spacenegative on “6kk”. sn

6kk

= pzM*i*kkziM = pzM*i*bx*by = pyS*bx*by*sinh(αS/M) + pzS*i*bx*by*cosh(αS/M) = pyS*kkyrS + pzS*i*kkziS

The “kkziS” component is the spacing of particle rows in the “x” direction, and “kkziS” includes a factor “by” for the density along the particle rows. The “kkyrS” component is the frequency of particle rows that pass a person standing on the roadside “S” (and “kkyrS” includes a factor “by” for the density of particles along the rows). The Lorentz Transformation used the below matrix equation. The matrix operator below is not a space-negative, as identified by the different location of the negative on the off-diagonal terms. (kkyrS + i*kkyiS)

cosh(αS/M) -i*sinh(αS/M) =

(kkzrS + i*kkziS)

(kkyrM + i*kkyiM) *

i*sinh(αS/M)

cosh(αS/M)

(kkzrM + i*kkziM)

The above matrix operator was also used for the Lorentz Transformation for the electromagnetic field and area invariants. EyS + i*KyS

coshαS/M

-i*sinhαS/M

= EzS + i*KzS ByS + i*AyS

i*sinhαS/M

coshαS/M

coshαS/M

-i*sinhαS/M

= BzS + i*AzS

EyM + i*KyM * EzM + i*KzM ByM + i*AyM *

i*sinhαS/M

coshαS/M

BzM + i*AzM

Per-Volume Differential Operator “4sn” (xi) has four terms. sn 4

= 1Msn*(3/(xM*yM*zM)) + qxMsn*(3/(ctM*yM*zM)) + qyMsn*(3/(ctM*xM*zM)) + qzMsn*(3/(ctM*xM*yM)) = 1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM

149 CHAPTER 3 – FIELDS A long rectangular prism array of particles sits on the floor of bus “M”. On each are three numbers: “x” (to the front of the bus), “y” (from right to left), and “z” (up). At the rear, right side and on the floor is the particle labelled “x = 0”, “y = 0” and “z = 0”. The count of particles is given by “1Count = (bx*xM)*(by*yM)*(bz*zM)”. (“bx > 0”, “by > 0”, and “bz > 0”) Per-volume differential operator “4sn” operates on particle count scalar “1Count = (bx*xM)*(by*yM)*(bz*zM)” to quantify the change in count relative to volume. sn 4 *(1Count)

= 1Msn*(3/(xM*yM*zM))*(1Count) + qxMsn*(3/(ctM*yM*zM))*(1Count) + qyMsn*(3/(ctM*xM*zM))*(1Count) + qzMsn*(3/(ctM*xM*yM))*(1Count)

= 1Msn*(3/(xM*yM*zM))*(bx*xM)*(by*yM)*(bz*zM) + qxMsn*(3/(ctM*yM*zM))*(bx*xM)*(by*yM)*(bz*zM) + qyMsn*(3/(ctM*xM*zM))*(bx*xM)*(by*yM)*(bz*zM) + qzMsn*(3/(ctM*xM*yM))*(bx*xM)*(by*yM)*(bz*zM) = 1Msn*(3/(xM*yM*zM))*(bx*xM)*(by*yM)*(bz*zM) = 1Msn*bx*by*bz “4sn*(1Count) = 1Msn*bx*by*bz” states the change in count with respect to volume equals “bx*by*bz”, as observed by a person seated on bus “M”. “bx*by*bz” is the density of counts: Counts per Volume or counted particles per volume. “4sn*(1Count)” is Lorentz Transformed. tS*(1Count) xS*(1Count)

tM*(1Count)

cosh(αS/M) -sinh(αS/M) =

* -sinh(αS/M)

cosh(αS/M)

xM*(1Count)

The right-side column vector has “tM*(1Count) = bx*by*bz” and “xM*(1Count) = 0”. The matrix operator is a space negative. The terms of the left side column vector, “tS*(1Count) = bx*by*bz*cosh(αS/M)” and “xS*(1Count) = -bx*by*bz*sinh(αS/M)”, are in the invariant expression below.

150 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY sn 4 *(1Count)

= 1Ssn*(tS*(1Count)) + qxSsn*(xS*(1Count)) = 1Ssn*bx*by*bz*cosh(αS/M) + qxSsn*(-bx*by*bz*sinh(αS/M))

“1Ssn*bx*by*bz*cosh(αS/M)” of “4sn*(1Count)” states there are more particles for the same amount of x*y*z volume if we use “xS” rather than “xM” for that x*y*z volume, by a factor “cosh(αS/M)”. “qxSsn*(-bx*by*bz*sinh(αS/M))” of “4sn*(1Count)” is a measure of perarea and per-time. The per-area is in the “y*z” direction, perpendicular to the direction of motion. The per-time is associated with the negative because the x-direction count as written on each of the particles decreases relative to a person standing on the roadside “S”. The complementary invariant is the current-density invariant “4” (rho). There is no space-negative on “4”. 4

= 1M*tM = 1M*bx*by*bz = 1S*bx*by*bz*cosh(αS/M) + qxS*bx*by*bz*sinh(αS/M) = 1S*tS + qxS*xS

“tS” is density. “xS” is flow per area in the “x” direction due to the movement of the bus “M” relative to the roadside “S”. There was no space-negative in the Lorentz Transformation matrix operator. 4

= 1S*bx*by*bz*cosh(αS/M) + qxS*bx*by*bz*sinh(αS/M)

compared to sn 4 *(1Count)

= 1Ssn*bx*by*bz*cosh(αS/M) + qxSsn*(-bx*by*bz*sinh(αS/M))

shows “4 = 4sn*(1Count)”. It applies to a homogeneous material. If “bx = by = bz”, then it is a solid homogeneous block. Electric Current Density Invariant. “Q” is the electric charge per electron particle, “1Charge = Q*1Count”. “4J = Q*4” is found using the per-volume operator “4sn” (for a homogeneous material): 4J

= 4sn*(1Charge)

151 CHAPTER 3 – FIELDS Per-World-Volume Differential Operator “1” (omega) has one term. 1

= 1*(4/(ctM*xM*yM*zM))

Inside a long rectangular prism on bus “M” is an array of lights that flash. Four numbers are printed onto each light. The first number is a count of flashes the light has had. The second number is a sequential number in the “x” direction, third number in the “y” direction, fourth “z”. The initial light flash in the rear of the bus, on the right side and on the floor corresponds to “c*t = x = y = z = 0”. (“bt, bx, by, bz > 0”) “1Count = (bt*c*tM)*(bx*xM)*(by*yM)*(bz*zM)”. “1” operates on “1Count = (bt*c*tM)*(bx*xM)*(by*yM)*(bz*zM)” to quantify the change in count relative to world-volume. 1*(1Count) 4

= 1*(4/(ctM*xM*yM*zM))*(1Count) = 1*( /(c*tM*xM*yM*zM))*((bt*c*tM)*(bx*xM)*(by*yM)*(bz*zM)) = bt*bx*by*bz

“1*(1Count) = bt*bx*by*bz” is the density of counts per worldvolume. The same result “1*(1Count)” is observed by a person standing on the roadside “S” because length contraction balances time dilation as “cosh(αS/M)/cosh(αS/M) = 1”. With respect to the roadside, the lights do not all flash at the same time, but, rather, appear to move forward as a pulse faster than the speed-of-light. Per-world-volume density “1” is not complementary to, but rather, is equal to “1*(1Count) = bt*bx*by*bz”.

3.11 Exercises Text Comprehension Exercises. 1) Prove “-6E = 4*jsnx(4V)”. Use “4*j” and “4V” expressed in “S”. Substitute in the “M” expressions for the compound-labelnumbers and components of “4*j” and “4V”. Reduce the expression to the result of “6E” expressed in “M”. 2) Prove the following identities.

152 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY a. b. c. d. e. f. g. h. i. j. k. l. m. n.

4

1

*(3*jsnx(3V)) + 3snx(1*jsn*(3V))  0 sn *jsn *(1V))  0 3 x(3 sn *jsn  •(  x( 3 3 3V))  0 sn *jsn x(4V))  0 4 ■(4 *jsn •(1sn*(1*jsn*(3V))) + 1*jsn*(3sn•(1*jsn*(3V)))  0 3 *jsn •(1sn*(3*jsn*(1V))) + 1*jsn*(3sn•(3*jsn*(1V)))  0 3 *jsn •(3snx(3*jsnx(3V)))  0 3 *jsn •(4sn♦(4*jsnx(4V)))  0 4 j (4* snx4sn)*(-6E)  0 *jsn x(3sn*2P)  0 3 *jsn *(3sn*2P) + 3*jsn*(1sn*2P)  0 1 *jsn x(4sn*2P)  0 4 *jsn x(3snx6Q)  (3*jsn•3sn)*6Q - 3*jsn*(3sn•6Q) 3 sn

•(4sn*(-6Q)) = 4*jsn•(4sn■(-6Q)) + 4*jsn•(4sn♦(-6Q)) = 1*jsn*(3sn•(-3Qi)) + 3*jsn•(1sn♦(-3Qi) + 3snx(-3Qr)) + 1*jsn*(3sn•(-3Qr)) + 3*jsn•(1sn♦(-3Qr) + 3snx(-3Qi))  0 *jsn

3) Confirm “1sn*(-6Ethird) + 3snx(-6Ethird) = 0”. 4) For “2P = a*(c*t)2 + b*x2”, find “4sn*2P”. Find “4*jsn•(4sn*2P)”, find “(4*jsn•4sn)*2P”, and identify the relationship between “a” and “b” for “(4*jsn•4sn)*2P = 0”. For any values of “a” and “b” show that “4*jsnx(4sn*2P) = 0”. 5) Confirm “6E = Eamp*(-jyM + qzM)*cos(kxM*(xM - c*tM))” complies with Maxwell’s Equations. Find the simplest representation of “4V” for “6E”. What is the effect on “6E” and on the Lorenz Condition of adding a “4Vconstants” (that has components that are each a constant relative to time and to space) to “4V”? Confirm that the specific scalar gauge function “2P = Pmax*exp(i*n*kM*(xM - c*tM))” satisfies the criteria “(4*jsn•4sn)*2P = 0”. Find “4sn*2P”. 6) Show the “-ftr” and “-fti” component equations conform to Maxwell’s Equations

153 CHAPTER 3 – FIELDS Select Exercises Solutions. 1) Not Given 2) Not Given 3)

1

sn

3

sn

*(-6Ethird) = (1Msn)*(-6Ethird)/(ctM) = (1M)*(kyM + pzM)*Eamp*(exp(-i*(kxM*(xM + c*tM))))/(ctM) = (-i*kxM)*Eamp*(jyM + qzM)*exp(-i*(kxM*(xM + c*tM)))

x(-6Ethird) = (qxMsn)*(-6Ethird)/(xM) = Eamp*(qxMsn*kyM + qxMsn*pzM)*(exp(-i*(kxM*(xM + c*tM))))/(xM) = (-i*kxM)*Eamp*(-qzM - jyM)*exp(-i*(kxM*(xM + c*tM)))

qxMsn*kyM = -qxM*kyM = -qzM qxMsn*pzM = -qxM*kzM/i = qzM*kxM/i = qyM*(-i) = -jyM

4)

1

sn

*(-6Ethird) + 3snx(-6Ethird) = 0 OK

2P

= a*(c*t)2 + b*x2

4

sn

4

*jsn

*2P = (1M*t + qxM*x + qxM*x + qxM*x)*(a*(c*t)2 + b*x2) = 1M*2*a*c*t + qxM*2*a*x •(4sn*2P) = (t*1M*jsn + x*qxM*jsn + y*qyM*jsn + z*qzM*jsn) •(1M*2*a*c*t + qxM*2*b*x) = (t*(2*a*c*t))*(1M*jsn*1M) + (x*2*b*x)*(qxM*jsn*qxM) = 2*a + 2*b

(4*jsn•4sn)*2P = (t2 - x2 - y2 - z2)*(a*(c*t)2 + b*x2) = 2*a + 2*b a = -b 4

x(4sn*2P) = (t*1M*jsn + x*qxM*jsn + y*qyM*jsn + z*qzM*jsn) x(1M*2*a*c*t + qxM*2*b*x)

*jsn

= (t*(2*b*x))*(1M*jsn*qxM) + (x*2*a*c*t)*(qxM*jsn*1M) = 0*pxM + 0*pxM = 0

154 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 5)

6E

= Eamp*(-jyM + qzM)*cos(kxM*(xM - c*tM))

KyM = -Eamp*cos(kxM*(xM - c*tM)) EzM = Eamp*cos(kxM*(xM - c*tM)) -tM*KyM + xM*EzM = -((-1)*(-kxM) + kxM)*Eamp*-sin(kxM*(xM - c*tM)) = 0 OK First attempt at “4V”: VxM = -z*KyM = z*Eamp*cos(kxM*(xM - c*tM)) VtM = -y*EzM = -y*Eamp*cos(kxM*(xM - c*tM)) 4V = 1M*VtM + qxM*VxM = 1M*VtM + qxM*VxM EzM = -zM*VtM = Eamp*cos(kxM*(xM - c*tM)) OK -KyM = -zM*VxM = Eamp*cos(kxM*(xM - c*tM)) OK *jsn •4V = tM*VtM + xM*VxM 4 = ((-y)*(-kxM) + z*kxM)*Eamp*-sin(kxM*(xM - c*tM)) ≠ 0 not OK Second attempt at “4V”: VzM = KyM/kxM = (1/kxM)*-Eamp*cos(kxM*(xM - c*tM)) VyM = EzM/kxM = (1/kxM)*Eamp*cos(kxM*(xM - c*tM)) 4V = qyM*VyM + qzM*VzM EzM = -tM*VzM = -(-kxM)*(1/kxM)*Eamp*cos(kxM*(xM - c*tM)) OK -KyM = xM*VzM = (kxM)*(1/kxM)*Eamp*cos(kxM*(xM - c*tM)) OK *jsn •4Vsn = yM*VyM + zM*VzM = 0 + 0 = 0 OK 4 4Vconstants

= 1M*Konstt + qxM*Konstx + qyM*Konsty + qzM*Konstz

“4*jsnx4Vconstants = 0” and “4*jsnx4V = 0”, therefore there is no effect on “6E” or on the Lorenz Condition if “4V + 4Vconstants” is substituted for “4V”. 2P

= Pmax*exp(i*n*kM*(xM - c*tM))

(4*jsn•4sn)*2P = (tM2 - xM2 - yM2 - zM2)*2P = (-i*n*kM)2*2P - (i*n*kM)2*2P = 0 OK

155 CHAPTER 3 – FIELDS 4

sn

4

*jsn

4

*jsn

*2P = (1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM)*2P = (1M*(-i*n*kM) - qxM*(i*n*kM))*2P = (1M + qxM)*(-i*n*kM)*2P

•(4sn*2P) = (tM*1M*jsn + xM*qxM*jsn + yM*qyM*jsn + zM*qzM*jsn)•(4sn*2P) *jsn = ((1M *1M)*(-i*n*kM)2 + (qxM*jsn*qxM)*(i*n*kM)*(-i*n*kM))*2P = ((1)*(-i*n*kM)2 + (1)*(i*n*kM)*(-i*n*kM))*2P = 0 OK x(4sn*2P) = (tM*1M*jsn + xM*qxM*jsn + yM*qyM*jsn + zM*qzM*jsn)x(4sn*2P) = (tM*1M*jsn + xM*qxM*jsn)x((1M + qxM)*(-i*n*kM)*2P) + (yM*qyM*jsn + zM*qzM*jsn)x((1M + qxM)*(-i*n*kM)*2P) *jsn = ((1M *qxM)*(-i*n*kM)2 + (qxM*jsn*1M)*(i*n*kM)*(-i*n*kM))*2P = ((pxM)*(-i*n*kM)2 + (pxM)*(i*n*kM)*(-i*n*kM))*2P = 0 OK 6) Solution is almost identical to the text given for “-fzr” and “-fzi”. Further Thought. 1) “KyS = -(q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*zS*sinhαS/M” of the BiotSavart Law reaches a maximum when the two particles are closest. Should the moving particle be a little past the other? 2) Can we violate the Lorenz Condition and still have a consistent mathematics for electromagnetism? 3) The author could not find an all-number identity that combined the below identities, and the reason is speculated to be that the force density math of the last chapter is needed. 3A 3

= Ax*ix + Ay*iy + Az*iz ;

3B

= Bx*ix + By*iy + Bz*iz

= x*ix + y*iy + z*iz = /x*ix + /y*iy + /z*iz ; 1 = t = /(c*t)

156 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 3•(3Ax3B)

 -(3x3A)•3B + (3x3B)•3A 1*(3A•3B)  (1*3A)•3B + (1*3B)•3A 1*(3Ax3B)  (1*3A)x3B - (1*3B)x3A x( Ax 3 3 3B)  ((3B •3)*3A) - ((3A•3)*3B) + (3•3B)*3A - (3•3A)*3B 3*(3A•3B)  ((3B •3)*3A) + ((3A•3)*3B) + (3x3B)x3A + (3x3A)x3B 7) Proposed invariant “4S” has time component “U = ǝ*(3E•3E + c2*3B•3B)/2” and space components “3S = ǝ*c*(3Ex3B - 3Bx3E)/2”. Try to apply compound label numbers and find “(pxM)*i ≠ -pxM” for “*i” applied to quaternions. Try to perform a Lorentz Transformation. Explain why “4S” is not an actual invariant. “-1f = 4sn•4S” with respect to components, but why “-4f  sn 4 *4S”? Use “6Efourth = -Eamp*(kyM - pzM)*exp(-i*(kxM*(xM - c*tM)))” and “4S = (3E - 3K)*(3E + 3K)/2”. Notice there’s no mathematically beautiful way to create “4S” from “6E”. 4) Try to find a linear combination of two of “qx”, “qy” or “qz” for “”. 5) How can anti-matter be worked into the macroscopic approximation theory that the inertial mass we measure is the electromagnetic field energy of an electron? 6) The triple-vector-product identity “4sn■(4*jsnx(4V))  0” includes two space-component identities that add together to create Faraday’s Law of Induction. Is there significance to “1sn*(-3K) + sn 3 x(-3E) = 0” being formed from two identities? 1

sn

*(3*jsnx(3V)) + 3snx(1*jsn*(3V))  0 ; 3snx(3*jsn*(1V))  0

1

sn

1

sn

1

sn

*(3*jsnx(3V)) + (3snx(1*jsn*(3V)) + 3snx(3*jsn*(1V)))  0 *(3*jsnx3V) + 3snx(1*jsn*3V + 3*jsn*1V)  0 *(-3K) + 3snx(-3E) = 0

157 CHAPTER 3 – FIELDS 7) To better understand the space-negative, review how the spacenegative behaves in the non-relativistic approximation example given below. A temperature gradient is given in “B” and is then Lorentz Transformed to “M” and to “S”. The very simple nonrelativistic approximation example is given so that the spacenegative is the only complexity. Is there a more optimal algebraic technique that can replace the messy space-negative and that performs the same function? In “B”: 1T = b*xB ; 2sn*1T = qxBsn*b = qxBsn*1T/xB (1T/ctB = 0) sn sn In “M”: 1T = b*(xM - vM*tM) ; 2 *1T = 1M *(-b*vM/c) + qxMsn*b In “S”: 1T = b*(xS - vS*tS) ; 2sn*1T = 1Ssn*(-b*vS/c) + qxSsn*b Because of the non-relativistic approximation: • • •

tB = tM = tS vS = vM + vS/M vM*vS/M/c2 = 0

1T/ctM 1T/xM

1

* -vM/c

1T/ctS

1

1

1T/xS

* -vS/M/c

1T/ctS

1T/xB 1T/ctM

-vS/M/c

= 1T/xS

1T/ctB

-vM/c

=

1

1T/xM

1

-vS/M/c

=

1

-vM/c

* -vS/M/c

1

-vM/c

* 1

1T/ctB 1T/xB

1

1T/ctB

-vS/c

=

* -vS/c

1

1T/xB

158 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

159 WAVES

Chapter 4 – Waves The Dirac Equation is our relativistic model for the dynamics of an electron. An electron is so small Newton’s Second Law (force equals mass times acceleration) does not apply. Newton’s Second Law was presented in Chapter 1 by simply writing the geometric-vector equation and then explaining force, mass, and acceleration. In contrast, the Dirac Equation is a set of four first order differential equations that needs to be developed slowly emphasizing a logical thought process.

4.1 Differential Operator Mechanical Energy-Momentum. An electron’s mechanical energy and momentum combine in the time-space momentum invariant “4p”. 4p

= exp(-*/2)*(EM/c + qx*pxM + qy*pyM + qz*pzM)*exp(-*/2)

“EM” is mechanical energy. “pxM”, “pyM”, and “pzM” are mechanical momentum components. “4p” is time-like because energy is modeled using “cosh” and momentum “sinh”. 4p

= exp(-*/2)*(mB*c*coshαM + q*mB*c*sinhαM)*exp(-*/2) = exp(-*/2)*mB*c*exp(q*αM)*exp(-*/2)

“αM” relates to electron speed by “vM = c*tanhαM”. Subscript “M” identifies the inertial reference frame of the observer. “B” is the rest frame of the particle so that “mB” is the rest mass. “c” is the speed-of-light. “q” is made general through use of knowable circular-angles “”. q = qx*cos(x/yz) + (qy*cos(y/z) + qz*sin(y/z))*sin(x/yz)

q*q = 1

“1M” and “qM” are compound-label-numbers. “” is the unknown and unknowable unspecified simple-label-number. “” contrasts with “q” because “q” is knowable.

160 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1M = exp(-*)

; qM = exp(-*/2)*q*exp(-*/2)

qM = qxM*cos(x/yz) + (qyM*cos(y/z) + qzM*sin(y/z))*sin(x/yz) 4p

qM*j*qM = -1

= 1M*mB*c*exp(q*αM) = 1M*mB*c*coshαM + qM*mB*c*sinhαM

Electrical Energy-Momentum. The electrical energy-momentum invariant is “4q = QB*4V”. “QB” is electron electric charge. “4V” is external voltage. The analogy for the energy component “QB*1V” is the potential energy of a car at the top of a hill. To visualize the electrical momentum component “QB*3V” think about what happens when a wire with direct current is cut. The magnetic field around the wire provides inertia to maintain the electric current, typically by ionizing the air to make the air conductive. Total Energy-Momentum. Per the de Broglie relations, total energy is proportional to frequency “” (or “1k”) and total momentum is proportional to wavenumber “k” (or “3k”) with the constant of proportionality Planck’s constant “ħ” (h-bar) (“ħ = h/(2*)”). ħ = 1.054571800(13)*10-34 Joule*seconds

(angular momentum)

Total “ħ*4k” equals mechanical plus electrical. ħ*4k = 4p + QB*4V The next task is to associate an actual wave to the particle. Gradient Substitution for Frequency/Wave-Number. The total energymomentum invariant “ħ*4k” is replaced with a differential gradient operator. ħ*4k = i**ħ*4sn

and

ħ*4ksn = i**ħ*(4sn)sn = i**ħ*4

161 WAVES The replacement of frequency “4k” by the gradient operator “i**4 ” is justified by use of an invariant wave function “1T” of the form below. Assume “” equals “+1”. sn

1T

= exp(i**(kxM*xM - M*tM))

-4k*j•4r = kxM*xM - M*tM (written for “kyM = kzM = 0”) -4k*jsn•4r = -kxM*xM - M*tM (written for “kyM = kzM = 0”) “4k = i**4sn” (and “4ksn = i**4”) 1M*M/c is replaced by i**1sn sn 3k is replaced by i**3

; 1Msn*M/c is replaced by i**1 sn ; 3k is replaced by i**3

Per the above equations, the space-negative alternative is redundant and so will be dropped along with the “”. “i**1sn” is substituted for the total energy-momentum invariant. i**ħ*(4sn) = 4p + QB*4V = 1M*mB*c*coshαM + qM*mB*c*sinhαM + QB*(1V + 3V) i**ħ*(4sn) = i**ħ*(1) - i**ħ*(3) 1M*mB*c*coshαM = i**ħ*(1) - QB*1V qM*mB*c*sinhαM = -i**ħ*(3) - QB*3V To make the above two equations component equations, divide them left and right by “exp(-*/2)”. 1*mB*c*coshαM = i*ħ**tM - QB*VtM q*mB*c*sinhαM = qx*(-i*ħ**xM - QB*VxM) + qy*(-i*ħ**yM - QB*VyM) + qz*(-i*ħ**zM - QB*VzM) “4sn” requires wave function “1T” so that the above component equations apply to something that is both a particle and a wave.

162 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Anti-Matter Visualized as the Space-Negative. Anti-matter was modeled with “αS/M = i*”. Here, anti-matter is suggested to be the spacenegative. The two models for anti-matter are independent, such that matter could be modeled with the space-negative.

4.2 Development of the Dirac Equation Algebraic Matrix Equation for the mechanical components. 12 = exp(-q*αM)*exp(q*αM) 12 = (coshαM - q*sinhαM)*(coshαM + q*sinhαM) 12 = cosh2αM - q2*sinh2αM 12 - cosh2αM = -q2*sinh2αM -(12 - cosh2αM) = q2*sinh2αM “-(12 - cosh2αM) = q2*sinh2αM” becomes four equations. +: +: -: -:

(1 + coshαM)*(-1 + coshαM) = (q*sinhαM)*(q*sinhαM) (1 - coshαM)*(-1 - coshαM) = (-q*sinhαM)*(-q*sinhαM) (1 - coshαM)*(-1 - coshαM) = (q*sinhαM)*(q*sinhαM) (1 + coshαM)*(-1 + coshαM) = (-q*sinhαM)*(-q*sinhαM)

“+” is redundant to “+”, and “-” to “-”. To address both “+” and “-” include a “” sign. The “” equation is split by introducing enabler functions “PPM” and “QQM”. ((1 + coshαM)*PPM)*((-1 + coshαM)*-QQM) = ((q*sinhαM)*-QQM)*((q*sinhαM)*PPM) ((-1 + coshαM)*-QQM) = ((q*sinhαM)*PPM) ((1 + coshαM)*PPM) = ((q*sinhαM)*-QQM) (1 + coshαM)*PPM + (q*sinhαM)*QQM = 0 (q*sinhαM)*PPM + (-1 + coshαM)*QQM = 0

163 WAVES (1 + coshαM)

(q*sinhαM)

PPM *

(q*sinhαM) (-1 + coshαM)

0 =

QQM

0

The “” sign in front of “q*sinhαM” (“q*sinhαM = q*sinh(αM)” and “coshαM = cosh(αM)”) relates to motion being right or left per “vM/c = tanhαM = tanh(αM)” because the “-” of “” refers to the space-negative. The “0”’s on the right make it a “singular” matrix equation. Algebraic Solutions to the Matrix Equation. In general, an algebraic 2x2 singular matrix equation has two independent solution pairs: a “1” pair and a “2” pair. Because of angle identities, there are three equivalent ways of writing the two pairs. cosh(αM/2) = cosh(αM - αM/2) = coshαM*cosh(αM/2) - sinhαM*sinh(αM/2) sinh(αM/2) = sinh(αM - αM/2) = sinhαM*cosh(αM/2) - coshαM*sinh(αM/2) (1 + coshαM)/sinhαM = sinhαM/(-1 + coshαM) = cosh(αM/2)/sinh(αM/2) ((PPM, QQM)1)not-half-anglesA = ((-q*sinhαM), 1 + coshαM) ((PPM, QQM)2)not-half-anglesA = (sinhαM, (-q)*(1 + coshαM)) ((PPM, QQM)1)not-half-anglesB = ((-q)*(-1 + coshαM), sinhαM) ((PPM, QQM)2)not-half-anglesB = (-1 + coshαM, (-q*sinhαM)) (PPM, QQM)1 = ((-q*sinh(αM/2)), cosh(αM/2)) (PPM, QQM)2 = (sinh(αM/2), (-q*cosh(αM/2))) “(-q)” is preferred to be a factor on “sinhαM”. But, as is obvious in the three forms above, this preference cannot be satisfied. “sinh(αM/2)” and “cosh(αM/2)” together form mechanical momentum, per “sinhαM = 2*sinh(αM/2)*cosh(αM/2)”. And, both together form mechanical energy, per “coshαM = cosh2(αM/2) + sinh2(αM/2)”. A general solution for a singular matrix equation is a linear combination formed by multiplying an arbitrary constant, “f” or “g”, by each of the two solutions.

164 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY PPM1 = (-q)*PPM2

and

QQM1 = (-q)*QQM2

(PPM, QQM) = f*(PPM, QQM)1 + g*(PPM, QQM)2 = f*(PPM, QQM)1  g*(-q)*(PPM, QQM)1 = (f  (-q)*g)*(PPM, QQM)1 The half angle is useful because the half is in the exponent, and a half in an exponent represents a square root operation. It implies a need for a square operation later in the analysis. The square operation, in the form of a complex number multiplied by its conjugate, is required for calculating a measurable particle property from the quantum mechanics wave function solution. Substitute Differential Operators into the Matrix Equation. First, multiply by rest mass “mB*c”. (mB*c + mB*c*coshαM) (q*mB*c*sinhαM)

(q*mB*c*sinhαM) PPM 0 * = (-mB*c + mB*c*coshαM) QQM 0

(mB*c + i**ħ*tM - QB*VtM) (-i**ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) PPM 0 * = (-i**ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) (-mB*c + i**ħ*tM - Q*VtM) QQM 0

“1T = exp(i**(kxM*xM - M*tM))” had a handedness specified by “ = +1”. Opposite handedness is specified by “ = -1”. For completeness, “” has remained in the analysis until now, when it will be identified as irrelevant. A change in handedness (in this development of the Dirac Equation) is a change from matter to anti-matter (or from anti-matter to matter). To make that change: Change “” to “-”. Multiply all four components of the above 2x2 matrix by “-1”. Swap the sign of the electric charge so that “QB” is replaced by its negative “-QB” and swap the sign of the rest mass so that “mB” is replaced by “-mB”. Look at what remains. See it is the same as what was started with. The choice of “” was irrelevant and therefore “ = +1” is used.

165 WAVES (mB*c + i*ħ*tM - QB*VtM) (-i*ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) PPM 0 * = (-i*ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) (-mB*c + i*ħ*tM - QB*VtM) QQM 0

The transformation of the mathematical model from particle to wave occurred when the mechanical energy and mechanical momentum were replaced by the total energy (minus the electrical) and the total momentum (minus the electrical), respectively, by use of differential operators. Now that the equation pertains to waves and not to particles, the either-or plus or minus separately sign “” is replaced by the both plus and minus but also neither plus nor minus separately sign “N”. (mB*c + i*ħ*tM - QB*VtM) N(-i*ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) PM 0 * = N(-i*ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) (-mB*c + i*ħ*tM - QB*VtM) QM 0

Substitute-in Matrix Isomorphs. “qx”, “qy” and “qz” have 2x2 matrix isomorph equivalents which are very similar to the Pauli Spin Matrices (with the difference being “R2” is negative). “R1 = G1/i”, “R2 = G2/i”, “R3 = G3/i”, and “1 = i/i”. 0 i G1 => i 0

0 -1 i G2 => G3 => 1 0 0

0 1 R1 =>

0 k R2 =>

1 0

1

0 -i 0

R3 => -k 0

1 0 1 =>

0

-1

0 1

(“k” is the same as “i” and is different symbolically for tracking.) The traditional substitution is “qx => R1”, “qy => R2”, and “qz => R3”. Enabler functions are replaced by column vectors. Dirac Equation:

166 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1000 1000 0001 0100 0100 0010 mB*c* + (i*ħ*/ctM - QB*VtM)* N (-i*ħ*/xM - QB*VxM)* 0 0 -1 0 0010 0100 0 0 0 -1 0001 1000 001 0 1M 0 0 0 -1 2M N (-i*ħ*/yM - QB*VyM)* N (-i*ħ*/zM - QB*VzM)* * = 0 k00 1 000 3M -k 0 0 0 0 -1 0 0 4M 000 k 0 0 -k 0

0 0 0 0

The “Dirac Spinor” is the column-vector with the four components “1M”, “2M”, “3M”, “4M”. These four component symbols are shorthand symbols for the more expanded symbols “z1M”, “z2M”, “z3M”, “z4M”. “z” identifies “qz” has the non-zero major diagonal elements. “” identifies “N” is included. An algebraic 4x4 singular matrix equation has four independent solutions. The Dirac Equation is a differential 4x4 singular matrix equation with eight solutions: Four for matter and four for anti-matter.

4.3 Solutions to the Dirac Equation In general, finding solutions to the Dirac Equation is difficult. But, finding solutions is easy for the specific case of motion in the positive “x” direction with no external voltage applied. For that case, the Dirac Equation is reduced to two simple pairs of equations of identical form. 1M + (iħ/mBc)*1M/ctM - N(iħ/mBc)*4M/xM = 0 -4M + (iħ/mBc)*4M/ctM - N(iħ/mBc)*1M/xM = 0 2M + (iħ/mBc)*2M/ctM - N(iħ/mBc)*3M/xM = 0 -3M + (iħ/mBc)*3M/ctM - N(iħ/mBc)*2M/xM = 0 The first pair has four Dirac Spinor solutions analogous, one for one, with the four electromagnetic spiral waves.

167 WAVES First Dirac Spinor Solution (for “z+1M-first” and “z+4M-first”) 1M-first = amp*cosh(αM/2)*exp(i*(NkxM-am*xM + M-am*tM)) 4M-first = amp*sinh(αM/2)*exp(i*(NkxM-am*xM + M-am*tM)) 4M-first = tanh(αM/2)*1M-first Second Dirac Spinor Solution (for “z+1M-second” and “z+4M-second”) 1M-second = -amp*sinh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM)) 4M-second = amp*cosh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM)) 4M-second = -coth(αM/2)*1M-second Third Dirac Spinor Solution (for “z+1M-third” and “z+4M-third”) 1M-third = amp*sinh(αM/2)*exp(-i*(NkxM-am*xM + M-am*tM)) 4M-third = amp*cosh(αM/2)*exp(-i*(NkxM-am*xM + M-am*tM)) 4M-third = coth(αM/2)*1M-third Fourth Dirac Spinor Solution (for “z+1M-fourth” and “z+4M-fourth”) 1M-fourth = amp*cosh(αM/2)*exp(-i*(NkxM-m*xM - M-m*tM)) 4M-fourth = -amp*sinh(αM/2)*exp(-i*(NkxM-m*xM - M-m*tM)) 4M-fourth = -tanh(αM/2)*1M-fourth (For engineering, “N” will be “+”. For theory development retain the “N”, and do not place it as in the below alternative.) Alternative First Dirac Spinor Solution 1M-first = amp*cosh(αM/2)*exp(i*(kxM-am*xM + M-am*tM)) 4M-first = Namp*sinh(αM/2)*exp(i*(kxM-am*xM + M-am*tM)) 4M-first = Ntanh(αM/2)*1M-first

Proof the First Solution is Correct. coshαM = ħ*M/(mB*c2) ;

sinhαM = ħ*kxM/(mB*c)

(for 4V = 0)

Substitute the first solution into the pair of differential equations. “amp” and “exp(i*(NkxM*xM + M*tM))” divide out.

168 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 0 = 1M-first + (iħ/mc)*1M-first/ctM - N(iħ/mc)*4M-first/xM = cosh(αM/2) + (iħ/mc2)*(i*M)*cosh(αM/2) - N(iħ/mc)*(i*NkxM)*sinh(αM/2) = cosh(αM/2) - coshαM*cosh(αM/2) + sinhαM*sinh(αM/2) = 0 0 = -4M-first + (iħ/mc)*4M-first/ctM - N(iħ/mc)*1M-first/xM = -sinh(αM/2) + (iħ/mc2)*(i*M)*sinh(αM/2) - N(iħ/mc)*(i*NkxM)*cosh(αM/2) = -sinh(αM/2) - coshαM*sinh(αM/2) + sinhαM*cosh(αM/2) = 0 Single Speed in this Simple Example. Regardless of there being only one speed represented by “αM”, the concept of an interference group applies. Matter and Anti-matter. The plus “+” or minus “-” sign (for “N” “+”) in front of “M” determines if the solution is anti-matter or is matter, respectively. “kxM-am” and “M-am” are for anti-matter. “kxM-m” and “M-m” are for matter. Anti-matter requires a space-negative in a Lorentz Transformation. The actual numerical value of “kxM” and “M” is not dependent on it being matter or anti-matter. kxM-am = kxM-m = kxM

;

M-am = M-m = M

Matter. The second and fourth solutions pertain to matter (for “N” “+”) with “mB > 0” and “QB < 0”. “kxM-m*xM - M-m*tM = -4k*j•4r” describes wave crests and nodes that move in the positive “xM” direction at a phase speed “vpM”. The phase speed “vpM/c”, for the simple case of no voltage, is the reciprocal of the electron particle’s group speed “vM/c = tanhαM”. vpM-m = M/kxM = M-m/kxM-m = c*coshαM/sinhαM vpM-m/c = c/vM-m = cothαM (matter) Anti-Matter. The first and third solutions (for “N” “+”) have “mB < 0” and “QB > 0” for an anti-matter electron (positron). “kxM-am*xM + M-am*tM = *jsn •4r” describes wave crests and nodes that move in the negative “xM” 4k direction at speed “vpM”. vpM-am/c = (M/c)/(-kxM) = (M-am/c)/(-kxM-am) = coshαM/-sinhαM = c/vM-am = -cothαM (anti-matter)

169 WAVES In the headlight/tail-light visualization both electrons and positrons have the headlights pointing toward more positive “x” (right). And, electric current for both electrons and positrons is the same. Spin. The “+”/“-” sign in front of “i” determines if angular momentum spin is right-hand or left-hand. In the car visualization a right-side steering wheel represents right-hand spin. A positron with a left steering wheel is the anti-matter counterpart to an electron with a right steering wheel, and that pairs first with fourth and second with third. Glove Visualization for Both Anti-matter and Spin. The second/fourth solutions (matter) are visualized as a left/right glove pair with the fingers pointing to positive “x”. Pull the gloves inside out, and now the gloves represent the first/third solutions (anti-matter) pointing to negative “x”. Second became first and fourth became third. Rotate these to point to positive “x” and place them inside the original gloves, first inside fourth and third inside second. Excess energy separates them with second/fourth moving to more positive “x” and third/first moving to more negative “x”, for equal and opposite linear momentum and angular momentum.

4.4 Particle Properties Because Dirac Spinor waves cannot be measured directly, the solution is post-processed to replace the wave nature with a particle nature, as proposed by Max Born for Schrödinger’s Equation solutions. Multiply the complex-conjugate of the Dirac Spinor solution wave function, “4*i”, by the Dirac Equation, and perform the opposite operation. Add the two and, alternatively, subtract the two. Subtraction drops the mathematically real terms. Addition drops imaginary terms. Subtraction Equation Resulting in Electric Current Density. The sum equals zero because of the zeros on the right side of the Dirac Equation. (The “M” subscript is dropped to not clutter the equation.)

170 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 0 = 1*i*(mc*1 + iħ*1/ct - QVt*1 - Niħ*4/x - NQVx*4 - Nk*iħ*4/y - Nk*QVy*4 - Niħ*3/z - NQVz*3) + 2*i*(mc*2 + iħ*2/ct - QVt*2 - Niħ*3/x - NQVx*3 - N(-k)*iħ*3/y - N(-k)*QVy*3 - N(-1)*iħ*4/z - N(-1)* QVz*4) + 3*i*(-mc*3 + iħ*3/ct - QVt*3 - Niħ*2/x - NQVx*2 - Nk*iħ*2/y - Nk*QVy*2 - Niħ*1/z - NQVz*1) + 4*i*(-mc*4 + iħ*4/ct - QVt*4 - Niħ*1/x - NQVx*1 - N(-k)*iħ*1/y - N(-k)*QVy*1 - N(-1)*iħ*2/z - N(-1)*QVz*2) - (mc*1*i - iħ*1*i/ct - QVt*1*i + Niħ*4*i/x - NQVx*4*i + N(-k)*iħ*4*i/y - N(-k)*QVy*4*i + Niħ*3*i/z - NQVz*3*i)*1 - (mc*2*i - iħ*2*i/ct - QVt*2*i + Niħ*3*i/x - NQVx*3*i + N(+k)*iħ*3*i/y - N(k)*QVy*3*i + N(-1)*iħ*4*i/z - N(-1)*QVz*4*i)*2 - (mc*3*i - iħ*3*i/ct - QVt*3*i + Niħ*2*i/x - NQVx*2*i + N(-k)*iħ*2*i/y - N(-k)*QVy*2*i + Niħ*1*i/z - NQVz*1*i)*3 - (mc*4*i - iħ*4*i/ct - QVt*4*i + Niħ*1*i/x - NQVx*1*i + N(k)*iħ*1*i/y - N(k)*QVy*1*i + N(-1)* iħ*2*i/z - N(-1)*QVz*2*i)*4

“mc” and “QV” terms subtract away. Only “iħ” terms remain. 0 = 1*i*(iħ*1/ct - Niħ*4/x - Nk*iħ*4/y - Niħ*3/z) + 2*i*(iħ*2/ct - Niħ*3/x - N(-k)*iħ*3/y - N(-1)*iħ*4/z) + 3*i*(iħ*3/ct - Niħ*2/x - Nk*iħ*2/y - Niħ*1/z) + 4*i*(iħ*4/ct - Niħ*1/x - N(-k)*iħ*1/y - N(-1)*iħ*2/z) - (-iħ*1*i/ct + Niħ*4*i/x + N(-k)*iħ*4*i/y + Niħ*3*i/z)*1 - (-iħ*2*i/ct + Niħ*3*i/x + Nk*iħ*3*i/y N -iħ*4*i/z)*2 - (-iħ*3*i/ct + Niħ*2*i/x + N(-k)*iħ*2*i/y + Niħ*1*i/z)*3 - (-iħ*4*i/ct + N(iħ*1*i/x + Nk*iħ*1*i/y N -iħ*2*i/z)*4 “iħ” terms are combined using the chain rule. For example: 4*i*(2/z) + (4*i/z)*2 = (4*i*2)/z 4*i*(-(-1)*iħ*2/z) + (-(-1)*iħ*4*i/z)*2 = -(-1)*iħ*(4*i*2)/z 0 = iħ*(1M*i*1M + 2M*i*2M + 3M*i*3M + 4M*i*4M)/ctM - Niħ*(1M*i*4M + 2M*i*3M + 3M*i*2M + 4M*i*1M)/xM

171 WAVES - Nk*iħ*(1M*i*4M - 2M*i*3M + 3M*i*2M - 4M*i*1M)/yM - Niħ*(1M*i*3M - 2M*i*4M + 3M*i*1M - 4M*i*2M)/zM Substitute “+” for “N” because the subtraction equation is a particle equation and is not a wave equation. 0 = tM*JtM + xM*JxM + yM*JyM + zM*JzM = JtM/ctM + JxM/xM + JyM/yM + JzM/zM JtM = QB*(1M*i*1M + 2M*i*2M + 3M*i*3M + 4M*i*4M) JxM = -QB*(1M*i*4M + 2M*i*3M + 3M*i*2M + 4M*i*1M) JyM = -QB*k*(1M*i*4M - 2M*i*3M + 3M*i*2M - 4M*i*1M) JzM = -QB*(1M*i*3M - 2M*i*4M + 3M*i*1M - 4M*i*2M) “” components are “(particle count per volume)” and “QB” is “electric charge per particle” to make the measurement units on “J” “electric charge per volume”. First Dirac Spinor Solution example: JtM = QB*(1M*i*1M + 2M*i*2M + 3M*i*3M + 4M*i*4M) = QB*amp2*(cosh2(αM/2) + sinh2(αM/2)) = QB*amp2*coshαM JxM = -QB*(1M*i*4M + 2M*i*3M + 3M*i*2M + 4M*i*1M) = -QB*amp2*(2*cosh(αM/2)*sinh(αM/2)) = -QB*amp2*sinhαM JyM = 0 4J

JzM = 0

= QB*amp2*(1M*coshαM - qxM*sinhαM) = QB*amp2*1M*exp(-qx*αM)

The first solution above was for a positron that moves to negative “x” for “αM > 0”, per the “-qx*αM”. “QB > 0” so net current is to negative “x”.

172 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Addition Equation. The “addition equation” has the last four lines of the subtraction equation started by a “+” rather than a “-”. It pertains to the field outside the classical radius of the electron. 0 = 1*i*(mc*1 - QVt*1 - NQVx*4 - Nk*QVy*4 - NQVz*3) + 2*i*(mc*2 - QVt*2 - NQVx*3 - N(-k)*QVy*3 - N(-1)*QVz*4) + 3*i*(-mc*3 - QVt*3 - NQVx*2 - Nk*QVy*2 - NQVz*1) + 4*i*(-mc*4 - QVt*4 - NQVx*1 - N(-k)*QVy*1 - N(-1)*QVz*2) + (mc*1*i - QVt*1*i - NQVx*4*i - N(-k)*QVy*4*i - NQVz*3*i)*1 + (mc*2*i - QVt*2*i - NQVx*3*i - Nk*QVy*3*i - N(-1)*QVz*4*i)*2 + (mc*3*i - QVt*3*i - NQVx*2*i - N(-k)*QVy*2*i - NQVz*1*i)*3 + (mc*4*i - QVt*4*i - NQVx*1*i - Nk*QVy*1*i - N(-1)*QVz*2*i)*4 + 1*i*(iħ*1/ct - Niħ*4/x - Nk*iħ*4/y - Niħ*3/z) + 2*i*(iħ*2/ct - Niħ*3/x - N(-k)*iħ*3/y - N(-1)*iħ*4/z) + 3*i*(iħ*3/ct - Niħ*2/x - Nk*iħ*2/y - Niħ*1/z) + 4*i*(iħ*4/ct - Niħ*1/x - N(-k)*iħ*1/y - N(-1)*iħ*2/z) + (-iħ*1*i/ct + Niħ*4*i/x + N(-k)*iħ*4*i/y + Niħ*3*i/z)*1 + (-iħ*2*i/ct + Niħ*3*i/x + Nk*iħ*3*i/y - Niħ*4*i/z)*2 + (-iħ*3*i/ct + Niħ*2*i/x + N(-k)*iħ*2*i/y + Niħ*1*i/z)*3 + (-iħ*4*i/ct + Niħ*1*i/x + Nk*iħ*1*i/y - Niħ*2*i/z)*4 = 2*(m*c)*(1*i*1 + 2*i*2 - 3*i*3 - 4*i*4) + 2*(-Q*Vt)*(1*i*1 + 2*i*2 + 3*i*3 + 4*i*4) + 2*(-QVx)*(1*i*4 + 2*i*3 + 3*i*2 + 4*i*1) + 2*k*(-QVy)*(+1*i*4 - 2*i*3 + 3*i*2 - 4*i*1) + 2*(-QVz)*(1*i*3 - 2*i*4 + 3*i*1 - 4*i*2) + 1*i*(iħ*1/ct - Niħ*4/x - Nk*iħ*4/y - Niħ*3/z) + 2*i*(iħ*2/ct - Niħ*3/x - N(-k)*iħ*3/y - N(-1)*iħ*4/z) + 3*i*(iħ*3/ct - Niħ*2/x - Nk*iħ*2/y - Niħ*1/z) + 4*i*(iħ*4/ct - Niħ*1/x - N(-k)*iħ*1/y - N(-1)*iħ*2/z) + (-iħ*1*i/ct + Niħ*4*i/x + N(-k)*iħ*4*i/y + Niħ*3*i/z)*1 + (-iħ*2*i/ct + Niħ*3*i/x + Nk*iħ*3*i/y - Niħ*4*i/z)*2 + (-iħ*3*i/ct + Niħ*2*i/x + N(-k)*iħ*2*i/y + Niħ*1*i/z)*3 + (-iħ*4*i/ct + Niħ*1*i/x + Nk*iħ*1*i/y - Niħ*2*i/z)*4

173 WAVES Specific to the Fourth Dirac Spinor Solution, the addition equation results in the following equation. 0 = 2*(m*c)*(1M-fourth*i*1M-fourth - 4 M-fourth*i*4 M-fourth) + iħ*1M-fourth*i*1M-fourth/ct - Niħ*1M-fourth*i*4M-fourth/x + iħ*4M-fourth*i*4M-fourth/ct - Niħ*4M-fourth*i*1M-fourth/x + -iħ*1M-fourth*i/ct*1M-fourth + Niħ*4M-fourth*i/x*1M-fourth + -iħ*4M-fourth*i/ct*4M-fourth + Niħ*1M-fourth*i/x*4M-fourth = 2*(m*c)*amplitude2*(cosh2(αM/2) - sinh2(αM/2)) - ħ*(M/c)*amplitude2*(cosh2(αM/2) + cosh2(αM/2)) + Nħ*(kxM)*amplitude2*(Ncosh(αM/2)*sinh(αM/2) + Nsinh(αM/2)*cosh(αM/2)) + Nħ*(kxM)*amplitude2*(Ncosh(αM/2)*sinh(αM/2) + Nsinh(αM/2)*cosh(αM/2))

- ħ*(M/c)*amplitude2*(sinh2(αM/2) + sinh2(αM/2)) = 2*(m*c)*amplitude2 + ħ*(M/c)*2*amplitude2*cosh(αM) - ħ*(kxM)*2*amplitude2*sinh(αM) = 2*(m*c)*amplitude2 + 2*(m*c)*amplitude2*(cosh2(αM) - sinh2(αM)) = 2*(m*c)*amplitude2*(1 - (cosh2(αM) - sinh2(αM)) =0

4.5 Two Alternative Arrangements The “x” Arrangement has “qy => R1”, “qz => R2”, and “qx => R3”. 1000 0100

1000 0100

0001 0010 m*c* + (i*ħ*/ctM - q*VtM)* N (-i*ħ*/yM - q*VyM)* 0 0 -1 0 0010 0100 0 0 0 -1 0001 1000 000 k 001 0 x1M 0 0 0 -k 0 0 0 0 -1 x2M 0 N (-i*ħ*/zM - q*VzM)* N (-i*ħ*/xM - q*VxM)* * = 0 k00 1 000 x3M 0 -k 0 0 0 0 -1 0 0 x4M 0

For the case of “x” direction motion and no external voltage:

174 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY x1M + (iħ/mc)*x1M/ctM - N(iħ/mc)*x3M/xM = 0 -x3M + (iħ/mc)*x3M/ctM - N(iħ/mc)*x1M/xM = 0 x2M + (iħ/mc)*x2M/ctM + N(iħ/mc)*x4M/xM = 0 -x4M + (iħ/mc)*x4M/ctM + N(iħ/mc)*x2M/xM = 0 First Dirac Spinor Solution x1M-first = amp*cosh(αM/2)*exp(i*(NkxM*xM + M*tM)) x3M-first = amp*sinh(αM/2)*exp(i*(NkxM*xM + M*tM)) x3M-first = x1M-first*tanh(αM/2) Second Dirac Spinor Solution x1M-second = amp*sinh(αM/2)*exp(i*(NkxM*xM - M*tM)) x3M-second = -amp*cosh(αM/2)*exp(i*(NkxM*xM - M*tM)) x3M-second = -x1M-second*coth(αM/2) Third Dirac Spinor Solution x1M-third = amp*sinh(αM/2)*exp(-i*(NkxM*xM + M*tM)) x3M-third = amp*cosh(αM/2)*exp(-i*(NkxM*xM + M*tM)) x3M-third = x1M-third*coth(αM/2) Fourth Dirac Spinor Solution x1M-fourth = amp*cosh(αM/2)*exp(-i*(NkxM*xM - M*tM)) x3M-fourth = -amp*sinh(αM/2)*exp(-i*(NkxM*xM - M*tM)) x3M-fourth = -x1M-fourth*tanh(αM/2) The “y” Arrangement has “qz => R1”, “qx => R2”, and “qy => R3”. 1000 0100

1000 0100

0001 0010 m*c* + (i*ħ*/ctM - q*VtM)* N (-i*ħ*/zM - q*VzM)* 0 0 -1 0 0010 0100 0 0 0 -1 0001 1000 000 k 001 0 0 0 -k 0 0 0 0 -1 N (-i*ħ*/xM - q*VxM)* N (-i*ħ*/yM - q*VyM)* 0 k00 1 000 -k 0 0 0 0 -1 0 0

y1M 0 y2M 0 * = y3M 0 y4M 0

175 WAVES For the case of “x” direction motion and no external voltage: y1M + (iħ/mc)*y1M/ctM - Nk*(iħ/mc)*y4M/xM = 0 -y4M + (iħ/mc)*y4M/ctM N k*(iħ/mc)*y1M/xM = 0 y2M + (iħ/mc)*y2M/ctM N k*(iħ/mc)*y3M/xM = 0 -y3M + (iħ/mc)*y3M/ctM - Nk*(iħ/mc)*y2M/xM = 0 First Dirac Spinor Solution y1M-first = amp*cosh(αM/2)*exp(i*(NkxM*xM + M*tM)) y4M-first = -k*amp*sinh(αM/2)*exp(i*(NkxM*xM + M*tM)) y4M-first = -k*y1M-first*tanh(αM/2) Second Dirac Spinor Solution y1M-second = amp*sinh(αM/2)*exp(i*(NkxM*xM - M*tM)) y4M-second = -(-k)*amp*cosh(αM/2)*exp(i*(NkxM*xM - M*tM)) y4M-second = k*y1M-second*coth(αM/2) Third Dirac Spinor Solution y1M-third = amp*sinh(αM/2)*exp(-i*(NkxM*xM + M*tM)) y4M-third = -k*amp*cosh(αM/2)*exp(-i*(NkxM*xM + M*tM)) y4M-third = -k*y1M-third*coth(αM/2) Fourth Dirac Spinor Solution y1M-fourth = amp*cosh(αM/2)*exp(-i*(NkxM*xM - M*tM)) y4M-fourth = -(-k)*amp*sinh(αM/2)*exp(-i*(NkxM*xM - M*tM)) y4M-fourth = k*y1M-fourth*tanh(αM/2) “z”, “x”, and “y” arrangements appear redundant because they result in the same measurable charge density space-time invariant “2J”.

4.6 Lorentz Transformation of a Dirac Spinor The Lorentz Transformation for a Dirac Spinor uses half angle “αM/2”. The proof of validity of the Lorentz Transformation is that the Dirac Spinor in “S” can be obtained by either the Lorentz Transformation of the

176 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Dirac Spinor directly, or by solving the Dirac Equation with its component variables specified in “S” rather than in “M”. “4J”, too, must be properly affected by the Lorentz Transformation. The fourth solution is the example because it is for matter. “1M-fourth” and “4M-fourth” are associated with compound-labelnumbers “e1M” and “e4M”, respectively, with “e1M = qx*e4M”. 1 0 e1 = qx*e4 =

0 0 0 1 0 0 1 0 =

0 0

0 0 *

0 1 0 0 1 0 0 0

0 1

In the next chapter is an explanation for the anti-commutative operations “e1M = -e4M*qx” and “e4M = -e1M*qx”. Condensing components into a single expression for “4-fourth”. 4-fourth

= e1M*1M + e4M*4M = e1M*1M-fourth + e4M*4M-fourth

= (e1M*amp*cosh(αM/2)*exp(-i*(NkxM-m*xM - M-m*tM))) + (e4M*amp*-sinh(αM/2)*exp(-i*(NkxM-m*xM - M-m*tM))) = (e1M*amp*cosh(αM/2)*exp(-i*(NkxM-m*xM - M-m*tM))) + (-e1M*qx*amp*-sinh(αM/2)*exp(-i*(NkxM-m*xM - M-m*tM))) = e1M*amp*exp(qx*αM/2)*exp(-i*(NkxM-m*xM - M-m*tM)) The form “e1M*amp*exp(qx*αM/2)*exp(-i*(NkxM*xM - M*tM))” is convenient because, for the Lorentz Transformation, the “e1M” changes to “e1S”, the “αM” changes to “αS = αM + αS/M”, and the invariant expression “NkxM-m*xM - M-m*tM” changes to “NkxS-m*xS - S-m*tS”. We begin with the general form of the Lorentz Transformation. 4-fourth

= e1M*amp*exp(qx*αM/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = e1M*amp*exp(qx*αM/2)*exp(-i*(NkxM-m*xM - M-m*tM)) *exp(qx*αS/M/2)/exp(qx*αS/M/2)

177 WAVES = (e1M/exp(qx*αS/M/2)) *amp*exp(qx*αM/2)*exp(qx*αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = e1S*amp*exp(qx*(αM + αS/M)/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = e1S*amp*exp(qx*αS/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = e1S*amp*exp(qx*αS/2)*exp(-i*(NkxS-m*xS - S-m*tS)) = e1S*1S-fourth + e4S*4S-fourth cosh(-αM/2)*cosh(-αS/M/2) + sinh(-αM/2)*sinh(-αS/M/2) = cosh(-αM/2 - αS/M/2) cosh(-αM/2)*sinh(-αS/M/2) + sinh(-αM/2)*cosh(-αS/M/2) = sinh(-αM/2 - αS/M/2) e1*1S-fourth + e4*4S-fourth = (e1*1M-fourth + e4*4M-fourth)*exp(qx*αS/M/2) = (e1*1M-fourth + e4*4M-fourth)*(cosh(αS/M/2) + qx*sinh(αS/M/2)) = e1*1M-fourth*cosh(αS/M/2) + e4*qx*4M-fourth*sinh(αS/M/2) + e1*qx*1M-fourth*sinh(αS/M/2) + e4*4M-fourth*cosh(αS/M/2) = e1*(1M-fourth*cosh(αS/M/2) - 4M-fourth*sinh(αS/M/2)) + e4*(-1M-fourth*sinh(αS/M/2) + 4M-fourth*cosh(αS/M/2)) 1S-fourth = 1M-fourth*cosh(αS/M/2) - 4M-fourth*sinh(αS/M/2) = amp*cosh(-αM/2)*cosh(αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) - amp*sinh(-αM/2)*sinh(αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = amp*cosh(-αM/2)*cosh(-αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) + amp*sinh(-αM/2)*sinh(-αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = amp*cosh(-αM/2 - αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = amp*cosh(-αS/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = amp*cosh(-αS/2)*exp(-i*(NkxS-m*xS - S-m*tS)) = amp*cosh(αS/2)*exp(-i*(NkxS-m*xS - S-m*tS))

178 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 4S-fourth = 1M-fourth*sinh(-αS/M/2) + 4M-fourth*cosh(-αS/M/2) = amp*cosh(-αM/2)*sinh(-αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) + amp*sinh(-αM/2)*cosh(-αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = amp*sinh(-αM/2 - αS/M/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = amp*sinh(-αS/2)*exp(-i*(NkxM-m*xM - M-m*tM)) = amp*sinh(-αS/2)*exp(-i*(NkxS-m*xS - S-m*tS)) = -amp*sinh(αS/2)*exp(-i*(NkxS-m*xS - S-m*tS)) e1M = e1*exp(-qx*/2) = e1*cosh(/2) - e1*qx*sinh(/2) = e1*cosh(/2) + e4*sinh(/2) e4M = e4*exp(-qx*/2) = -e4*qx*sinh(/2) + e4*cosh(/2) = e1*sinh(/2) + e4*cosh(/2) e1S = e1M/exp(qx*αS/M/2) = e1M*cosh(αS/M/2) + e4M*sinh(αS/M/2) e4S = e4M/exp(qx*αS/M/2) = e1M*sinh(αS/M/2) + e4M*cosh(αS/M/2) e1S = -e4S*qx e4S = -e1S*qx The complex-conjugate is needed for the electric current density. 4-fourth

*i

= (exp(-i*(NkxM-m*xM - M-m*tM)))*i*(exp(-qx*αM/2))*i*amp*i*e1M*i

1S-fourth*i = 1M-fourth*i*cosh(-αS/M/2) + 4M-fourth*i*sinh(-αS/M/2) 4S-fourth*i = 1M-fourth*i*sinh(-αS/M/2) + 4M-fourth*i*cosh(-αS/M/2) JtS = QB*(1S*i*1S + 2S*i*2S + 3S*i*3S + 4S*i*4S) = QB*(1S-fourth *i*1S-fourth + 4S-fourth*i*4S-fourth) = QB*((1M-fourth*i*cosh(-αS/M/2) + 4M-fourth*i*sinh(-αS/M/2)) *(1M-fourth*cosh(-αS/M/2) + 4M-fourth*sinh(-αS/M/2)) + (1M-fourth*i*sinh(-αS/M/2) + 4M-fourth*i*cosh(-αS/M/2)) *(1M-fourth*sinh(-αS/M/2) + 4M-fourth*cosh(-αS/M/2))) = QB*(1M-fourth*i*1M-fourth*cosh2(αS/M/2) + 1M-fourth*i*4M-fourth*cosh(αS/M/2)*sinh(-αS/M/2)

179 WAVES + 4M-fourth*i*4M-fourth*sinh2(αS/M/2) + 4M-fourth*i*1M-fourth*cosh(αS/M/2)*sinh(-αS/M/2) + 1M-fourth*i*1M-fourth*sinh2(αS/M/2) + 1M-fourth*i*4M-fourth*cosh(αS/M/2)*sinh(-αS/M/2) + 4M-fourth*i*4M-fourth*cosh2(αS/M/2) + 4M-fourth*i*1M-fourth*cosh(αS/M/2)*sinh(-αS/M/2) = QB*(1M-fourth*i*1M-fourth*(cosh2(αS/M/2) + sinh2(αS/M/2)) + 1M-fourth*i*4M-fourth*2*cosh(αS/M/2)*sinh(-αS/M/2) + 4M-fourth*i*4M-fourth*(sinh2(αS/M/2) + cosh2(αS/M/2)) + 4M-fourth*i*1M-fourth*2*cosh(αS/M/2)*sinh(-αS/M/2) = QB*(1M-fourth*i*1M-fourth + 4M-fourth*i*4M-fourth)*coshαS/M + QB*(1M-fourth*i*4M-fourth + 4M-fourth*i*1M-fourth)*sinh(-αS/M) = QB*(amp2*(coshαM/2*coshαM/2 + sinh(-αM/2)*sinh(-αM/2))*coshαS/M +QB*(amp2(coshαM/2*sinh(-αM/2) + sinh(-αM/2)*coshαM/2)*sinh(-αS/M) = QB*amp2*coshαM*coshαS/M + QB*amp2*sinh(-αM)*sinh(-αS/M) = QB*amp2*cosh(-(αM + αS/M)) = QB*amp2*cosh(-αS) JxS = -QB*(1S*i*4S + 2S*i*3S + 3S*i*2S + 4S*i*1S) = -QB*(1S-fourth *i*4S-fourth + 4S-fourth*i*1S-fourth) = -QB*((1M-fourth*i*cosh(αS/M/2) + 4M-fourth*i*sinh(-αS/M/2)) *(1M-fourth*sinh(-αS/M/2) + 4M-fourth*cosh(αS/M/2)) + (1M-fourth*i*sinh(-αS/M/2) + 4M-fourth*i*cosh(αS/M/2)) *(1M-fourth*cosh(αS/M/2) + 4M-fourth*sinh(-αS/M/2))) = -QB*(1M-fourth*i*4M-fourth*cosh2(αS/M/2) + 1M-fourth*i*1M-fourth*cosh(αS/M/2)*sinh(-αS/M/2) + 4M-fourth*i*1M-fourth*sinh2(αS/M/2) + 4M-fourth*i*4M-fourth*cosh(αS/M/2)*sinh(-αS/M/2) + 1M-fourth*i*4M-fourth*sinh2(αS/M/2) + 1M-fourth*i*1M-fourth*cosh(αS/M/2)*sinh(-αS/M/2) + 4M-fourth*i*1M-fourth*cosh2(αS/M/2) + 4M-fourth*i*4M-fourth*cosh(αS/M/2)*sinh(-αS/M/2)

180 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = -QB*(1M-fourth*i*4M-fourth*(cosh2(αS/M/2) + sinh2(αS/M/2)) + 1M-fourth*i*1M-fourth*2*cosh(αS/M/2)*sinh(-αS/M/2) + 4M-fourth*i*1M-fourth*(sinh2(αS/M/2) + cosh2(αS/M/2)) + 4M-fourth*i*4M-fourth*2*cosh(αS/M/2)*sinh(-αS/M/2) = -QB*(1M-fourth*i*4M-fourth + 4M-fourth*i*1M-fourth)*coshαS/M - QB*(1M-fourth*i*1M-fourth + 4M-fourth*i*4M-fourth)*sinh(-αS/M) = -QB*(amp2(coshαM/2*sinh(-αM/2) + sinh(-αM/2)*coshαM/2)*coshαS/M + QB*(amp2(coshαM/2*coshαM/2 + sinh(-αM/2)*sinh(-αM/2))*sinh(-αS/M) = -QB*amp2*sinh(-αM)*coshαS/M + q*amp2*coshαM*sinh(-αS/M) = -QB*amp2*sinh(-(αM + αS/M)) = -QB*amp2*sinh(-αS) 4J

= QB*amp2*(1S*coshαS - qxS*sinh(-αS)) = QB*amp2*1S*exp(qx*αS) (N becomes + for a particle)

In “4J = QB*amp2*1S*exp(qx*αS)”, “amp2 > 0” and “QB < 0”. “exp(qx*αS)” represents motion to positive “x” (right) for positive “αS” and the “QB” makes the current-density of that motion negative. Similarly, “QB < 0” makes the charge density time term negative. The conclusion is that the Lorentz Transformation was correct. Anti-Matter. For anti-matter there is an “-am” subscript. S-am/c

coshαS/M

-sinhαS/M

-sinhαS/M

coshαS/M

= kxS-am S-m/c

*

coshαS/M

kxM-am M-m/c

sinhαS/M

= kxS-m

M-am/c

* sinhαS/M

coshαS/M

kxM-m

The above process can be repeated for the First Dirac Spinor Solution (that is, for anti-matter), to the result “αS = αM - αS/M”.

181 WAVES

4.7 Exercises Text Comprehension Exercises. 1) Start with “4p” and “QB*4V”, and end with 1*mB*c*coshαM = i*ħ**tM - QB*VtM q*mB*c*sinhαM = qx*(-i*ħ**xM - QB*VxM) + qy*(-i*ħ**yM - QB*VyM) + qz*(-i*ħ**zM - QB*VzM) 2) Start with “12 = exp(-q*αM)*exp(q*αM)” and end with (mB*c + i*ħ*tM - QB*VtM) N(-i*ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) PM 0 * = N(-i*ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) (-mB*c + i*ħ*tM - QB*VtM) QM 0

3) Write the Dirac Equation as four first order differential equations. 4) Prove the Fourth Dirac Spinor Solution to the Dirac Equation. 5) Make the analogy that a matter electron with left-hand spin is a car with its steering wheel on the left, as in France. Two very high energy photons collide over the channel between Dover and Dunkerque to create a matter car and an anti-matter car. Explain the analogy between cars and electrons. 6) Find the electric charge current density space-time invariant “4J” for the Second Dirac Spinor Solution. 1M-second = -amp*sinh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM)) 4M-second = amp*cosh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM)) 7) Find the four solutions for the second pair of equations for “x” direction motion for the “x”-arrangement and “y”-arrangement Dirac Equations. Prove correct and calculate current density.

182 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 8) Write the general form of the Lorentz Transformation for the Second Dirac Spinor Solution. “e1M = -e4M*qx” and “e4M = -e1M*qx”. 1M-second = -amp*sinh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM)) 4M-second = amp*cosh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM)) Select Exercise Solutions. 1)

4p

= exp(-*/2)*(EM/c + qx*pxM + qy*pyM + qz*pzM)*exp(-*/2) = exp(-*/2)*(mB*c*coshαM + q*mB*c*sinhαM)*exp(-*/2) = 1M*mB*c*coshαM + qM*mB*c*sinhαM

ħ*4k = 4p + QB*4V = i**ħ*4sn = i**ħ*(1) - i**ħ*(3) ħ*4ksn = 4psn + QB*4Vsn = i**ħ*(4sn)sn = i**ħ*4 = i**ħ*(1) + i**ħ*(3) 1M*m*c*coshαM = i**ħ*(1) - QB*1V 1*mB*c*coshαM = i*ħ**tM - QB*VtM qM*m*c*sinhαM = -i**ħ*(3) - QB*3V q*mB*c*sinhαM = qx*(-i*ħ**xM - QB*VxM) + qy*(-i*ħ**yM - QB*VyM) + qz*(-i*ħ**zM - QB*VzM) 2) 12 = exp(-q*αM)*exp(q*αM) 12 = (coshαM - q*sinhαM)*(coshαM + q*sinhαM) 12 = cosh2αM - q2*sinh2αM 12 - cosh2αM = -q2*sinh2αM -(12 - cosh2αM) = q2*sinh2αM (1 + coshαM)*(-1 + coshαM) = (q*sinhαM)*(q*sinhαM) ((1 + coshαM)*PM)*((-1 + coshαM)*-QM) = ((q*sinhαM)*-QM)*((q*sinhαM)*PM) (1 + coshαM)*PM = (q*sinhαM)*(-QM) + 0 (-1 + coshαM)*(-QM) = (q*sinhαM)*PM - 0

183 WAVES

(1 + coshαM)

(q*sinhαM) PM * = (q*sinhαM) (-1 + coshαM) QM

0 0

(mB*c + mB*c*coshαM) (q*mB*c*sinhαM)

(q*mB*c*sinhαM) PM 0 * = (-mB*c + mB*c*coshαM) QM 0

(mB*c + i*ħ*tM - QB*VtM) N(-i*ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) PM 0 * = N(-i*ħ*(qx*xM+qy*yM+qz*zM)-QB*(qx*VxM+qy*VyM+qz*VzM)) (-mB*c + i*ħ*tM - QB*VtM) QM 0

3) Solution: mB*c*1M + i*ħ*1M/ctM - QB*VtM*1M N -i*ħ*4M/xM N -QB*VxM*4M N -i*ħ*k*4M/yM N -QB*VyM*k*4M N -i*ħ*3M/zM N -QB*VzM*3M = 0 mB*c*2M + i*ħ*2M/ctM - QB*VtM*2M N -i*ħ*3M/xM N -QB*VxM*3M N -i*ħ*-k*3M/yM N -QB*VyM*-k*3M N -i*ħ*-4M/zM N -QB*VzM*-4M = 0 mB*c*-3M + i*ħ*3M/ctM - QB*VtM*3M N -i*ħ*2M/xM N -QB*VxM*2M N -i*ħ*k*2M/yM N -QB*VyM*k*2M N -i*ħ*1M/zM N -QB*VzM*1M = 0 mB*c*-4M + i*ħ*4M/ctM - QB*VtM*4M N -i*ħ*1M/xM N -QB*VxM*1M N -i*ħ*-k*1M/yM N -QB*VyM*-k*1M N -i*ħ*-2M/zM N -QB*VzM*-2M = 0

4) 1M-fourth + (iħ/mBc)*1M-fourth/ctM - N(iħ/mBc)*4M-fourth/xM = 0 cosh(αM/2) + (iħ/mBc2)*(--i*M)*cosh(αM/2) - N(iħ/mBc)*(-i*NkxM)*-sinh(αM/2) = 0 cosh(αM/2) - coshαM*cosh(αM/2) + sinhαM*sinh(αM/2) = 0 0=0

184 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY -4M-fourth + (iħ/mBc)*4M-fourth/ctM - N(iħ/mBc)*1M-fourth/xM = 0 --sinh(αM/2) + (iħ/mBc2)*(--i*M)*-sinh(αM/2) - N(iħ/mBc)*(-i*NkxM)*cosh(αM/2) = 0 sinh(αM/2) + coshαM*sinh(αM/2) - sinhαM*cosh(αM/2) = 0 0=0 5) Car Visualization for Both Anti-matter and Spin: In the pair production event, a left-hand spin matter electron is modeled as a car made of matter with the steering wheel on the left. The car faces Dunkerque, France. In the same physical space at the instant of production a reverse-parity second car is also produced. This second car made of anti-matter also has the steering wheel on the left and faces Dunkerque, France. Any energy in excess of the rest energy of the two cars is equally applied as kinetic energy that pushes the matter car toward Dunkerque and the anti-matter car toward Dover. The steering wheel on the left of both cars means both cars have a left-hand spin, for zero total angular momentum (per the analogy of a nut and bolt that unscrew and separate, retaining their spin), just as there needs to be zero total linear momentum. The anti-matter car has reverse-parity. To un-reverse the reverse-parity of the antimatter car, flatten it front to back and go further to stretch it full length, so that the front points toward Dover, England, and see the steering wheel is on the right, as cars are in England. The right steering wheel means right-handed spin. Reverse parity has reverse spin. 6) 1M-second*i = -amp*sinh(αM/2)*exp(-i*(NkxM-m*xM - M-m*tM)) 4M-second*i = amp*cosh(αM/2)*exp(-i*(NkxM-m*xM - M-m*tM)) JtM = QB*(1M*i*1M + 2M*i*2M + 3M*i*3M + 4M*i*4M) = QB*amp2*(sinh2(αM/2) + cosh2(αM/2)) = QB*amp2*coshαM JxM = -QB*(1M*i*4M + 2M*i*3M + 3M*i*2M + 4M*i*1M) = -QB*amp2*(-2*cosh(αM/2)*sinh(αM/2)) = QB*amp2*sinhαM

185 WAVES JyM = 0 4J

JzM = 0

= QB*amp2*(1M*coshαM + qxM*sinhαM) = QB*amp2*1M*exp(qx*αM)

7) Solution not given. See text for the similar solutions. 8)

4-second

= e1M*1M + e4M*4M = e1M*1M-second + e4M*4M-second

= (e1M*-amp*sinh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM))) + (e4M*amp*cosh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM))) = (-e4M*qx*-amp*sinh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM))) + (e4M*amp*cosh(αM/2)*exp(i*(NkxM-m*xM - M-m*tM))) = e4M*amp*exp(qx*αM/2)*exp(i*(NkxM-m*xM - M-m*tM)) = e4M*amp*exp(qx*αM/2)*exp(i*(NkxM-m*xM - M-m*tM)) *exp(qx*αS/M/2)/exp(qx*αS/M/2) = e4S*amp*exp(qx*(αM + αS/M)/2)*exp(i*(NkxM-m*xM - M-m*tM)) = e4S*amp*exp(qx*αS/2)*exp(i*(NkxM-m*xM - M-m*tM)) = e4S*amp*exp(qx*αS/2)*exp(i*(NkxS-m*xS - S-m*tS)) = e1S*1S-second + e4S*4S-second Further Thought 1) Perform the Lorentz Transformation of the Fourth Dirac Spinor Solution with “αS/M = (i - jx)*/2” (for the transformation of timelike to space-like), and again with “αS/M = i*/2” (for sub-lightspeed motion of a Dirac Spinor wave crests and nodes) with a factor “-i” applied to the hyperbolic-radius, and then again with “αS/M = i*” (for anti-matter) with a factor “-1” applied to the hyperbolic-radius. Explain what the results represent physically and speculate if anything in nature fits that description.

186 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2) The equation that relates the current-density invariant to the Dirac Spinor invariant has the components of the Dirac Spinor invariant multiplied by each other. The implication is that Dirac Spinor space is a square root of geometric space. If this implication is valid, then other four-component invariants of our geometric-vector space should also be able to be expressed as a form of a square of Dirac Spinor space. Try to quantify this thought by creating a more general theory that relates Dirac Spinor space to geometric space. 3) Geometry of Dirac Spinor Space. The mathematical existence of the four “” components of the Dirac Equation suggest there is a translation from numbers to geometry that creates a physically real “” space. We expect some sort of physical reality to “” space because the intention of theory-development-algebra is to mathematically model the physics in all its intricate detail, and physics is real, a reality that we can suppose is geometrically real. In this book we ignore the possible physical reality of “” space per the excuse of the Process from Descartes, in which we do not revert to the geometry of step 3 until prepared to take a measurement. Forget the Process from Descartes and make a guess at what the geometry of “” Dirac Spinor Space is. How might the guess be verified by experiment? If there is a geometry applicable to Dirac Spinor “” space, then a translation to geometry is needed inside Dirac Spinor “” space. Think about this further after reading the next chapter in which electromagnetic field theory is combined into the Dirac Equation through the requirement that precision resolution is restricted to a finite value in geometry, any geometry, including Dirac spinor space.

187 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS

Chapter 5 – Proposed Theory The proposed Theory of Non-Finite Numbers replaces real numbers with “local-real numbers”. A set of local-real numbers is local to a finite maximum count of known or knowable place-value digits before and after the decimal point. •

Local-real numbers are derived from the new proposed

•

Real numbers are dropped after proving Cantor’s Continuum Hypothesis is incompatible with the proposed new axiom

•

Local-real numbers are applied using the Lorentz Transformation to create “the proposed Theory of Special Relativity with NonFinite Numbers”

•

The application is verified because electromagnetic force density is derived from the Dirac Equation

reciprocal-of-zero axiom

5.1 Local-Real Numbers Finite. A finite number results from addition, subtraction, multiplication and division (not by zero) operations that can be counted. An irrational number is finite because it is bounded by rational numbers. Three Dots and Six Dots. Three dots “…” represents a finite count using natural numbers. Three dots does not apply to the quantity of zeros after the decimal point for an integer because there are more zeros than can be counted, and that’s because wherever a count ends one more can be added. Six dots “……” is a new symbol to represent the quantity of zeros after the decimal point for an integer. To create an integer, the number is written in place-value digit form, and then the number is cut at the

188 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY decimal point and all contributions after the decimal point are discarded. This process, called truncation, is a bulk process in which place-value digits are not addressed individually, and therefore there is no count. “……” represents a quantity larger than “…”. Three dots represents a finite count that is indefinite in its magnitude, what has traditionally been called the finite potential infinity, “L”. In contrast, six dots is given the symbol “”. L=1+1+1+…

;

 = 1 + 1 + 1 + ……

Place-Value Digit Notation. A place-value digit notation number, for example “56.45” may be rewritten as a power series, “5*101 + 6*100 + 4*10-1 + 5*10-2”. In general …… + an*sn + … + a1*s1 + a0*s0 + a-1*s-1 + … + a-n*s-n + …… The outer two “……” represent a quantity that extends beyond what can be counted. It applies to the zeros after the decimal for an integer, and it applies to the non-pattern of an irrational number, for example, the square-root-of-two, “2 = 1.414213562373……”. At “< n” or “= n” the “a”’s all become zero up to “an”. Irrational Numbers. An irrational number is proven to not be rational. The example is “log23 = ln3/ln2 = 1.09861……/0.69314…… = 1.58496……”, as used in “21.5849625…… = 3”. Assume “log23 = p/q” with “p, q  N”. log23 = p/q ; q*log23 = p ; log2(3^q) = p ; 2^log2(3^q) = 2^p ; 3^q = 2^p An odd number “3^q” cannot equal an even number “2^p”. Therefore, “p, q  N” is impossible, and “log23 = p/q” is not rational. “q*log23 = p” and “3^q = 2^p” are valid if “p = q = 0”: “0*log23 = 0” and “3^0 = 2^0 = 1” to suggest a relationship between the irrational numbers “p/q = 0/0” and “log23”. The sequence of equations “log23 = p/q ; (1/p)*log23 = 1/q ; 3^(1/p) = 2^(1/q)” with “p = q = 0” gives “(1/0)*log23 = 1/0” and “3^(1/0) = 2^(1/0)”, to include the irrational number “1/0” in that relationship.

189 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS The New Reciprocal-of-Zero Axiom. Division by zero is prohibited per the definition of rational numbers. In contrast, division by zero has a relationship to irrational numbers. Because there is no division by zero in the ZFC axiomatic set theory on which algebra is based, we introduce an axiom called the new proposed reciprocal-of-zero axiom. •

No operation that includes a division by zero may possibly result in a non-zero finite number. Per the new axiom, we define properties for “1/0”:

•

.1. The selection of “0” in “1/0” being positive or else negative is unknown and unknowable

•

.2. “1/0” is so large it represents the never-ending feature of large number magnitude

•

.3. Permitted Operation: Addition of a finite number “q” to “1/0”. 1/0 + q = 1/0 + (0*q)/0 = (1 + 0*q)/0 = 1/0

•

qQ

.4. Prohibited Operation: Addition of “1/0” to another “1/0”. Each “1/0” has an independent selection of being positive or negative. 1/0 + 1/0 = 1/0 - 1/0 ≠ a result

•

.5. Permitted Operation: Distributive property of multiplication over addition using “1/0” as the common factor but with the condition the two “1/0”s on the right have dependent (the same) positive or negative property. (1/0)*2 = (1/0 + 1/0) Both “0”’s have the same positive or negative (1/0)*(a + b) = (1/0)*a + (1/0)*b a, b  Q a, b ≠ 0

•

.6. Permitted Operation: Distributive property for the addition of zero. It requires “0/0” not equal “1/0”. (1/0)*(1 + 0) = (1/0) + (1/0)*0 = 1/0 + 0/0 = 1/0

qQ

190 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •

.7. Permitted Operation: Multiplication of finite “q” to “1/0”. (1/0)*q = q/0 = 1/(0/q) = 1/0

•

q  Q and q ≠ 0

.8. Prohibited Operation: Multiplication of zero “0” by “1/0”. (1/0)*0 = 0/0 ≠ a result

•

.9. Permitted Operation: Multiplication of “1/0” to another “1/0”. (1/0)*(1/0) = 1/0

•

.10. Permitted Operation: “1/0” base to “n” exponent. (1/0)^n = 1/0

•

nN

(n = 0 is Prohibited)

.11. Permitted Operation: Reciprocal of “1/0”. (1/0)^(-1) = 1/(1/0) = 0

•

.12. Permitted Operation: Base “n” to “1/0” exponent. The visualization is “n*n*n*n*……” with the count represented by “……” being “1/0”, in analogy to “1 + 1 + 1 + ……” being “1/0”. n^(1/0) = 1/0 or else 0 with the selection unknown and unknowable n^(-1/0) = 0 or else 1/0 with selection unknown and unknowable n  N or n = e

•

.13. Permitted Operation: Logarithm of zero. 1/0 = -logn(0) or else +logn(0) with the selection being unknown and unknowable n  N or n = e

•

.14. Permitted Operation: Logarithm of an exponent. logn(n^1/0) = (1/0)*(logn(n)) = (1/0)*1 = 1/0 , n  N or n = e

191 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS •

.15. Permitted Operation: A finite “n” cannot equal “1/0”. nN

0 < n < 1/0 and n ≠ 1/0 and 1/0 < -n < 0 •

.16. Prohibited Operation: The inequality of “1/0” to itself. “1/0 + q = 1/0 + (0*q)/0 = (1 + 0*q)/0 = 1/0”. Therefore,“1/0” cannot be greater than or less than “1/0”.

(n = e” means “exp(x)” applies, not “e^x”.) An algebra specifies a logical system that always gives the same calculated result. If a violation is found, then insert another prohibition to avoid it. +

0

Q

1/0

*

0

Q

1/0

0

0

Q

1/0

0

0

0

No

Q

Q

2*Q

1/0

Q

0

Q^2

1/0

1/0

1/0

1/0

No

1/0

no

1/0

1/0

-

0

Q

1/0

/

0

Q

1/0

0

0

-Q

1/0

0

no

0

0

Q

Q

0

1/0

Q

1/0

1

0

1/0

1/0

1/0

No

1/0

1/0

1/0

No

^

0

N

1/0

log

0

n

1/0

0

1

0

no

0

No

0?

no

n

1

n^n

1/0 or 0

N

1/0 or -1/0

1

-1/0 or 1/0

1/0

no

1/0

no

1/0

No

no

no

Table 4. Crude non-number algebra operations involving “1/0” (“Q  Q”).

Applying the Crude Non-Number Algebra of “1/0” to “log23”. Use the power series of “log23” to evaluate “(1/0)*log23 = 1/0”. Per the six dots, the tail-end terms are multiplied by “1/0” in one bulk operation to a sum result of “1/0”. Therefore, “(1/0)*log23 = 1/0”. There is no end to the

192 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY place-value digits of “log23” after the decimal point, and there is no end to the magnitude of “1/0”. Because “1/0” cannot be less than “1/0”, the two concepts counter each other so that “(1/0)*log23 = 1/0” is valid, such that the “1/0” on the right does not have a contribution of place-value digits after the decimal point. Also, per the new axiom, “3^(1/p) = 2^(1/q)” is valid for “1/p = 1/q = 1/0” but not for smaller values of “1/p” and “1/q”. Because the six dots “……” has been used to represent the quantity of non-pattern place-value digits for an irrational number, the six dots represent “l/0”, “ = 1 + 1 + 1 + …… = 1/0”. Before we had the new proposed reciprocal-of-zero axiom, we had no formal means of stating in a positive sense that an irrational number had no end to its non-repeating pattern. Equations for proven irrational numbers, “q*log23 = p”, “q*2 = p”, “q* = p”, and “q*e = p”, are each satisfied only if “q” and “p” equal “l/0” and nothing less, except “0”. The association of “1/0” with irrational numbers per the six dots is theoretical because, in our geometric, physical world, we are only able to put measurement to finite quantities, per the three dots. Only a rational approximation can be realized. Base Two Representation of Numbers rather than base ten is preferred. •

Base two has only ones and zeros for the place-value digits

•

The quantity of sets “P(a)” created from the quantity of possible members “a” in a set is found using a base of two, “P(a) = 2^a”

•

Base two has “1 + 20 + 21 + 22 = 23 = 8”

Truncated Numbers. A power series representation is divided into two portions. •

The first portion is a known or knowable rational number, called a truncated number, identified with the three dots to a count of “Lmax  N” place-value digits before and after the decimal point

193 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS •

The second portion, imprecision term “” (xi), is the unknown and unknowable portion split between what is larger and smaller than the rational number, as identified with the six dots

Truncated Numbers “TN”: •

Written in base two and positive or negative or zero and is specific to the “Lmax” count of possibly non-zero place-value digits before and after the decimal point, and “TN  Q”

•

“1/(2^Lmax)” is the smallest, “2^Lmax” the largest, and the quantity is “((2^Lmax)*(2^Lmax) + 1)*2 - 1”

For “Lmax = 0” the truncated numbers are “-1”, “0”, and “1”. The eight (“2^Lmax = 8”) truncated numbers between zero and one for “Lmax = 3” are on the left. The nine “((2^Lmax)*(2^Lmax) + 1)*2 - 1 = 9” truncated numbers for “Lmax = 1” are on the right. 0.1110000…… 0.1100000…… 0.1010000…… 0.1000000…… 0.0110000…… 0.0100000…… 0.0010000…… 0.0000000……

10.0…… 1.1…… 1.0…… 0.1…… 0.0…… -0.1…… -1.0…… -1.1…… -10.0……

A rational number with a denominator not a power of two has a non-zero repeating pattern and is not in a set of truncated numbers. Local-Zero, “AminA0”, is the unknown and unknowable place-value digits smaller than a truncated number. •

All zeros before the decimal point and all zeros after the decimal point to a finite count of “Lmax” (in base two)

194 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •

“d = b - b” for each place-value digit after a count of “Lmax” after the decimal point (in base two) AminA0 = 0.000…(zeros to a count of Lmax)…000ddddd…… AminA0 = 0.000dddd…… for Lmax = 3 AminA0 = 0.dddd…… for Lmax = 0

“b” represents “box” (or “both”). We do not know if the box contains a “0” or a “1” until the box is opened. An unopened box is both a “0” and a “1” and is not a “0” or a “1” separately. “b” is the mathematical analogy for the box containing Schrödinger’s Cat. Until we open the box, we do not know if Schrödinger’s Cat is dead or alive. Inside the box is a radioactive nucleus. If the nucleus decays, then the emitted particle breaks a container of poison and the cat dies. The decay both happened (state = “1”) and did not happen (state = “0”) until observed by the box being opened. That is the condition of each “b”. The value of “b” is both one and zero, and the value of “b” is neither-separately one nor zero until the place-value digit is observed, after which it becomes a one or else a zero. Each place-value digit symbol “b” is a separate box that is opened independently of any other box. “AminA0 = 0.dddd……” is the result of “0minB = 0.bbbbb……” subtracted from “0minC = 0.bbbbb……”. Because both “0minC” and “0minB” are positive, their difference “AminA0 = 0minC - 0minB” can be either positive or negative, depending on the first non-zero value of “d” in the sequence. “A” represents the concept of positive and negative, both and neitherseparately. “minA0” has the “min” to emphasize it is smaller than truncated numbers. Local-Infinity. There is a “0.25” chance a “d” will be “-1 (= 0 - 1)”, “0 (= 0 - 0)”, “0 (= 1 - 1)”, or “1 = (1 - 0)”. Therefore, a “0.5 = 1/2” chance the first “d” will become a zero “0” and “0.52 = 1/(22) = 0.25” chance, both it and the next “d” will become a zero. The quantity of “d”’s has no end, for “1/(21/0)”, which equals either “0” or “1/0” using the proposed new reciprocal-of-zero axiom. “0” is the relevant solution. Because a local-zero cannot equal zero, a division reciprocal, called a local-infinity, “AmaxA = 1/AminA0”, exists. It is larger in magnitude than a truncated number. In the general power series given above, it is the six dots on the left.

195 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS Local-Real Numbers. A local-real number consists of a truncated number added to an imprecision term “” (xi), which is either a local-zero or else a local-infinity. “C” is used because “A” pertains to “” being either “CmaxC” (“A” “+”) or else “CminC0” (“A” “-”). Continuum quantities in the geometrically real world are local-real numbers.  = CmaxC or else CminC0 Two physically real quantities cannot be equal. A geometric continuum quantity, for example, length, subtracted from a geometrically independent version of itself, does not equal zero, and that means the four sides of a unit length square are not the ideal of the integer one, but, rather, each have a local-zero adder for finite imprecision. And, a “2” diagonal is not perfectly “2” because it has a local-zero adder. The local-zeros added to each of two sides of a unit length square are proven to be different: There is a finite chance the first “d” placevalue digit will be the same between the two local-zeros, and that quantity squared both it and the next “d” will be the same for zero chance for all the “d”’s. A “1” followed by a “-1” has the same result as a “0” followed by a “1”, and that is accounted for. Continuum. There is an even chance a local-real number with placevalue digits “d” randomly selected to be “1”, “0”, “0”, or “-1” will become any (irrational) number. Because there is an even chance, the local-real numbers form a “continuum”. This is a different definition of “continuum” when compared to the continuum in Cantor’s theory of infinite sets for real numbers, but the concept is the same, and, therefore, the same word “continuum” is used. No Positive Actual Infinity. Local-real numbers have been defined without the use of a positive actual infinity. The positive actual infinity, as a count of place-value digits after the decimal point, was proposed by Cantor as required for real numbers. Therefore, local-real numbers are different from real numbers.

196 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

5.2 Cantor’s Theory of Infinite Sets Cantor made the formal assumption that a positive infinity was the quantity of members in the set of natural numbers and a larger positive infinity was the quantity of members in the set of real numbers. The two infinities were forced to be actual infinities and not large finite number potential infinities because, per his Continuum Hypothesis, no set can have a quantity of members between the two infinities. Cantor proposed his theory of infinite sets late in the 1800’s, several years before 1905, when Einstein developed his theory of Special Relativity. Special Relativity, like every other mathematical model of physics, does not directly use Cantor’s two infinities. The disconnect between Cantor’s infinities and Special Relativity prompted the author’s search for a proper replacement in applied mathematics for the positive actual infinity. Cantor’s infinity theory was critically important. •

It formed the foundation for evolving more theory on how numbers are constructed

•

It showed the importance of defining numbers using place-value digits as opposed to ratios (for the rational numbers) or nebulously (for the irrational numbers)

•

It used base two to an exponent, and that suggested a switch to base two place-value digit nomenclature

•

It placed the positive actual infinity into a theory against which alternative theories of infinity could be contrasted

•

It generated plenty of literature so that several perspectives on the theory could be found by reading, rather than by inventing

•

It became an axiom in set theory, which suggested an alternative to Cantor’s Continuum Hypothesis would be an alternative axiom

•

It lacked an algebra by which to calculate with the positive actual infinities

197 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS •

It was absent from mathematical models of physics and that meant it probably wasn’t real and highlighted that an improvement in the theory of infinity was needed

Cantor’s Theory of Infinite Sets – Countable Sets. The positive actual infinity “N0” (aleph null) is the quantity of members in “N”. N = {1, 2, 3, …, N0} Cantor counted integers to prove the set of integers “Z” has the same quantity “N0” as does the set of natural numbers “N”: “0” was first, “1” was second, “-1”, was third, “2” was fourth, “-2” was fifth, etc. The person reading the proof must extrapolate from a finite count to an actual infinity quantity. The word “countable” means the members of the set can be identified one-to-one with the set of natural numbers. Examples of countable sets are integers, prime numbers, rational numbers (per Cantor’s diagonal proof), squares of natural numbers, even numbers, and products of two integers. The set of real numbers is not “countable”. Dedekind Cut. The Dedekind Cut was the precedent to Cantor’s theory of infinite sets. The Dedekind Cut is best described by Dedekind himself in a quote from Essays on the Theory of Numbers by Richard Dedekind, 1963 (originally 1901) by Dover Publications, Inc. Page 15: “From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts.” If the cut is at a rational number, then the high side number is the rational number and the low side number is infinitely close, but just less. Alternatively, if the cut is at an irrational number, then both the high side and low side numbers are infinitely close to each other, and the irrational number is between them. Per the definition of the Dedekind Cut, we ignore the removed material of a saw cut, and we pretend the cut is a scissors cut (which does not remove material) by making the two numbers “essentially” the same number.

198 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY A true scissors cut would be an idealized cut because the high and low numbers differ by the integer zero. It appears Dedekind was attempting to make the cut non-idealized, as if the cut were physical in our geometric world where the number-line exists as an actual line. To visualize the Dedekind cut, use “Lmax = 3” for the count of base two place-value digits after the decimal point. A Dedekind Cut at half (“.100” in base two) has “.100” on the high side and “.100 - 2^-Lmax = .100 - .001 = .011” on the low side. To satisfy the word “essentially”, the difference “2^-Lmax” must be smaller than a rational number, and that requires “Lmax” to be a positive actual infinity, “N0”, for “2^- N0”. This contrasts with the idealized extreme case of “Lmax = 1/0” for which “2^-(1/0) = 0” and both numbers are “.100”. A Dedekind Cut at the irrational number square-root-of-two “2 = 1.4142135……” has high/low number pairs at the cut (1.4, 1.5), (1.41, 1.42), (1.414, 1.415), (1.4142, 1.4143), etc., for the largest count number “Lmax” increasing through “1”, “2”, “3”, “4”, etc., respectively. When “Lmax” reaches an actual infinity “N0”, then the high and low numbers and all numbers between them, including the irrational number squareroot-of-two, are essentially the same number, because the span between the low and high numbers is the positive infinitesimal “10^-N0”. If sequential rational numbers “1/Lmax” and “1/(Lmax - 1)” differ by “1/(Lmax*(Lmax - 1))”, then a difference “2^(-Lmax)” of sequential real numbers is smaller, and more so if “2^(-Lmax)” becomes “2^-N0”. Uncountable Sets. The “cardinality” of a set is the quantity of members in the set. The cardinality is given operator notation of two “|” lines. A textbook for Cantor’s theory of infinite sets is Mathematical Proofs, A Transition to Advanced Mathematics, Second Edition by Chartrand, Polimeni, and Zhang, Pearson Addison Wesley, 2008, beginning on Page 221. From the top of Page 236: “Indeed, if A is any denumerable set, then |A| = N0. The set R of real numbers is also referred to as the continuum and its cardinality is denoted by c. Hence |R| = c and from what we have seen, N0 < c. It was the German mathematician Georg Cantor who helped to put the theory of sets on a firm foundation.” Cantor proved “N0 < c” using a quantity “c” of real numbers spanning from zero (inclusive) to one (exclusive) and using a quantity

199 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS “N0” of place-value digits after the decimal point. He created several different real numbers by writing random place-value digits after the decimal point. 0.5928609813… 0.5232… 0.7290165316… 0.3991… 0.7831994831… 0.8482401809… Cantor took the first place-value digit from the first number, the second from the next, and so on, to form a new number. And he varied the process to create other new numbers. Because new numbers could be formed by this process, he concluded “N0 < c”. The person must extrapolate from finite to infinite. As a refinement to “N0 < c”, Cantor proposed “c = 2^N0”. To visualize “c = 2^N0”, substitute “Lmax = 3” for “N0”. The set of three members “{.1, .01, .001}” implies eight sets per “8 = 23”: {}, {.1}, {.01}, {.001}, {.1, .01}, {.1, .001}, {.01, .001}, {.1, .01, .001} There are “2 = 2^1” numbers “0.0” and “0.1” if we consider only the first place-value digit after the decimal point. “4 = 2^2” numbers “0.00”, “0.01”, “0.10” and “0.11” for the first two. “8 = 2^3” as given above for the first three. “2^Lmax” numbers for the first “Lmax”. This visualization requires base two, and that’s why base two is in “c = 2^N0”. “c = 2^N0” as a quantity of real numbers applies to the span from negative infinity “-N0” to positive infinity “N0”, in analogy to “N0” applying to any countable set. Properties of the two Actual Infinities. From Cantor’s proofs, we identify the following properties for “N0” and “c”. Finite numbers satisfy these properties. •

“N0” and “c” have no contribution after the decimal point, in analogy to the natural numbers used in counting

•

“N0” and “c” are positive, also in analogy to the natural numbers used in counting, and as required by “N0 < c” and “c = 2^N0”

•

“N0 < c” per Cantor’s proof, and “c = 2^N0”

200 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Perhaps these other properties also apply: •

“c” is an even number because of the equation “c = 2^N0”, by furthering the analogy of Cantor’s infinities with finite numbers

•

“0 < N0 < c < 1/0”, “N0 ≠ 1/0” and “c ≠ 1/0” if we accept the proposed new reciprocal-of-zero axiom

Attempt to place a “1” at the “N0”th position before the decimal point by writing an ever-increasing string of zeros to form “c = 2^N0”. Per this visualization, “c = 2^N0” cannot be negative and therefore cannot equal “1/0”. Analogously, “1/c = 2^-N0” cannot equal zero. Cantor’s Continuum Hypothesis. Included in Cantor’s theory of infinite sets is the conjecture that there is no set “S” for which N0 < |S| < c The inequality “N0 < |S| < c” states the quantity of members in a set may equal “N0”, as applies to countable sets, and may equal “c”, as applies to uncountable sets, but the quantity of members in a set cannot be “|S|” between “N0” and “c”. There cannot be a count up from “N0” toward “c” because any number in a count is the maximum number in the set of numbers from one to that number. There is no “N0 + 1”, “2*N0”, “c/2”, or “c - 1”. The Continuum Hypothesis (that “N0 < |S| < c” is impossible) cannot apply if “N0” and/or “c” are finite. Therefore, it is the Continuum Hypothesis that creates mathematically the property of positive actual infinity for “N0” and for “c”. And, it is the Continuum Hypothesis that justifies the multi-set-applicability property of “N0” and of “c”. The Continuum Hypothesis is only a conjecture because it has not been proven. Attempts have been made. In the referenced textbook (middle of page 236) is a description of the attempts: “However, in 1931 the Austrian mathematician Kurt Gödel proved that it was impossible to disprove the Continuum Hypothesis from the axioms on which the theory of sets is based. In 1963 the American

201 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS mathematician Paul Cohen took it one step further by showing that it was also impossible to prove the Continuum Hypothesis from these axioms. Thus the Continuum Hypothesis is independent of the axioms of set theory.” An almost identical quote is given on Page 137 of THE

PHILOSOPHY OF SET THEORY, An Historical Introduction to Cantor’s Paradise, by Mary Tiles, from Dover Publications, Inc, 2004 (Originally

1989 from Basil Blackwell Ltd.). A very readable summary of the effect the Continuum Hypothesis has had on axiomatic set theory is given in the article “Dispute over Infinity Divides Mathematicians – To determine the nature of infinity, mathematicians face a choice between two new logical axioms. What they decide could help shape the future of mathematical truth” by Natalie Wolchover, Quanta Magazine, December 3, 2013. The “axioms of set theory” are listed in Mary Tiles’ book, Pages 121-123.

Axiom of extensionality Null set axiom Pair set axiom Sum set axiom Axiom of infinity

Axiom of foundation Subset axiom and replacement axiom Power set axiom Power Axiom of choice

Per page 125, the axiom of infinity identifies the unboundedness of natural numbers because one can always be added. Unboundedness pertains to the finite potential infinity (defined per Aristotle) and not to the completed (defined per Aristotle) or actual infinity “N0”. Rather, “N0” is created when the Continuum Hypothesis is used as an axiom, as is suggested as okay by the two proofs (by Gödel and by Cohen). The book and article emphasize that axiomatic set theory with the actual infinity forms “holes” that pure mathematicians attempt to close. Applying Binary Operations to an Actual Infinity. If the calculation “c = 2^N0” is accepted, then, it follows “N0 = ln(c)/ln(2)” should be, too. But equations such as “N0*ln(2) +  = 1 + 1/2 + 1/3 + … + 1/(c - 1)” require a violation of the Continuum Hypothesis and suggest any attempt at an algebra will fail.

202 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY In the tables below, binary operations consistent with the Continuum Hypothesis are on the left, and binary operations consistent with finite numbers are on the right. If, on the right-side tables, the finite numbers “N0” and “c” are both replaced by “1/0”, then the crude nonnumber algebra applies. + 0 Q N0 C

0 0 Q N0 C

Q Q 2*Q N0 C

N0 N0 N0 N0 c

C C C C C

+ 0 Q N0 c

0 0 Q N0 c

Q Q 2*Q N0+Q c+Q

N0 N0 N0+Q 2*N0 c+N0

c c c+Q c+N0 2*c

* 0 Q N0 C

0 0 0 0 0

Q 0 Q*Q N0 C

N0 0 N0 N0 c

C 0 C C C

* 0 Q N0 c

0 0 0 0 0

Q 0 Q*Q N0*Q c*Q

N0 0 N0*Q N0^2 c*N0

c 0 c*Q c*N0 c^2

^ 0 2 N0 c

0 1 1 1 1

2 0 2^2 N0 C

N0 0 c ? ?

C 0 2^c ? ?

^ 0 2 N0 c

0 1 1 1 1

2 0 2^2 N0^2 c^2

N0 0 2^N0 N0^N0 c^N0

c 0 2^c N0^c c^c

Table 5. A comparison of binary operations. Cantor’s Continuum Hypothesis is on the left. In contrast, actual infinities being finite are on the right (“Q  Q”).

The table does not include inverse operations because “log2(N0)” (in analogy to “log2(c) = N0”) does not seem to have a result. The tables for actual infinities are not sufficient for calculations. An algebra might exist, but we presently do not know the rules of that algebra. Higher Order Actual Infinities. Cantor replaced “c” with “N1” (aleph one) because “N1 = 2^N0” suggests “N2 = 2^N1”, “N3 = 2^N2”, …. Is there an actual infinity subscript? Perhaps that question illustrates the holes.

203 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS Incompatibility of “N0” with Irrational Numbers. If “2^c” is even and “3^c” is odd, then “log23” is not in the set of real numbers. That result was expected because in the discussion on the Dedekind cut an irrational number resided inside a “2^-N0” interval, and the Dedekind cut appears to be the basis of Cantor’s “c = 2^N0” real numbers over an interval. Because an irrational number resides inside a “1/c” interval, “c*log23 ≠ c”, by which is stated a positive number “c” (without contribution after the decimal point) cannot be multiplied by “log23” to equal any other “c” (also without contribution after the decimal point), and that’s because the nonpattern for “log23” cannot end at a positive quantity “N0” of place-value digits (per the proposed new reciprocal-of-zero axiom by which “log23” has more place-value digits after the decimal point than “c” has before the decimal point). Because irrational numbers are removed from real numbers, infinities “N0” and “c” lose relevance and the Continuum Hypothesis does not apply. From another perspective: Because “N0” and “c” are positive, we can speculate “0*N0 = 0” and “0*c = 0”, and 1/0 + N0 = 1/0 + 0*N0/0 ; 1/0 + c = 1/0 + 0*c/0 = (1 + 0*N0)/0 = (1 + 0)/0 = 1/0 = (1 + 0*c)/0 = (1 + 0)/0 = 1/0 Replacing Real Numbers with Local-Real Numbers. In our geometric world of the number-line, there is no “1/0”, and so no irrational numbers, and, per the argument above, there is no positive actual infinity “N0”, and so no real numbers, either. What remains is the finite potential infinity “Lmax” and the proposed local-real numbers. But, to dispose of the positive actual infinity “N0”, its mysteriousness had to be tamed by defining it using finite number properties, after which it was proven incompatible with a proposed new axiom. The properties and even the axiom could be over-simplified or otherwise in error, and so, it’s inappropriate to celebrate. Instead, we honor Cantor because his innovations provided necessary historical precedent. And, most humbling, we expect another new theory to come along soon because the proposed new axiom intentionally ignored identified complexity at infinity (see a “Further Thought” question) and did so per the excuse we only needed enough self-consistency for some fairly mundane applied mathematics.

204 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Infinity in Mathematical Models of Physics. Finite imprecision of localreal numbers applies where a local-infinity replaces a division by zero in a mathematical model of physics. Motion at the speed-of-light is a division-by-zero singularity in Special Relativity. But first we need to place the local-infinity into an algebra field, or, at least, have defined enough properties to use it.

5.3 Algebra Field for Local-Real Numbers Finished-Calculation and Final-Result. Local-real numbers with finite precision pertain to the geometric world of continuum quantities in which the problem to be solved is set up. The problem is then translated into an all number algebra after which operations are not inhibited by the finite precision. In preparation for a measurement, the results are translated back into geometry and restricted to the finite precision. •

Finished-Calculation. Find the “finished-calculation” at the end of the all-number second step in the Process from Descartes

•

Final-Result. Impose finite precision to find the geometric “finalresult” in the third step in the Process from Descartes

For example: “AminA0” times “AminA0” has “Lmax” zeros after the decimal point in each of the two factors and has “2*Lmax + 1” zeros in the positive product as the finished-calculation. The translation into a final-result is limited to “Lmax” zeros and is written as another local-zero “CminC0”. Probability Functions. Swap “maximum” for “minimum” for “Lmax = 0”. AmaxA0 = 0.dddd……

AminA = 1/AmaxA0

Probability Distribution for “AmaxA0”. For “s = AmaxA0”, the calculated probability distribution “dPmax0/ds” is given below.

205 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS dPmax0/ds = 0 dPmax0/ds = s + 1 dPmax0/ds = 1 - s dPmax0/ds = 0

Pmax0 = 0 Pmax0 = (s + 1)2/2 Pmax0 = (1 + 2*s - s2)/2 Pmax0 = 1

for s < -1 for -1 < s < 0 for 0 < s < 1 for 1 < s

Probability Curve for the Local-Infinity. “s = 1/AminA0” dPmin/ds(s) = s-2 + s-3 dPmin/ds(s) = 0 dPmin/ds(s) = s-2 - s-3

Pmin(s) = -s-1 - s-2/2 Pmin(s) = 1/2 Pmin(s) = 1 - s-1 + s-2/2

Probability Curve for the Quotient. “s = AminA0/BminB0” Pmin0/min0 = -1/(3*s) - 1/(12*s2) Pmin0/min0 = 1/2 + s/3 + s2/12 Pmin0/min0 = 1/2 + s/3 - s2/12 Pmin0/min0 = 1 - 1/(3*s) + 1/(12*s2)

for s < -1 for -1 < s < 0 for 0 < s < 1 for 1 < s

dPmin0/min0/ds = 1/(3*s2) + 1/(6*s3) dPmin0/min0/ds = 1/3 + s/6 dPmin0/min0/ds = 1/3 - s/6 dPmin0/min0/ds = 1/(3*s2) - 1/(6*s3)

for s < -1 for -1 < s < 0 for 0 < s < 1 for 1 < s

Figure 35. Probability Distributions.

for - < s < -1 for -1 < s < 1 for 1 < s < 

206 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Figure 36. On the right side is the Probability Distribution for the ratio of two local-zeros, “s = AminA0/BminB0”.

Base Two Exponential Function. Assuming only a real number result, “2^(AmaxA)” is calculated by counting zeros as place-value digits left and right of the decimal point, in base two: “ > 2^Lmax” (lambda) for “A” “+”, and “1/ < 2^-Lmax” for “A” “-”, both outside truncated numbers. 2^(AmaxA) = *(1/2 + A1/2) + (1/)*(1/2 - A1/2) The final-result of “2^(AmaxA)” is the truncated number zero added to imprecision term “” (xi). “” equals either a local-infinity “CmaxC” or else a local-zero “CminC0 = 1/CmaxC” depending on “A” being “+” or “A” “-”, respectively. A local-real number is a truncated number with “” added to it. Also: exp(AmaxA) = ()*(1/2 + A1/2) + (1/)*(1/2 - A1/2)

207 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS Imprecision Term “” for Special Relativity. “αM” of “αM = atanh(vM/c)” is a truncated number to which “” is added using a Lorentz Transformation. A hyperbolic-radius is also a truncated number to which “” is added. Both are included in “αS/M” using the “(1 - qx)” factor. αS/M = (1 - qx)* = (1 - qx)*CmaxC or else = (1 - qx)*CminC0

First Case Second Case

First Case pertains to “CmaxC”. For “C” “+”, “CmaxC” becomes “calcC”. exp(qx*αS/M) = exp(qx*(1 - qx)*) = exp(qx*(1 - qx)*CmaxC) First Case = exp(qx*(1 - qx)*calcC) “C” positive, “+” = exp((-1 + qx)*calcC) = exp(-calcC)*exp(qx*calcC) = exp(-calcC)*(cosh(calcC) + qx*sinh(calcC)) = exp(-calcC)*(exp(calcC) - sinh(calcC) + qx*sinh(calcC)) = exp(-calcC)*(exp(calcC) - (1 - qx)*sinh(calcC)) = 1 - (1 - qx)*exp(-calcC)*sinh(calcC) = 1 - (1 - qx)*exp(-calcC)*(exp(calcC) - exp(-calcC))/2 = 1 - (1 - qx - (1 - qx)*exp2(-calcC))/2 = (1 + qx + (1 - qx)*exp2(-calcC))/2 = (1 + qx)/2 + (1 - qx)*exp2(-calcC)/2  (1 + qx)/2 “exp(qx*(1 - qx)*calcC)  (1 + qx)/2” conforms to an exponent with a singular matrix having a unit magnitude result. Division Reciprocal (which is, effectively, “C” negative, “-”): 1/exp(qx*αS/M) = exp(-qx*αS/M) = exp(-qx*(1 - qx)*) = exp(-qx*(1 - qx)*calcC) = exp((1 - qx)*calcC) = exp(calcC)*exp(-qx*calcC) = exp(calcC)*(cosh(-calcC) + qx*sinh(-calcC)) = exp(calcC)*(exp(-calcC) - sinh(-calcC) + qx*sinh(-calcC)) = exp(calcC)*(exp(-calcC) - (1 - qx)*sinh(-calcC))

208 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = 1 - (1 - qx)*exp(calcC)*sinh(-calcC) = 1 - (1 - qx)*exp(calcC)*(exp(-calcC) - exp(calcC))/2 = 1 - (1 - qx - (1 - qx)*exp2(calcC))/2 = (1 + qx)/2 + (1 - qx)*exp2(calcC)/2  (1 - qx)*exp2(calcC)/2 “exp(-qx*(1 - qx)*calcC)  (1 - qx)*exp2(calcC)/2” is a reciprocal of the portion of “exp(qx*(1 - qx)*calcC)” that deviated away from “(1 + qx)/2”. “((1 + qx)/2)*exp(qx*α) = ((1 + qx)/2)*exp(α)” applied. Second Case pertains to “CminC0”. For “C” “+”, “CminC0” becomes “0calcC”. exp(qx*αS/M) = exp(qx*(1 - qx)*) = exp(qx*(1 - qx)*CminC0) Second Case = exp(qx*(1 - qx)*0calcC) “C” positive, “+” = exp((-1 + qx)*0calcC) = exp(-0calcC)*exp(qx*0calcC) = exp(-0calcC)*(cosh(0calcC) + qx*sinh(0calcC)) = exp(-0calcC)*(exp(0calcC) - sinh(0calcC) + qx*sinh(0calcC)) = exp(-0calcC)*(exp(0calcC) - (1 - qx)*sinh(0calcC)) = 1 - (1 - qx)*exp(-0calcC)*sinh(0calcC)  1 - (1 - qx)*(1 - 0calcC)*0calcC 1 Division Reciprocal (which is, effectively, “C” negative, “-”): 1/exp(qx*αS/M) = exp(-qx*αS/M) = exp(-qx*(1 - qx)*) = exp(-qx*(1 - qx)*0calcC) = exp((1 - qx)*0calcC) = exp(0calcC)*exp(-qx*0calcC) = exp(0calcC)*(cosh(-0calcC) + qx*sinh(-0calcC)) = exp(0calcC)*(exp(-0calcC) - sinh(-0calcC) + qx*sinh(-0calcC)) = exp(0calcC)*(exp(-0calcC) - (1 - qx)*sinh(-0calcC)) = 1 - (1 - qx)*exp(0calcC)*sinh(-0calcC)  1 - (1 - qx)*(1 + 0calcC)*(-0calcC) 1 “exp(qx*αS/M)  1” and “1/exp(qx*αS/M)  1”.

209 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS

Fields and Waves use a Lorentz Transformation with division by two in the argument. First Case Finished-Calculation: exp(AmaxA) = calcC “C” “+” exp(qx*αS/M/2) = exp(qx*(1 - qx)*calcB/2) First Case, “C” “+” = (1 + qx)/2 + (1 - qx)*exp2(-calcC/2)/2  (1 + qx)/2 1/exp(qx*αS/M/2) = exp(-qx*(1 - qx)*calcC/2) = (1 + qx)/2 + (1 - qx)*exp2(calcC/2)/2  (1 - qx)*exp2(calcC/2)/2 exp(qx*αS/M/2) = exp(qx*(1 - qx)*0calcC/2) Second Case, “C” “+” = 1 - (1 - qx)*exp(-0calcC/2)*sinh(0calcC/2)  1 - (1 - qx)*(1 - 0calcC/2)*0calcC/2  1 1/exp(qx*αS/M/2) = exp(-qx*(1 - qx)*0calcC/2) Second Case, “C” “+” = 1 - (1 - qx)*exp(0calcC/2)*sinh(-0calcC/2)  1 + (1 - qx)*(1 + 0calcC/2)*(-0calcC/2)  1 Criteria of a Group. Finished-Calculations use group theory as it pertains to rational numbers. The inverse operations do not apply to the finalresult due to deletion of the portion of the finished-calculation that was smaller and larger than truncated numbers.

5.4 Lorentz Transformation with Non-Finite Numbers Lorentz Transformation for Proposed Theory of Special Relativity with Non-Finite Numbers. Imprecision term “” is added to truncated number “αM” using “αS/M = (1 - qx)*” in “αS = αM + αS/M”. The “(1 - qx)” factor makes First Case components finite. αS/M = (1 - qx)*

;

vS/M/c = tanhαS/M = tanh((1 - qx)*)

210 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY vS/M-FirstCase/c = tanh((1 - qx)*calcC) First Case, “C” “+” = (tanh(calcC) - qx*tanh(calcC))/(1 - qx*tanh(calcC)*tanh(calcC)) = ((1 - qx)*tanh(calcC))/(1 - qx*tanh(calcC)*tanh(calcC))  (1 - qx)*(1 - exp2(-calcC))/(1 - qx*(1 - exp2(-calcC))2)  (1 - qx)*(1 - exp2(-calcC))/(1 - qx*(1 - 2*exp2(-calcC))) 1 “vS/M-FirstCase/c” equals one minus a number smaller in magnitude than the smallest positive truncated number “2^-Lmax” so that the finalresult equals one plus a local-zero. It means the First Case bus “M” moves at the speed-of-light. Our visualization is the photon. vS/M-SecondCase/c = tanh((1 - qx)*0calcC) Second Case, “C” “+” = (tanh(0calcC) - qx*tanh(0calcC))/(1 - qx*tanh(0calcC)*tanh(0calcC)) = (1 - qx)*tanh(0calcC)/(1 - qx*tanh(0calcC)*tanh(0calcC))  (1 - qx)*0calcC/(1 - qx*0calcC*0calcC)  0 “vS/M-SecondCase/c” final-result equals a local-zero for no motion. The bus is not moving. Our visualization is the electron. General Form of the Lorentz Transformation. 2r

= 1M*(c*tB)*exp(qx*αM) = 1M*(c*tB)*exp(qx*αM)*1 = 1M*(c*tB)*exp(qx*αM)*exp(qx*αS/M)/exp(qx*αS/M) = 1M*(c*tB)*exp(qx*αM) *exp(qx*(1 - qx)*)/exp(qx*(1 - qx)*) = (1M/exp(qx*(1 - qx)*)) *(c*tB)*exp(qx*αM)*exp(qx*(1 - qx)*) = 1S*(c*tB)*exp(qx*αS) = 1S*(c*tS + qx*xS)

1S = 1M/exp(qx*(1 - qx)*) (c*tB)*exp(qx*αS) = (c*tB)*exp(qx*αM)*exp(qx*(1 - qx)*)

211 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS Component Transformations. standard process.

“c*tS” and “xS” are found using the

c*tS + qx*xS = (c*tB)*exp(qx*αM)*exp(qx*(1 - qx)*) First Case Finished-Calculation:  = calcC (with “C” “+”) c*tS + qx*xS = (c*tB)*exp(qx*αM)*exp(qx*αS/M) = (c*tB)*exp(qx*αM)*exp(qx*(1 - qx)*) = (c*tB)*exp(qx*αM)*exp(qx*(1 - qx)*calcC) First Case and “C” “+” 2 = (c*tB)*exp(qx*αM)*((1 + qx)/2 + (1 - qx)*exp (-calcC)/2) = (c*tB)*(exp(qx*αM)*(1 + qx)/2 + exp(qx*αM)*(1 - qx)*exp2(-calcC)/2) = (c*tB)*(exp(αM)*(1 + qx)/2 + exp(-αM)*(1 - qx)*exp2(-calcC)/2) c*tS = (c*tB)*(exp(αM)/2 + exp(-αM)*exp2(-calcB)/2) First Case  (c*tB)*exp(αM)/2 xS = (c*tB)*(exp(αM)/2 - exp(-αM)*exp2(-calcB)/2)  (c*tB)*exp(αM)/2

First Case

“c*tS  xS” because “c*tS” and “xS” have the same finite truncated number portion “(c*tB)*exp(αM)/2” in the finished calculation, and that is because the hyperbolic-radius “(c*tB)*exp(-calcC)” is small, effectively zero, and the other factor “exp(qx*calcC)” is large (infinite), to compensate. Roadside observer “S” measures the time “c*tS” when an object of speed “vM” inside a bus passes location “xS”. For the First Case, the bus travels at the speed-of-light, “vS/M-FirstCase = c” and per “c*tS  (c*tB)*exp(αM)/2”, the measured time “tS” increases proportionally to the advancement of time “tB” on the object. For example, if “αM = 0”, then “c*tS  c*tB/2”. Observing an Object that Moves at the Speed-of-Light. c*tS  (c*tB)*exp(αM)/2 ; = (c*tB)*(coshαM + sinhαM)/2 = (c*tM + xM)/2

xS  (c*tB)*exp(αM)/2 = (c*tB)*(coshαM + sinhαM)/2 = (c*tM + xM)/2

212 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY If “xM” identifies the back of the baseball and “xM + xM” identifies the front of the baseball, then, because “(c*tM + xM)back” equals “(c*tM + xM)front”, front is earlier than back by “xM/c”. c*tS = xS = (c*tM + xM)back/2 = (c*tM + xM)front/2 c*tMfront = c*tMback - (xMfront - xMback) = c*tMback - xM All locations in “M” back to front along the baseball are at one location “xS” at one time “c*tS” because each location is at a different “tM” time, per length contraction. Time “tB” displayed on the clock mounted on the moving object (baseball) is not stopped when observed from “S”. This special feature of the mathematical model contrasts with our traditional expectation that there is no passage of time on a photon. Second Case Finished-Calculation:  = 0calcC (with “C” “+”) c*tS + qx*xS = (c*tB)*exp(qx*αM)*exp(qx*(1 - qx)*) = (c*tB)*exp(qx*αM)*exp(qx*(1 - qx)*0calcC) Second Case and “C” “+” = (c*tB)*exp(qx*αM)*(1 - (1 - qx)*exp(-0calcC)*sinh(0calcC)) = (c*tB)*exp(qx*αM) - (c*tB)*exp(qx*αM)*(1 - qx)*exp(-0calcC)*sinh(0calcC) = (c*tB)*exp(qx*αM) - (c*tB)*exp(-αM)*(1 - qx)*exp(-0calcC)*sinh(0calcC)  (c*tB)*exp(qx*αM) - (c*tB)*exp(-αM)*(1 - qx)*(1 - 0calcC)*0calcC  (c*tB)*exp(qx*αM) Second Case c*tS = c*tM - (c*tB)*exp(-(αM + 0calcC)*sinh(0calcC)  c*tM - (c*tB)*exp(-(αM + 0calcC)*0calcC  c*tM Second Case xS = xM + (c*tB)*exp(-(αM + 0calcC)*sinh(0calcC)  xM + (c*tB)*exp(-(αM + 0calcC)*0calcC  xM Second Case For the Second Case, the components as measured in “S” are the same as the components as measured in “M”.

213 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS There is only one roadside “S” where the observer stands. Also, there is only one bus “M” where a different observer sits. There are two “S/M” speeds for the bus: A “S/M-FirstCase” and a “S/M-SecondCase”. Because of the two “S/M” speeds, there are two separate observations from the one roadside “S” of the one bus “M”. Compound-Label-Numbers. 1S = 1M/exp(qx*αS/M) = 1M*exp(-qx*αS/M) = 1M*exp(-qx*(1 - qx)*) First Case Finished-Calculation:  = calcC (with “C” “+”) 1S = 1M*exp(-qx*(1 - qx)*) = 1M*exp(-qx*(1 - qx)*calcC) First Case with “C” “+” = 1M*((1 + qx)/2 + (1 - qx)*exp2(calcC)/2) = 1M*(1 + qx)/2 + 1M*(1 - qx)*exp2(calcC)/2  1M*(1 - qx)*exp2(calcC)/2 qxS = qx*1S = qx*1M*(1 + qx)/2 + qx*1M*(1 - qx)*exp2(calcC)/2 = 1M*(1 + qx)/2 - 1M*(1 - qx)*exp2(calcC)/2  -1M*(1 - qx)*exp2(calcC)/2 = -1S The translation from the First Case finished-calculation “1S” to the final-result “itS” has the “exp2(calcC)/2” replaced by a local-infinity. And, for the First Case, “ixS = -itS”. Second Case Finished-Calculation:  = 0calcC (with “C” “+”) 1S = 1M*exp(-qx*(1 - qx)*) = 1M*exp(-qx*(1 - qx)*0calcC) Second Case with “C” “+” = 1M*(1 - (1 - qx)*exp(0calcC)*sinh(-0calcC))  1M*(1 - (1 - qx)*(1 + 0calcC)*(-0calcC))  1M qxS = qx*1S  qxM

214 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Relative to a measurement (that is, when translated into geometry), the Second Case compound-label-numbers create the same geometric-unit-vectors for “S” as for “M”. Two Locations at One Time. Two people stand on the roadside “S”. A person at the speed limit sign observed the Second Case at the same time a person down the road observed the First Case. Per the proposed theory, the two observations are of the same particle because the photon is a projection of the electron. If “C” is negative “-”, then the First Case (“ = CmaxC”) component magnitudes are infinite and the compound-label-numbers are finite. There might not be theoretical significance to “C” “-” if “” is the exclusively positive finished-calculation “2^(AmaxA) = *(1/2 + A1/2) + (1/)*(1/2 - A1/2)” rather than the final-result of “2^(AmaxA)”, which is “CmaxC” or “CminC0” for “A” “+” or “A” “-”, respectively. Other General Form. 2r

= 1M*(c*tB)*exp(qx*αM)*exp(qx*αS/M)/exp(qx*αS/M) = 1M*T*exp(ln(c*tB/T)*exp(qx*αM)*exp(qx*αS/M)/exp(qx*αS/M) = 1S*T*exp(ln(c*tB/T)*exp(qx*αM)*exp(qx*(1 - qx)*) = 1S*T*exp(ln(c*tB/T) - )*exp(qx*(αM + )) = 1S*T*exp(ln(c*tB/T))*exp(qx*αM)*exp(-)*exp(qx*) = 1S*T*exp(ln(c*tB/T))*exp(qx*αM)*(cosh()/exp() + qx*sinh()/exp()) = 1S*(c*tB)*exp(qx*αM)*(cosh()/exp() + qx*sinh()/exp()) = 1S*(c*tB)*exp(qx*αM)*((1 + exp2(-)) + qx*(1 - exp2(-))/2 = 1S*(c*tB)*exp(qx*αM)*((1 + qx)/2 + (1 - qx)*exp2(-)/2)

Other Invariants. Other invariants also have a First Case. Location (time-like) Location (space-like) Frequency (time-like) Wave-number (space-like)

2r

= 1S*c*tB*exp(qx*αS) s 2 = 1S*qx*sxB*exp(qx*αS) 2/c

= 1S*(B/c)*exp(qx*αS) k = 1 2 S*qx*kxB*exp(qx*αS)

215 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS Energy-Momentum Charge Density

2p 2J

= 1S*mB*c*exp(qx*αS) = 1S*B*exp(qx*αS)

Wave-number and Frequency Observed at the Speed-of-Light. On the “2r” hypercomplex-plane “2/c = 1B*(B/c)” is plotted as stationary horizontal evenly spaced parallel “wave crest” lines that extend left and right for all of “xB” space. The lines are closer for higher frequency “B”. As a visualization, two long rods along the floor of the bus, front to back, bounce side to side to make a bang sound everyone seated on the bus hears. The bang sound is the horizontal lines. And people standing along the roadside raise their hands in unison for each bang. Then, the bus moves forward with “αS/M > 0”. 2/c

= 1S*(B/c)*exp(qx*αS/M) = 1S*(B/c)*cosh(αS/M) + qxS*(B/c)*sinh(αS/M) = 1S*(S/c) + qxS*kxS

Because the bus is moving, the wave crest lines slope up and to the right and are spaced closer together by factor “cosh(αS/M)” (because of time dilation and “S/c = (B/c)*cosh(αS/M)”). People on the roadside each raise their hand when they hear the bang sound of the rods. The speed of the hands is faster than the speed-of-light. The bus speeds up to (nearly) the speed-of-light so that time dilation is nearly infinite. The hands raise only once and the motion of raising the hand moves at the speed-of-light.

Figure 37a. “2/c = 1S*(B/c)” is illustrated on the left. On the right the invariant “2/c = 1S*(B/c)*exp(qx*αS)” is illustrated, for “αS > 0”.

216 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Figure 37b. “2/c = 1S*(B/c)*exp(qx*αS)” for “αS = +” is illustrated on the left. Right has “2/c = 1S*(B/c)*exp(qx*αS)” for the First Case “αS = (1 - qx)*”, “C” “+”.

For the First Case, frequency and wavenumber components are finite and equal, “S/c  kS/c  (B/c)/2”. The hands raise at the speedof-light and do so repeatedly. S/c = (B/c)*exp(-calcC)*cosh(calcC) First Case = (B/c)*exp(-calcC)*(exp(calcC) + exp(-calcC))/2 = (B/c)*(1 + exp(-2*calcC))/2  (B/c)/2 The people on the roadside also hear the Second Case, for which the bus is stationary. For the Second Case they all raise their hands simultaneously. Both the First Case and Second Case together are visible in the raising of their hands. Momentum Observed at the Speed-of-Light. First Case energy “ES/c” and “pxS” are finite. For “αM = 0”: ES/c = (mB*c)*exp(-calcC)*cosh(calcC) First Case = (mB*c)*exp(-calcC)*(exp(calcC) + exp(-calcC))/2 = (mB*c)*(1 + exp(-2*calcC))/2  (mB*c)/2 pxS = (mB*c)*exp(-calcB)*sinh(calcC) First Case = (mB*c)*exp(-calcB)*(exp(calcC) - exp(-calcC))/2 = (mB*c)*(1 - exp(-2*calcC))/2  (mB*c)/2

217 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS For “αM  0” the energy and momentum as observed from the roadside “S” equal the mean average of the energy and momentum as observed inside the bus “M”. ES/c  (mB*c)*exp(αM)/2 First Case = (EB/c)*exp(αM)/2 = ((EB/c)*cosh(αM) + (EB/c)*sinh(αM))/2 = (EM/c + pxM)/2 pxS  (EM/c + pxM)/2

First Case

Photons have the property of energy equal to momentum. For the case of very low energy photons, “αM 2^Lmax”) from the hyperbolic-angle, for a product that is a finite finished-calculation result. Therefore, the final-result components “S/c” and “kxS” in “4 = 1S*S/c + qxS*kxS” are finite, so that the photon’s precession frequency “S” observed through measurements is finite and is not zero (regardless of the hyperbolic-radius “B*(QB*/(mB*c)))” being so small it may be considered to be a zero). The precession frequency “S” is the frequency of the electromagnetic wave. Imagine a bar magnet with its magnetic field spiraling through space pointing radially outward to create the magnetic field component of a spiral electromagnetic wave. Per this proposed model, the spin of a photon due to precessing motion has a factor of two compared to the frequency of an electron. That appears to be why a photon has a spin (angular momentum) of one unit (of Planck’s constant) and an electron has a spin half of one unit. More mathematical rigor is needed, but the point is made that photon spin is not required to be zero (or a half) by the proposed Theory of Special Relativity with Non-Finite Numbers.

248 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

5.8 Exercises Exercises for Text Comprehension 1) Prove the square-root-of-two “2” is irrational. Apply the crude algebra for “1/0” to prove “1/0*2 = 1/0”. 2) Find the rational approximation for “log23” in base two for thirtytwo place-value digits after the decimal point. 3) What are the truncated numbers for “Lmax = 0” and “Lmax = 1”? How many truncated numbers are for “Lmax = 9”, and what are the largest and smallest? 4) Write the local-zero “AminA0” for “Lmax = 9”. 5) Prove a local-zero cannot equal the integer zero. 6) Write Cantor’s proof that both the set of natural numbers and the set of integers have the same quantity of members. 7) Write Cantor’s proof that the quantity of real numbers from zero (inclusive) to one (exclusive) is a larger quantity than the quantity of natural numbers. Use place-value digit notation in base ten with at least five representative strings of place-value digits. 8) Write Cantor’s Continuum Hypothesis and explain how it requires the two infinities to be actual infinities and not finite, and for real numbers to have infinitesimal precision with an infinite quantity over a finite interval. 9) Review the equations for “dPmax0(s)/ds” and “Pmax0(s)”. Write the equations for “dPmin0(s)/ds” and “Pmin0”. What is the area under the curve “dPmin0(s)/ds” from negative one to positive one?

249 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS 10) Prove local-real numbers form a continuum per the criteria that there be an equal chance of a randomly selected number being in any interval of the same length along the number-line. 11) In the Dirac Equation form for Maxwell’s Wave Equation there is a “N”. What does the “N” represent? Answers to Select Exercises. 1) Per “p2 = 2*q2”, “p2” has a factor of “2” and “p” has a factor of “2”, and therefore “q2” has a factor of “2” and “q” has a factor of “2”. Our first observation is that both “p” and “q” must be even numbers. Our second observation is that at least one of “p” or “q” must be able to be odd, and that is because “p” and “q” are in a ratio such that they both can be divided by two until one of them is odd. The two observations are incompatible. Therefore, the original assumption “2 = p/q”, with “p” and “q” formed by starting with one and adding one repeatedly, is incorrect. Set “p = 1/0” and “q = 1/0” and apply the crude non-number algebra so that both observations are satisfied by “p” and “q” both being even and, because “1/0 = 1/0 + 1” one of them can be odd, too. 2) log1011  1.10010101110000000001101000111001…… 3) For “Lmax = 0” the count of truncated numbers is “((1)*(1) + 1)*2 - 1 = 3”, and the truncated numbers are “1”, “0” and “-1”. For “Lmax = 1” the count of truncated numbers is “((2)*(2) + 1)*2 - 1 = 9”, and the truncated numbers (written in base two) are “10.0”, “1.1”, “1.0”, “0.1”, “0.0”, “-0.1”, “-1.0”, “-1.1”, “-10.0”. For “Lmax = 9” the count of truncated numbers is “((2^9)*(2^9) + 1)*2 - 1 = 534389”, and the largest truncated number is (written in base two) “1000000000.0” (“512”), and the positive smallest truncated number is “0.000000001” (“1/512”). 4) AminA0 = 0.000000000dddddd…… 5) See book text 6) See book text

250 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 7) Use the table below to form new numbers, by selecting a placevalue digit from each number given in the table. .224744871 .581138830 .870828693 .121320344 .345207880

.549409757 .738612788 .915475947 .082207001 .240370349

New: .280309741 .571202909

8) The Continuum Hypothesis requires no sets that have a quantity of members between the quantity of members in the set of natural numbers and the quantity of members in the set of real numbers. Any number is the quantity of members in a set for which it is the largest, and so to forbid those intermediate numbers from existing, the two identified quantities must not be finite, but, rather, actual infinity. And, the two quantities cannot be the reciprocal-of-zero, per the crude algebra, because they are not equal and must be positive. For the set of real numbers to have a quantity base two to actual infinity, real numbers have an infinitesimal difference one to the next. Because real numbers are defined with a dependency on the actual infinity, and because the property of an actual infinity is derived from the Continuum Hypothesis, the Continuum Hypothesis forms the basis of the set of real numbers. 9) Write the equations for “dPmin0(s)/ds” and “Pmin0”. dPmin0/ds = 0 dPmin0/ds = (s/(2^-Lmax) + 1)*2^Lmax dPmin0/ds = (1 - s/(2^-Lmax))*2^Lmax dPmin0/ds = 0 Pmin0 = 0 Pmin0 = (s/(2^-Lmax) + 1)2/2 Pmin0 = (2 - (1 - s/(2^-Lmax))2)/2 Pmin0 = 1

for s/(2^-Lmax) < -1 for -1 < s/(2^-Lmax) < 0 for 0 < s/(2^-Lmax) < 1 for 1 < s(2^-Lmax)

for s/(2^-Lmax) < -1 for -1 < s/(2^-Lmax) < 0 for 0 < s/(2^-Lmax) < 1 for 1 < s/(2^-Lmax)

The area under the “dPmin0/ds” curve equals one, “Pmin0(s = ) = 1”. 10) Answer not provided.

251 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS 11) “N” being “+” models movement in the positive “x”-direction. “N” “-” models movement in the negative “x”-direction. Because of the “N”, “N” says both possibilities apply together, not one or the other. Further Thought. 1)

Attempt to prove “2” is included in Cantor’s real numbers by “c*2” equaling “c”. Is this a correct check? Must “c” must be even and so cannot be odd? Mathematics is unambiguous yet asking to prove “2” is in Cantor’s set of real numbers introduces subjectivity and wishy-washy ambiguity that fails independent verification. Do you agree?

2)

Prove two local-zeros cannot be the same by using a proof similar to the proof a local-zero cannot equal the integer zero.

3)

Set “T” equal to the infinite “……” sum of reciprocal natural numbers and subtract “T/2” from it, twice to derive “ln(2) - ln(2) = 0”. The second “ln(2)” is a quantity “1/0” of “0”’s. How can the crude algebra be expanded to include theory for this second “ln(2)”? If we don’t do this, then we are ignoring what could be a necessary expansion of our mathematical tools.

4)

Einstein’s two theories of Relativity each include a division by zero, a singularity. What other mathematical models of physics include a division by zero? Can the proposed Theory of Non-Finite Numbers be applied to those mathematical models of physics?

5) Apply the proposed Theory of Non-Finite Numbers to the singularity at the center of a black hole. The General Theory of Relativity is written as “Rab - (R*gab)/2 + *gab = 8**G*Tab”. 6)

Unknown and unknowable place-value digits become known or knowable as time progresses, along with Entropy increase, cause and effect, and the collapse of the wave function. Is “now” (and “here”) specific to a value of “Lmax”? How might the transition from unknowable to knowable with regard to numbers be related

252 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY to the wave function collapse of quantum mechanics, and to Hugh Evert’s 1957 proposed “many worlds interpretation” of quantum mechanics in which all possibilities occur in a divergence of reality. How is the “now” unique, or isn’t it in a theory, yet? 7)

With regard to the above question, if “Lmax” increases with time, then how do we reconcile “Lmax” with anti-matter if time is in reverse for anti-matter ?

8)

How does the new proposed reciprocal of zero axiom pertain to the exotic Lorentz Transformation for motion faster than the speed-of-light with respect to division by zero for the speed?

9)

Using this chapter’s new theory for force density, can we find a theory by which the energy and momentum for the electric field around an electron are derived through mathematics, rather than empirically derived through observations of experiments?

10) One value of “Lmax” applies, per the above models of quantum mechanics, when one particle observes another particle. Does each particle have a unique value of “Lmax” with respect to each other particle? Or, perhaps, does “Lmax” apply to a collection of particles, or to an inertial reference frame? 11) If a photon is one in the same particle as the electron that emitted it, then time-space is distorted by there being two locations for the one particle. And, the simultaneous validity of the First Case with the Second Case appears to create an equivalence between a set of label-numbers in “S” with singular-label-numbers in “M”. How is space structured? 12) Try to define properties so that “2^c” and “3^c” are equal, so that the irrational number “log23” is a ratio of two versions of “c”. Retain “c < 2^c < 1/0”. Try to define a positive actual infinity or two for which “log23” or “2” equals the ratio of that infinity or of those infinities. The important point is to make the mathematics unambiguous, and to not depend on the mysterious, vague, and remote properties of a positive actual infinity.

253 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS 13) If the new proposed reciprocal of zero axiom is included in axiomatic set theory, then the Continuum Hypothesis must be rejected. Do you agree? 14) What is the definition of a real number per Descartes? What is the definition of a real number per Cantor? What is next? 15) The speed-of-light in Special Relativity is mathematically analogous to the event horizon of a non-rotating black hole in General Relativity. The analogy is more complete if the radius of the black hole is infinite, to create a flat space event horizon. Per the theory by Hawking / Bekenstein, the temperature of a black hole’s surface decreases with the radius of the black hole, so that the more complete analogy of an infinite radius has zero temperature at the event horizon. Per the evolving theory of loop gravity in which General Relativity is reconciled with quantum mechanics, as developed by Eugenio Bianchi for black holes, the quantum fluctuations on the surface of the black hole decrease as the radius is increased. The quantum fluctuations generate the temperature. Might the “Lmax” be very large for cold nearly flat space and very small for the curved event horizon of a hot small radius black hole? To keep “Lmax” finite must there be curvature in the universe? What about the other extreme, for which the smallest black hole has a diameter of the event horizon on the scale of Matvei Bronstein’s Planck length, “LP = (ħ*G/c3)”? 16) Electrons are fermions of half spin that repel each other, per the Pauli Exclusion Principle, with the example of the shells of an atom. Photons are as opposite as possible. Photons are bosons of full spin that have a tendency to coincide, as in a laser. Electrons are the material and photons are the force field. Because they are different to an extreme, combining electrons with photons into one mathematical model feels strange, feels incorrect, feels like it should be impossible. And, photons are generated by protons, and photons are created by pair annihilation / pair production. How can the proposed Theory of Special Relativity with Non-Finite Numbers be generalized to include protons and pair production? What are your thoughts?

254 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 17) The theory of the Dirac Spinor matter-wave was developed from rest mass inertia which, per the classical radius model of the electron, results from the electromagnetic field. It’s not surprising the electromagnetic field is also a Dirac Spinor, per the new theory presented in this book, because the two phenomena are made of the same stuff, it seems. Where next to take this theory? 18) A proposed experiment is to simultaneously measure the photon particle and the electron particle that emitted it, to check for correlated properties, in analogy with the EPR experiment. How could such an experiment be set up? What would it measure? 19) In the book Reality is Not What It Seems – The Journey to Quantum Gravity by Carlo Rovelli, Riverhead Books, 2017 (2014 in Italian), Page 245/246, is the quote below. In the two points, the positive actual infinity is removed from mathematical relevance, and the quantity of information from one particle observing another particle increases with time. How can the proposed Theory of Non-Finite Numbers fit into quantum mechanics per this quote? “In fact, the entire structure of quantum mechanics can be read and understood in terms of information, as follows. A physical system manifests itself only in interacting with another. The description of a physical system, then, is always given in relation to another physical system, the one with which it interacts. Any description of a system is therefore always a description of the information a system has about another system, that is to say, the correlation between the two systems. The mysteries of quantum mechanics become less dense if interpreted this way, as the description of the information that physical systems have about one another. The description of a system, in the end, is nothing other than a way of summarizing all the past interactions with it, and using them to predict the effect of future interactions. The entire formal structure of quantum mechanics can be in large measure expressed in two simple postulates: 1. The relevant information in any physical system is finite 2. You can always obtain new information on a physical system”

255 CHAPTER 5 - PROPOSED THEORY OF NON-FINITE NUMBERS 20) Anti-Matter Photon. Per the First Case Lorentz Transformation, an observer “S” measures a passage of time “tS” proportional to the passage of time “tB” for the photon. Regardless of the passage of time, the photon has complete length contraction. Also, per experiments in which two photons annihilate each other, it appears a photon is its own anti-matter particle. How can the photon be its own anti-matter particle if the direction of its passage of time is dependent on the electron that emitted it being either matter or anti-matter? 21) The two zeros on the right side of the First Case Dirac Equation were replaced by the non-zero value “a”. Try to find a justification to create a theory for what electric charge really is. 22) As time progresses, a photon gets absorbed and disappears, so that the First Case disappears as the hyperbolic-angle becomes more precise with respect to an observing particle. The photon disappears because the hyperbolic-angle is small relative to “2^Lmax” and stays small. A succession of electron and photon emission and absorption events occurs through an increase in “Lmax”. Try to make this theory quantitative to make it useful.

256 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

257 APPENDIX A - OCTONIONS AND SEDONIONS

Appendix A – Octonions and Sedonions Octonions were proposed by Caylay immediately after Hamilton proposed quaternions, in 1843. Quaternions anti-commute “jx*jy = -jy*jx = jz”. Likewise, octonions anti-commute. When Hamilton selected “jx*jy = jz”, he selected against equating “jx*jy” to “-jz”. For octonions, there is more than one selection required when setting up the multiplication scheme. One possible multiplication scheme for octonions is selected below. Each row is a “triple”. j1*j2 = j3 ; j2*j4 = j6 ; j3*j6 = j5 ; j4*j3 = j7 ; j5*j1 = j4 ; j6*j7 = j1 ; j7*j5 = j2

Figure 40. The traditional triangle model for octonion multiplication. *

1

j1

j2

j3

1

1*1 = 1

1*j1 = j1

1*j2 = j2

1*j3 = j3

j1

j1*1 = j1

j1*j1= -1

j1*j2 = j3

j1*j3 = -j2

j2

j2*1 = j2

j2*j1 = -j3

j2*j2 = -1

j2*j3 = j1

j3

j3*1 = j3

j3*j1 = j2

j3*j2 = -j1

j3*j3 = -1

j4

j3*1 = j3

j4*j1 = -j5

j4*j2 = -j6

j4*j3 = j7

j5

j3*1 = j3

j5*j1 = j4

j5*j2 = j7

j5*j3 = j6

j6

j3*1 = j3

j6*j1 = -j7

j6*j2 = j4

j6*j3 = -j5

j7

j3*1 = j3

j7*j1 = j6

j7*j2 = -j5

j7*j3 = -j4

258 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY *

j4

j5

j6

j7

1

1*j4 = j4

1*j5 = j5

1*j6 = j6

1*j7 = j7

j1

j1*j4 = j5

j1*j5 = -j4

j1*j6 = j7

j1*j7 = -j6

j2

j2*j4 = j6

j2*j5 = -j7

j2*j6 = -j4

j2*j7 = j5

j3

j3*j4 = -j7

j3*j5 = -j6

j3*j6 = j5

j3*j7 = j4

j4

j4*j4 = -1

j4*j5 = j1

j4*j6 = j2

j4*j7 = -j3

j5

j5*j4 = -j1

j5*j5 = -1

j5*j6 = -j3

j5*j7 = -j2

j6

j6*j4 = -j2

j6*j5 = j3

j6*j6 = -1

j6*j7 = j1

j7

j7*j4= j3

j7*j5 = j2

j7*j6 = -j1

j7*j7 = -1

Table 6. Multiplication table for octonions.

0 1 2 3 4 5 6 7

1 -0 -3 2 -5 4 -7 6

2 3 -0 -1 -6 7 4 -5

3 -2 1 -0 7 6 -5 -4

4 5 6 -7 -0 -1 -2 3

5 -4 -7 -6 1 -0 3 2

6 7 -4 5 2 -3 -0 -1

7 -6 5 4 -3 -2 1 -0

Table 7. Short form of the multiplication table for octonions.

Criteria of an algebraic group for octonions “O = {1, j1, j2, j3, j4, j5, j6, j7}” with respect to multiplication, “{O, *(anti-commute)}”: •

Closure: There are no holes in the 16x16 multiplication table

•

Identity: The identity element is the positive number one

•

Commutative Property: Applies, but with two different of the seven octonions anti-commuting

•

Associative Property: Applies, but with three different octonions (that are not all three in the same triple) anti-associating

259 APPENDIX A - OCTONIONS AND SEDONIONS •

Inverse: The ratio of any two numbers is in the set of numbers

The inverse property example is “j1” divided by “j4”. First, substitute two factors for the numerator, then, second, apply the (anti-) commutative law (if needed) followed by the (anti-)associative law (if needed) to form a ratio of like numbers. The ratio of like numbers is then replaced with the number one. Each combination must equal “j1*-j4”. j1/j4 = (j2*j3)/j4 = (j2*(j7*j4))/j4 = -((j2*j7)*j4)/j4 = -(j2*j7)*(j4/j4) = -(j2*j7) = -j5 j1/j4 = (j4*j5)/j4 = -(j5*j4)/j4 = -j5*(j4/j4) = -j5 j1/j4 = (j6*j7)/j4 = (j6*(j4*j3))/j4 = -(j6*(j3*j4))/j4 = (j6*j3)*(j4/j4) = (j6*j3) = -(j3*j6) = -j5 Degradations. Each higher order of hypercomplexity includes a new break-down in symmetry through the introduction of another negative. A break-down in symmetry is called (in this book) a “degradation”. The pattern of degradations only goes to the fourth order, if only because the degradation of the fourth order is too severe to go on. •

Real numbers (hypercomplex order “N = 0”) have no degradation (other than their unboundedness which implies a lack of closure)

•

Anti-Identity. Complex numbers (“N = 1”) have the complex number label-number which squares to negative one i2 = -1

•

Anti-Commutative. Quaternions (“N = 2”). jy*jz = jx = -(-jx) = -jz*jy

•

Anti-Associative. Octonions (“N = 3”). (j1*j2)*j4 = j3*j4 = -j7 = -j1*j6 = -j1*(j2*j4)

260 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •

Anti-Inverse. Sedonions (“N = 4”) j14s/j6s = (j8s*j6s)/j6s = j8s*(j6s/j6s) = j8s = -(-j8s) = -(j13s*j5s) = -(j13s*j5s)*(j6s/j6s) = -((j13s*j5s)*j6s)/j6s = (j13s*(j5s*j6s))/j6s = -(j13s*(j6s*j5s))/j6s = -(j13s*j3s)/j6s = -j14s/j6s

Because “j14s/j6s = -j14s/j6s”, it is said sedonion algebra is not a division algebra. (Alternatively, we may say sedonion algebra is not yet properly specified.) Matrix Isomorphs for the Octonions. Pauli Spin Matrices, multiplied or divided by “i”, result in the below traditional set of 2x2 matrix isomorphs for Hamilton’s quaternions. 1 0

0 i

1=

jx = 0 1

0 -1 jy =

i 0

i

0

jz = 1 0

0 -i

The above four matrices are (traditionally) placed into a general matrix multiplication scheme using the complex-conjugate “*i” (“i*i = -i”) on the right-side column-vector. a

-b*i

b

a*i

c

-d*i

d

c *i

*

=

a*c - b*i*d

-a*d*i - b*i*c*i

b*c + a*i*d

-b*d*i + a*i*c*i

e =

-f*i

f e*i

The letters “a” and “b” (also “c” and “d”, and “e” and “f”) correspond to the terms of the 2x2 matrix isomorphs of the quaternions by these substitutions: 1: a = 1, b = 0 ;

jx: a = 0, b = i ;

jy: a = 0, b = 1 ;

jz: a = i, b = 0

Factors are rearranged because complex numbers commute. a

-b*i

b

a*i

c

-d*i

d

c *i

*

=

a*c - d*b*i

-d*i*a - b*i*c*i

c*b + a*i*d

-b*d*i + c*i*a*i

e =

-f*i

f e*i

261 APPENDIX A - OCTONIONS AND SEDONIONS The arrangement above is more correct because “e*i = -b*d*i + c *a ” has the factors in each term reversed compared to “e = a*c - d*b*i”. The new arrangement, with the reversed order of factors and terms in the product 2x2 matrix, specifies a new multiplication scheme between two 2x2 matrices. The new multiplication scheme is necessary for factors that do not commute, that is, if the factors are quaternions. The hypercomplexconjugate operation “*j” replaces the complex-conjugate operation “*i”. “jx*j = -jx”, “jy*j = -jy”, and “jz*j = -jz” with the factors reversed in order. The below matrix multiplication operation applies to matrices with quaternion hypercomplex numbers as elements/terms of the matrix. *i

*i

a

-b*j

b

a*j

c

-d*j

d

c*j

*

1: a = 1, b = 0 ; j1: a = 0, b = 1 ;

=

a*c - d*b*j

-d*j*a - b*j*c*j

c*b + a*j*d

-b*d*j + c*j*a*j

j2: a = jx, b = 0 ; j3: a = 0, b = jx ;

j4: a = jy, b = 0 ; j5: a = 0, b = jy ;

e =

-f*j

f e*j

j6: a = jz, b = 0 j7: a = 0, b = jz

(The above 2x2 matrix equation for octonion matrix isomorphs was discovered by the author. A search did not find a previously known set of matrix isomorphs for octonions.) Examples for “j2*j4 = j6” and “j4*j3 = j7”, respectively. jx

0

jy

0

* 0

-jx

jy

0

0

- jy

0

jx

* 0

-jy

jx*jy

0

=

0

0

0

- jz

0 -(-jy*jx) 0

jx*jy

= jx

jz =

0

jz

jz

0

= -jy*jx

0

“jx” can be replaced with its 2x2 matrix isomorph and “i” and “1” by their 2x2 matrix isomorphs to create 8x8 matrices. 0

-1

i =

1 0 1=

1

0

0 1

262 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY General Rule for Matrix Multiplication with non-Commuting Elements. A C

E

G

* B D

A*E + F*C

G*A + C*H

F*B + D*E

B*G + H*D

= F

H

Vector-Aft Multiplication: A C

E *

B D

A*E + F*C =

F

E*B + D*F

Vector-Front Multiplication: E

A C *

F

E*A + C*F =

B

D

B*E + F*D

Vector-front multiplication is required so that matrices anticommute in a vector-front multiplication compared to a vector-aft multiplication. For example: In “j5*j3 = j6” use the left column of “j3” to result in the left column of “j6”. Now, reverse the order of the factors to “j3*j5”. Again, only use the left column of “j3”. Per the vector-front multiplication operation, the result of “j3*j5” is the negative of the left column of “j6”. As a side note, we will not use the below alternative. A C

E

G

F

H

* B

D

E*A + C*F

A*G + H*C

B*F + E*D

G*B + D*H

Do not use

=

Rotation of a Seven-Dimensional Object. Seven-dimensional ultra-space modeled with octonions is organized as seven three-dimensional spaces so that there are three planes of rotation around each axis of rotation. Rotation around the “j1” axis has rotation in the “j2”/ “j3”, “j4”/ “j5”, and “j6”/ “j7” planes. Rotation angles in each plane are “23”, “45”, and “67”, respectively. “23” begins at the positive “j2” axis and is measured towards the positive “j3” axis. (Rotations in 7-d space might be new with this book.)

263 APPENDIX A - OCTONIONS AND SEDONIONS 0

-1

1

0

j1o1 = jx*cos23

jx*sin23

j2o1 =

; jx*sin23

-jx*cos23

jy*cos45

jy*sin45

j4o1 =

-jx*sin23

jx*cos23

jx*cos23

jx*sin23

j3o1 =

-jy*sin45

jy*cos45

jy*cos45

jy*sin45

-jz*sin67

jz*cos67

jz*cos67

jz*sin67

; j5o1 = jy*sin45

-jy*cos45

jz*cos67

jz*sin67

j6o1 =

; j7o1 = jz*sin67

-jz*cos67

Multiplications for “j1o1*j2o1 = j3o1”, “j5o1*j1o1 = j4o1”, and “j6o1*j7o1 = j1o1” do not involve angle addition, and, therefore, are relatively trivial. The other four triples “j2o1*j4o1 = j6o1”, “j3o1*j6o1 = j5o1”, “j4o1*j3o1 = j7o1”, and “j7o1*j5o1 = j2o1” involve the addition of angles. “j3o1*j6o1”: -jx*sin23

jx*cos23

jx*cos23

jx*sin23

j3o1*j6o1 =

jz*cos67

jz*sin67

* jz*sin67

-jz*cos67

(-jx*jz)*sin23*cos67 + (jz*jx)*cos23*sin67 (jz*-jx)*sin23*sin67 + (jx*-jz)*cos23*cos67 j3o1*j6o1 = (jz*jx)*cos23*cos67 + (jx*jz)*sin23*sin67 (jx*jz)*cos23*sin67 + (-jz*jx)*sin23*cos67

jy*(sin23*cos67 + cos23*sin67) -jy*(sin23*sin67 - cos23*cos67) j3o1*j6o1 = jy*(cos23*cos67 - sin23*sin67) -jy*(cos23*sin67 + sin23*cos67)

264 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY jy*sin(23 + 67) j3o1*j6o1 = jy*cos(23 + 67)

jy*cos(23 + 67) -jy*sin(23 + 67)

-jy*sin45

jy*cos45

jy*cos45

jy*sin45

j3o1*j6o1 =

= j5o1

The above matrix multiplication that resulted in “j3o1*j6o1 = j5o1” required “23 + 45 + 67 = 0”. “23 + 45 + 67 = 0” is also valid for “j2o1*j4o1 = j6o1”, “j4o1*j3o1 = j7o1”, and “j5o1*j2o1 = j7o1”. jz*(cos23*cos45 - sin23*sin45) -jz*(-cos23*sin45 - sin23*cos45) j2o1*j4o1 = jz*(-sin23*cos45 - cos23*sin45) -jz*(sin23*sin45 - cos23*cos45) jz*cos(23 + 45) j2o1*j4o1 = -jz*sin(23 + 45) jz*cos67 j2o1*j4o1 = jz*sin67

-jz*sin(23 + 45) -jz*cos(23 + 45)

jz*sin67 = j6o1 -jz*cos67

…… …… …… …… …… …… …… …… …… …… …… …… …… jy*cos45

jy*sin45

jy*sin45

-jy*cos45

j4o1*j3o1 =

-jx*sin23

jx*cos23

jx*cos23

jx*sin23

*

(jy*-jx*cos45*sin23 + jx*jy*sin45*cos23) (jx*jy*cos45*cos23 + jy*jx*sin45*sin23) j4o1*j3o1 = (-jx*jy*sin45*sin23 + -jy*jx*cos45*cos23) (jy*jx*sin45*cos23 + jx*-jy*cos45*sin23)

265 APPENDIX A - OCTONIONS AND SEDONIONS

jz*(cos45*sin23 + sin45*cos23) jz*(cos45*cos23 - sin45*sin23) j4o1*j3o1 = jz*(-sin45*sin23 + cos45*cos23) jz*(-sin45*cos23 - cos45*sin23) jz*sin(45 + 23) j4o1*j3o1 = jz*cos(45 + 23) -jz*sin67 j4o1*j3o1 = jz*cos67

jz*cos(45 + 23) -jz*sin(45 + 23) jz*cos67 = j7o1 jz*sin67

…… …… …… …… …… …… …… …… …… …… …… …… …… jz*(sin45*cos23 + cos45*sin23) jz*(-sin45*sin23 + cos45*cos23) j5o1*j2o1 = jz*(cos45*cos23 - sin45*sin623) jz*(-cos45*sin23 - sin45*cos23) jz*sin(45 + 23) j5o1*j2o1 = jz*cos(45 + 23) -jz*sin67 j5o1*j2o1 = jz*cos67

jz*cos(45 + 23) -jz*sin(45 + 23)

jz*cos67 = j7o1 jz*sin67

The seven-dimensional object is modeled as seven sticks.

266 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Figure 41. Seven-dimensional space as seven three-dimensional spaces.

Some sticks rotate ninety degrees and comply with “23 + 45 + 67 = 0”, per the illustration below.

Figure 42. Rotation in 2-3 plane causes a counter rotation in 4-5 plane.

The seven-dimensional object can also rotate around an axis in three-dimensional space, for example, the y-axis. The below substitutions may be visualized using the 4x4 matrix isomorphs for octonions. 0 -1 jyqy =

i*coszx

i*sinzx

; jzqy = 1 0

-i*sinzx

i*coszx

i*coszx

i*sinzx

; jxqy = i*sinzx -i*coszx

267 APPENDIX A - OCTONIONS AND SEDONIONS Unlike the octonions and the quaternions, there is no rotation possible for complex numbers, because there is only one label number for complex numbers. General Model for Label-Numbers. •

Real numbers “N = 0” have the trivial 1x1 matrix isomorph.

•

Complex numbers “N = 1” have “2N = 21 = 2” label-numbers, the “1” and the “i”, and a 2x2 matrix isomorph.

•

Quaternions “N = 2” have “2N = 22 = 4” label-numbers and a 4x4 matrix isomorph. (1 + jx)*(1 + jy) = 1 + jx + jy + jx*jy = 1 + jx + jy + jz

•

Octonions “N = 3” have “2N = 23 = 8” label-numbers and an 8x8 matrix isomorph. (1 + j1 + j5 + j6)*(1 + j2) = 1 + j1 + j2 + j3 + j4 + j5 + j6 + j7 (1 + j2 + j4 + j5)*(1 + j3) = 1 + j1 + j2 + j3 + j4 + j5 + j6 + j7 There are twenty-one restructures of the two octonion equations above, one restructure for each number in the 7x3 twenty-one number table.

•

Sedonions “N = 4” have sixteen label-numbers. The multiplication table for sedonions is not yet finalized.

Vector-Space. Vector space is so far useful in applied mathematics to “N = 2”. A more complex notion of vector space may be needed for “N = 3” and higher orders. •

“N = 1” introduced the complex label-number “i”. Subtract away the real number so the quantity of label-numbers in an “N = 1” vector-space is “2N - 1 = 1”.

268 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •

Four-dimensional time-space of Special Relativity was created by combining “N = 1” with “N = 2” to form the “N = 2” vector-space: “(21 - 1) + (22 - 1) = 1 + 3 = 4” terms. An example number is “658*i + 89*jx + 57*jy + 456*jz”.

•

The natural extension with the octonions is eleven-dimensional vector-space: “11 = 1 + 3 + 7 = (21 - 1) + (22 - 1) + (23 - 1)”. The seven dimensions of ultra-space are as different from space as space is from time. The example number is “658*i + 89*jx + 57*jy + 456*jz + 26*j1 + 44*j2 + 785*j3 + 963*j4 + 76*j5 + 659*j6 + 154*j7”.

•

“N = 4” creates a twenty-six-dimensional vector-space “26 = 1 + 3 + 7 + 15”.

•

After that is “N = 5” for “57 = 1 + 3 + 7 + 15 + 31”, for a fifty-sevendimensional vector-space, and so on.

Sedonions. A simple attempt at a sedonion algebra applies the same matrix multiplication operation applied to octonion algebra. The octonion-conjugate operation “*jo” requires the negative of the octonions, “j1*jo = -j1”, …, “j7*jo = -j7”, and requires the reverse of the order of factors. a

-b*jo

b

a*jo

c

-d*jo

d

c*jo

*

=

a*c - d*b*jo

-d*jo*a - b*jo*c*jo

c*b + a*jo*d

-b*d*jo + c*jo*a*jo

e

-f*jo

f

e*jo

=

1: a = 1, b = 0 ; j2s: a = j1, b = 0 ; j4s: a = j2, b = 0 ; j6s: a = j3, b = 0 j1s: a = 0, b = 1 ; j3s: a = 0, b = j1 ; j5s: a = 0, b = j2 ; j7s: a = 0, b = j3 j8s: a = j4, b = 0 ; j10s: a = j5, b = 0 ; j12s: a = j6, b = 0 ; j14s: a = j7, b = 0 j9s: a = 0, b = j4 ; j11s: a = 0, b = j5 ; j13s: a = 0, b = j6 ; j15s: a = 0, b = j7 j1s*j2s = j3s j1s*j4s = j5s j1s*j6s = j7s j1s*j8s = j9s j1s*j10s = j11s j1s*j12s = j13s j1s*j14s = j15s

; ; ; ; ; ; ;

j2s*j4s = j6s ; j2s*j8s = j10s ; j2s*j12s = j14s ; j4s*j8s = j12s ; j12s*j10s = j6s ; j14s*j8s = j6s ; j14s*j10s = j4s ;

j3s*j6s = j5s ; j3s*j10s = j9s ; j3s*j14s = j13s ; j5s*j12s = j9s ; j13s*j6s = j11s ; j15s*j6s = j9s ; j15s*j4s = j11s ;

j4s*j3s = j7s ; j8s*j3s = j11s ; j12s*j3s = j15s ; j8s*j5s = j13s ; j10s*j13s = j7s ; j8s*j15s = j7s ; j10s*j15s = j5s ;

j7s*j5s = j2s j11s*j9s = j2s j15s*j13s = j2s j13s*j9s = j4s j7s*j11s = j12s j7s*j9s = j14s j5s*j11s = j14s

269 APPENDIX A - OCTONIONS AND SEDONIONS 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1

-0

3

-2

5

-4

7

-6

9

-8

11

-10

13

-12

15

-14

2

-3

-0

1

6

-7

-4

5

10

-11

-8

9

14

-15

-12

13

3

2

-1

-0

-7

-6

5

4

-11

-10

9

8

-15

-14

13

12

4

-5

-6

7

-0

1

2

-3

12

-13

-14

15

-8

9

10

-11

5

4

7

6

-1

-0

-3

-2

-13

-12

15

14

9

8

-11

-10

6

-7

4

-5

-2

3

-0

1

-14

15

-12

13

10

-11

8

-9

7

6

-5

-4

3

2

-1

-0

15

14

13

12

-11

-10

-9

-8

8

-9

-10

11

-12

13

14

-15

-0

1

2

-3

4

-5

-6

7

9

8

11

10

13

12

-15

-14

-1

-0

-3

-2

-5

-4

7

6

10

-11

8

-9

14

-15

12

-13

-2

3

-0

1

-6

7

-4

5

11

10

-9

-8

-15

-14

-13

-12

3

2

-1

-0

7

6

5

4

12

-13

-14

15

8

-9

-10

11

-4

5

6

-7

-0

1

2

-3

13

12

15

14

-9

-8

11

10

5

4

-7

-6

-1

-0

-3

-2

14

-15

12

-13

-10

11

-8

9

6

-7

4

-5

-2

3

-0

1

15

14

-13

-12

11

10

9

8

-7

-6

-5

-4

3

2

-1

-0

Table 8. Short-hand multiplication table for sedonions (first attempt).

For rotation around the “j1s” axis, define seven rotation angles “23s”, “45s”, “67s”, “89s”, “1011s”, “1213s” and “1415s”. These angles are analogous to the previously defined angles “23”, “45”, and “67” in the discussion on octonions. 0

-1

1

0

j1s1 = j1*cos23s

j1*sin23s

j2s1 =

-j1*sin23s

j1*cos23s

j1*cos23s

j1*sin23s

-j2*sin45s

j2*cos45s

j2*cos45s

j2*sin45s

; j3s1 = j1*sin23s

-j1*cos23s

j2*cos45s

j2*sin45s

j4s1 =

; j5s1 = j2*sin45s

-j2*cos45s

270 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY j3*cos67s

j3*sin67s

j6s1 =

-j3*sin67s

j3*cos67s

; j7s1 = j3*sin67s

-j3*cos67s

j3*cos67s

j3*sin67s

j4*cos89s

j4*sin89s

-j4*sin89s

j4*cos89s

j4*sin89s

-j4*cos89s

j4*cos89s

j4*sin89s

j5*cos1011s

j5*sin1011s

-j5*sin1011s

j5*cos1011s

j5*cos1011s

j5*sin1011s

-j6*sin1213s

j6*cos1213s

j6*cos1213s

j6*sin1213s

-j7*sin1415s

j7*cos1415s

j7*cos1415s

j7*sin1415s

j8s1 =

; j9s1 =

j10s1 =

; j11s1 = j5*sin1011s

-j5*cos1011s

j6*cos1213s

j6*sin1213s

j12s1 =

; j13s1 = j6*sin1213s

-j6*cos1213s

j7*cos1415s

j7*sin1415s

j14s1 =

; j15s1 = j7*sin1415s

Octonions:

Sedonions:

-j7*cos1415s

x-y-z

jx jy jz 23 45 67 1-2-3 j2 j4 j6 2-4-6 j3 j5 j7 3-6-5 4-3-7 j1 j2 j3 j4 j5 j6 j7 5-1-4 23s 45s 67s 89s 1011s 1213s 1415s 6-7-1 j2s j4s j6s j8s j10s j12s j14s 7-5-2 j3 j5s j7s j9s j11s j13s j15s

The octonion/sedonion table above shows that the multiplication of “j1*j2 = j3” internal to the sedonion 2x2 matrices is analogous to “jx*jy = jz” internal to the octonion 2x2 matrices. Also “j2*j4 = j6”, “j5*j1 = j4”, and “j6*j7 = j1” are analogous to “jx*jy = jz” because the numbers increase in value: 1-2-3, 2-4-6, 1-4-5, and 1-6-7. The other three triples are not: 7-4-3, 6-5-4, and 7-5-2, because the numbers decrease in value. Each of the seven triples pertains to a plane perpendicular to the “1” axis. Each plane is associated with four triples.

271 APPENDIX A - OCTONIONS AND SEDONIONS 23s + 45s + 67s = 0, j2s1*j4s1 = j6s1 ;

(1-2-3) j3s1*j6s1 = j5s1 ; j4s1*j3s1 = j7s1 ;

23s + 89s + 1011s = 0, (1-4-5) j2s1*j8s1 = j10s1 ; j3s1*j10s1 = j9s1 ; j8s1*j3s1 = j11s1 ;

j7s1*j5s1 = j2s1 j11s1*j9s1 = j2s1

23s + 1213s + 1415s = 0, (1-6-7) j2s1*j12s1 = j14s1 ; j3s1*j14s1 = j13s1 ; j12s1*j3s1 = j15s1 ; j15s1*j13s1 = j2s1 45s + 89s + 1213s = 0, (2-4-6) j4s1*j8s1 = j12s1 ; j5s1*j12s1 = j9s1 ; j8s1*j5s1 = j13s1 ; j13s1*j9s1 = j4s1 1213s + 1011s + 67s = 0, (6-5-3) j12s1*j10s1 = j6s1 ; j13s1*j6s1 = j11s1 ; j10s1*j13s1 = j7s1 ; j7s1*j11s1 = j12s1 1415s + 89s + 67s = 0, (7-4-3) j14s1*j8s1 = j6s1 ; j15s1*j6s1 = j9s1 ; j8s1*j15s1 = j7s1 ; j7s1*j9s1 = j14s1 1415s + 1011s + 45s = 0, (7-5-2) j14s1*j10s1 = j4s1 ; j15s1*j4s1 = j11s1 ; j10s1*j15s1 = j5s1 ; j5s1*j11s1 = j14s1 The angles “s23”, “s45”, “s67”, “s89”, “s1011”, “s1213” and “s1415” must all be zero for the seven angle equations to each equal zero, and that means the math is incorrect as a model for rotation in fifteendimensional ultra-ultra space. The four normal planes can be thought of as a right-hand rotation. The three anti-normal planes can be thought of as a left-hand rotation. The switch of the hands made rotation impossible. To investigate the left-hand planes: “j6s*j9s = j15s” uses “j4*j3 = j7” because “j6s*j9s” has the product “j3*j4 = -j7” and “j4*j3” has the product “jy*jx = -jz”. “j6s*j9s”:

j3*cos67s

j3*sin67s

j3*sin67s

-j3*cos67s

j6s1*j9s1 =

-j4*sin89s

j4*cos89s

j4*cos89s

j4*sin89s

*

272 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY (j3*-j4*cos67s*sin89s + j4*j3*sin67s*cos89s) (j4*j3*cos67s*cos89s + j3*j4*sin67s*sin89s) j6s1*j9s1 = (-j4*j3*sin67s*sin89s + -j3*j4*cos67s*cos89s) (j3*j4*sin67s*cos89s + j4*-j3*cos67s*sin89s)

j7*(cos67s*sin89s + sin67s*cos89s) j7*(cos67s*cos89s - sin67s*sin89s) j6s1*j9s1 = j7*(-sin67s*sin89s + cos67s*cos89s) -j7*(sin67s*cos89s + cos67s*sin89s) j7*sin(67s + 89s)

j7*cos(67s + 89s)

j7*cos(67s + 89s)

-j7*sin(67s + 89s)

j6s1*j9s1 =

-j7*sin1415s

j7*cos1415s

j7*cos1415s

j7*sin1415s

j6s1*j9s1 =

= j15s1

Based on that example, the angle equations apply: “1213s + 1011s + 67s = 0”, “1415s + 89s + 67s = 0”, and “1415s + 1011s + 45s = 0”, such that there should not be negative.

Figure 43. A failed attempt at a tetrahedron triangle diagram for “N = 4” multiplication-division. Fill in the circles and arrows and find the task impossible.

The anti-inverse degradation of the above sedonion algebra may be experienced algebraically by attempting to create a 15x7 table for

273 APPENDIX A - OCTONIONS AND SEDONIONS sedonion 7-spaces analogous to the 7x3 table for octonion 3-spaces. A 15x7 table for sedonions would divide fifteen-dimensional space into fifteen different seven-dimensional spaces. The anti-inverse degradation is illustrated visually by attempting to create a sedonion tetrahedron diagram analogy to the octonion triangle diagram. The lack of a division algebra for this simple attempt at sedonion algebra is proven in the Hurwitz Theorem. See the last “Thought Exercises” question for the beginning of a second attempt at a 2x2 matrix formulation of sedonion algebra. Thought Exercises 1)

Write the 2x2 matrix isomorph for octonion “a + b*j1 + c*j2 + d*j3 + e*j4 + f*j5 + g*j6 + h*j7”.

2)

Write a matrix multiplication table for the octonions “j1”, “j2”, “j3”, “j4”, “j5”, “j6”, “j7” and the complex number factor “i”. Use “q1”, “q2”, “q3”, “q4”, “q5”, “q6”, “q7” with “q1 = j1/i”, etc. Address the negatives with a note.

3)

Write the 2x2 matrix isomorph of the octonion-complex number “a + b*q1 + c*q2 + d*q3 + e*q4 + f*q5 + g*q6 + h*q7” in which the components “a”, “b”, “c”, “d”, “e”, “f”, “g”, and “h” are mathematically complex.

4)

In Chapter 1 is a section titled “Quaternion-ComplexHypercomplex Numbers”. Follow the format of that section for octonions and octonions-complex. Why can’t octoniansquaternions or sedonions be placed into that format?

5)

“j1”, “j2”, “j3”, “j4”, “j5”, “j6”, “j7” were each assigned to a 2x2 matrix isomorph, per the text above. Is there an alternative matching of 2x2 matrix isomorphs to “j1”, “j2”, “j3”, “j4”, “j5”, “j6”, “j7” holding to the criteria that the same 7x3 table of triples applies?

6)

There is no algebra by which to find the real number magnitude of “658*i + 89*jx + 57*jy + 456*jz + 26*j1 + 44*j2 + 785*j3 + 963*j4 +

274 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 76*j5 + 659*j6 + 154*j7” because quaternion-octonion products have no specified algebra. This issue will likely get resolved through an application in applied mathematics. Find that application. 7)

Find octonion matrix isomorphs that are 2x2x2 matrices that use the complex label-numbers “1” and “i” for terms/elements. The major diagonal of a 2x2x2 matrix would likely be the elements “(ar + i*ai)” and “(ar + i*ai)*i”, in analogy to the 2x2 matrix isomorphs for quaternions. The three minor diagonals would likely each be analogous to the one minor diagonal of the 2x2 matrix isomorphs for quaternions. The challenge is to develop a 2x2x2 matrix multiplication scheme for which the 7x3 table of triples for octonion multiplication is satisfied. (ar + i*ai) (br + i*bi) (cr + i*ci)

(dr + i*di) -(cr + i*ci)*i -(dr + i*di)*i -(br + i*bi)*i (ar + i*ai)*i

1: ar = 1 ; j2: cr = 1 ; j4: ai = 1 ; j6: ci = 1 j1: br = 1 ; j3: dr = 1 ; j5: bi = 1 ; j7: di = 1 8)

Try to find a 2x2x2x2 matrix structure for sedonions. Four corners are adjacent to each of the two corners of the major diagonal. And six corners are not adjacent to the two corners.

9)

Propose an expansion of the Dirac Equation that uses octonion matrix isomorphs in addition to, or as a substitute for, the quaternion 2x2 matrix isomorphs. A clue in the text was the justification for “e4*qx = -qx*e4”. Another clue is that quaternions are inside the matrix isomorphs of the octonions.

10) The 7x3 table for octonions implies the first column is an “x” dimension, the second “y”, and the third “z”. But that implication is not used in the algebra. It suggests the algebra is incomplete.

275 APPENDIX A - OCTONIONS AND SEDONIONS 11) Using triples “j13s = j3s*j14s”, “j5s = j3s*j6s”, “j13s = j8s*j5s” and “j6s = j14s*j8s”, the violation of the anti-associative property that led to the anti-inverse property is: j14s = -(j3s)*j13s = -(j6s*j5s)*j13s j14s = -j6s*(j8s) = -j6s*(j5s*j13s) = +(j6s*j5s)*j13s Can the matrix multiplication operation be modified to remove the anti-associative violation? See the matrix multiplication operation below, in which new groupings apply (for example, “F(CI)” rather than “(FC)I”). The new groupings would affect octonions and not quaternions, and the operations become trinary and not binary. A C ((

E G )*(

B D

I K

))*( F H

(AE+FC)I+J(GA+CH) K(AE+FC)+(GA+CH)L )= J L I(EB+DF)+(BH+HD)J (EB+DF)K+L(BG+HD)

(AE)I+(FC)I+J(GA)+J(CH) K(AE)+K(FC)+(GA)L+(CH)L = I(EB)+I(DF)+(BH)J+(HD)J (EB)K+(DF)K+L(BG)+L(HD)

276 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

277 APPENDIX B - SPOOKY ACTION AT A DISTANCE

Appendix B – Spooky Action at a Distance EPR Experiment Set Up. Two photons created as a pair have coordinated properties and are called “entangled”. The two entangled photons travel a macroscopic distance, perhaps a meter or further in opposite directions, and are each detected. Each detector is comprised of a polarizing film followed a photographic film. The two polarizing films are parallel and both photographic films detect the photons, to show us the polarity of the photons was the same, as expected, because they are entangled. To explain this expected experiment result, we might venture to guess the polarity of the two photons was determined at the time of creation at the emission source that sits between the two detectors. Other phases of the EPR experiment have the polarizing films at a different angles. The results are interpreted using Bell’s Inequality (from year 1964) and lead to the conclusion the polarization of the two entangled photons is determined at the moment of detection, not emission. For that to happen, the direction of polarization is coordinated over the macroscopic distance that separates the two detectors and coordinated instantly over that macroscopic distance: Not at the speedof-light but faster, instantly. Einstein called the instant communication “spooky action at a distance”.

Figure 44. The EPR Experiment for photons.

278 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The EPR experiment may also be performed with electrons. Electrons do not have polarization, but, rather, electrons have spin. The axis of spin (using the right-hand rule) is measured as a direction perpendicular to the plane of rotation. Two entangled electrons have the axis of spin of one electron anti-parallel with respect to the other electron, as required for a total of zero angular momentum for the pair.

Figure 45. The EPR Experiment for electrons.

For our visualization, we substitute baseballs for electrons. Two baseballs were resting beside each other. A little expanding spring between the two baseballs sent the baseballs in opposite directions. The little expanding spring was not perfectly aligned between the two centers of the baseballs, and, therefore, created a spin for the two baseballs. The axis of spin must be equal and opposite, so that the angular momentum of both baseballs together equals zero. We watched the release of the expanding spring. We saw the spin of the two baseballs as they moved in opposite directions. We watched the two baseballs each pass their spin direction detector. The spin detector display confirmed what we had been seeing. The baseball is a particle, always a particle. The baseball is a particle before its spin is detected, and the baseball is a particle when its spin is detected. Electrons are different from baseballs because electrons have particle/wave duality. The electron is a wave until it is detected as a particle. Unlike particles, waves do not have spin. Therefore, waves do not have an axis for the spin. Per the particle/wave duality of quantum mechanics, the axis for the spin of the electron can only become specified when the electron transitions from being an unobserved wave into being an observed particle, a transition that occurs at the detector.

279 APPENDIX B - SPOOKY ACTION AT A DISTANCE Photons, too, have particle/wave duality. The polarization of a photon is a property of the particle. For a visualization, imagine the wave of a single photon approaching a polarizing film. The polarizing film is comprised of long stretched molecules that have thin gaps between the molecules. The photon’s electric field can only pass those long stretched molecules if the electric field is exactly parallel to the gaps. In contrast, if the electric field is exactly perpendicular to the gaps, then the stretched molecules absorb the photon by vibrating back and forth (like a guitar string) due to the energy gained by absorbing the photon’s electric field energy. This vibration energy is the energy of the photon, now fully absorbed. And then, the vibration energy is lost to heat. On the back side of the polarizing film is a photographic film detector with an electron that absorbs the photon with a jiggle vibration, if the photon passed the polarizing film to reach it. The jiggle of the electron causes an exposure spot on the photographic film. The polarizing film forces the single nondivisible photon to be completely polarized with the gaps (such that it passes) or completely polarized perpendicular to the gaps (such that it is absorbed). The photon, as a particle, cannot be a combination of perpendicular and parallel, because the photon is quantized as an all-ornothing particle. Half the photons pass the polarizing film, and half are absorbed. The electron, when used in the EPR experiment, likewise, is a single particle at the detector. The electron spin sensed by the detector forces the spin to become parallel or else anti-parallel to the detector device (analogous to the photon electric field being forced to be parallel or else perpendicular to the polarizing film of the detector device) with no intermediate spin direction possible. The lack of the particle property of polarization for photons and the lack of the particle property of spin direction for electrons, up until the moment of detection, was a debated issue, until the EPR experiment with Bell’s Inequality settled the debate. Bell’s Inequality is explained with an example. Consider a set of bikes in a large garage. Each bike has (A) an engine or not, has (B) five pounds of fuel or not, and has (C) blue paint or not. Notice that correlated properties may be included. For example, it doesn’t matter that bikes without an engine would naturally not have any fuel.

280 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The three properties “A”, “B”, and “C” each have values of “Yes” or “No”, for eight combination possibilities per bike (because 8 = 23). Each quantity (QTY) is found by counting. QTY1(YYY), QTY2(NYY), QTY3(YNY), QTY4(YYN) QTY5(YNN), QTY6(NYN), QTY7(NNY), QTY8(NNN) (YYN) means A=Yes, B=Yes, and C=No, and (NYN) means A=No, B=Yes, and C=No. If a letter A, B, or C is in the parentheses, then that property can be either Yes or No. For example, QTY(AYY) has A=Yes or No, B=Yes, and C=Yes. If there are ten bikes in the garage, then the eight quantities listed above sum (as QTY(ABC)) to ten. As an example, consider a set of ten objects: (YNN), (YYN), (YNN), (NYN), (NNN), (YYY), (YNN), (NYN), (YYN), (NNY) In this example set QTY1(YYY)=1 QTY3(YNY)=0

QTY5(YNN)=3 QTY7(NNY)=1

QTY2(NYY)=0 QTY4(YYN)=2

QTY6(NYN)=2 QTY8(NNN)=1

Bell’s Inequality is QTY(YNC) + QTY(AYN)  QTY(YBN) In words: The quantity of objects with A=Yes and B=No plus the quantity of objects with B=Yes and C=No is greater than or equal to the quantity of objects with A=Yes and C=No. The first term QTY(YNC) has no consideration as to the state of ‘C’. It can be expanded, as can the other two terms: QTY(YNC) = QTY3(YNY) + QTY5(YNN) QTY(AYN) = QTY4(YYN) + QTY6(NYN) QTY(YBN) = QTY4(YYN) + QTY5(YNN) Using these expanded terms:

281 APPENDIX B - SPOOKY ACTION AT A DISTANCE QTY(YNC) + QTY(AYN)  QTY(YBN) QTY3(YNY) + QTY5(YNN) + QTY4(YYN) + QTY6(NYN)  QTY4(YYN) + QTY5(YNN) QTY3(YNY) + QTY6(NYN)  0 The last statement is true because all quantities, including QTY3(YNY) and QTY6(NYN), are greater than or equal to zero. Therefore, Bell’s Inequality is proven. Using the ten objects of our example, we have: QTY(YNC) = (0+3) = 3 QTY(AYN) = (2+2) = 4 QTY(YBN) = (2+3) = 5 Because 3 + 4 = 7  5, Bell’s Inequality is satisfied. In words per the example: The number of bikes with an engine and less that five pounds of fuel plus the number of bikes with more than five pounds of fuel and no blue paint is greater than or equal to the number of bikes with an engine and no blue paint. In words the inequality is not obvious or intuitive. That is why we need the math. The important point about Bell’s Inequality is that it must be satisfied if the objects have properties. If Bell’s Inequality is not satisfied, then the objects do not have properties. Predicting How Bell’s Inequality Applies to a Baseball. In the plane perpendicular to the direction of motion is the center point of the ball and through that center point is a line for the projection of the axis of rotation onto the plane. Use the right-hand rule so that the axis of rotation projected onto this plane has an arrowhead designating the direction of the thumb. From the center point of the ball, draw the “x”-axis and, at ninety degrees, the “y”-axis on the plane. The angle “” from the “x”-axis to the arrowhead is always known for the baseball because, for a baseball, the direction of the axis of rotation exists even when we are not detecting it.

282 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Figure 46. Properties A, B, C used in Bell’s Inequality.

Figure 47. Combined properties used in Bell’s Inequality: QTY(YNC), QTY(AYN), QTY(YBN), respectively. QTY(YNC) + QTY(AYN)  QTY(YBN).

Figure 48. EPR experiment results with baseballs and not with electrons.

283 APPENDIX B - SPOOKY ACTION AT A DISTANCE “A” is “Yes” if the axis of rotation is in the hemisphere of “+y”, 0   < 180o, and “No” corresponds to 180o   < 360o. “B” is “Yes” if the axis of rotation is in the hemisphere with both “+x” and “+y”, -45o   < 135o, and “No” corresponds to 135o   < 315o. “C” is “Yes” for “+x”, -90o   < 90o, and “No” for 90o   < 270o. Each baseball pair is given “Yes” or “No” for each of A, B, and C. Baseballs that satisfy A=Yes and B=No have 135o   < 180o (one eighth of a circle). Baseballs that satisfy B=Yes and C=No have 90o   < 135o (one eighth of a circle). The baseballs that satisfy A=Yes and C=No have 90o   < 180o (one quarter of a circle). One eighth plus one eighth equals one quarter, and therefore Bell’s Inequality is satisfied. We do the experiment (hypothetically) three times. The first set of 1,000,000 has 125,032 A=Yes with B=No. The second set of 1,000,000 has 124,992 B=Yes with C=No. The third set of 1,000,000 has 250,005 A=Yes with C=No. QTY(YNC) + QTY(AYN)  QTY(YBN) 125,032 + 124,992 = 250,024  250,005 Bell’s Inequality was satisfied. Because the three numbers were from three different experiment events, there was a chance a statistically explainable variation could have made “QTY(YNC) + QTY(AYN)” slightly less than “QTY(YBN)”. That possibility alerts us to the need to have a very large number of runs in the experiment. Predicting How Bell’s Inequality Does Not Apply to an Electron. The right detector (which is the first to make a detection) reads if the axis of rotation of the electron is parallel with the “+y”-axis. The left detector has the same reading as the right detector (after compensating for the expected axis of rotation to be anti-parallel).

Figure 49. Both detectors set to “A”. Electrons are detected 50% for “A” equal to “Yes” in both.

284 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Now, in the second phase of the experiment, the left detector (second detector) is rotated so that it reads if the axis of rotation is parallel (or else anti-parallel) with the “+x”-axis. This reading is instantaneously after the right detector (“+y”-axis) reading. The word “instantaneous” means a signal at the speed-of-light would not have time to reach the second detector. Per quantum mechanics, the event of reading “A” (if the axis of rotation is in the “+y” direction) forces the electron’s axis of rotation to be exactly in the “+y” direction, or else in the “-y” direction, and nothing else. The second detection for reading “C” (on the left, an instant later) forces the rotation axis of the entangled electron to now move parallel or else anti-parallel to the “+x”-axis. The “+x”-axis is perpendicular to both the “+y”-axis and the “-y”-axis. Therefore, half the detections for the second detection (on the left) read the “+x”-axis and half the detections read the “-x”-axis.

Figure 50. Left detector set to “C”. Electrons are detected 50% for “C” equal to “Yes”. QTY(YBY) equals one quarter of the total.

Now, as a third phase in the experiment, have the first (right) detector aligned with the “+y”-axis, and have the second (left) detector aligned with the “-y”-axis. In this case, the left detector reads a “No” for each “Yes” on the right detector. This is an extreme case for which the angle “” equals zero. The general rule is that the second (left) detector has a probability of “sin2(/2)” with “” measured from the (opposite) direction of the previous measurement of the entangled pair of electrons. If the two detectors are parallel (the first phase), then “=180o” so that “sin2(/2) = sin2(180o/2) = sin2(90o) = 12 = 1”. If the two detectors are anti-parallel (the third phase), then “=0o” so that “sin2(/2) = sin2(0o/2) = 0”. And, if the

285 APPENDIX B - SPOOKY ACTION AT A DISTANCE two detectors are perpendicular (the second phase), then “=90o” so that “sin2(/2) = sin2(90o/2) = sin2(45o) = (1/2)2 = 1/2”. The last two phases of the experiment have “=45o” (so that “sin2(/2) = sin2(45o/2) = sin2(22.5o) = .382682 = 0.146…”) and “=135o” (so that sin2(/2) = sin2(135o/2) = sin2(67.5o) = .920002 = 0.854…”). All the particles are accounted for between what is counted as inconformance and not-in-conformance, per “cos2(/2) + sin2(/2) = 1”. The EPR Experiment with Electrons. Do the experiment (hypothetically) with pairs of electrons. There are three experimental set ups.

Figure 51. EPR Experiment with electrons. Bell’s Inequality is not satisfied because 7% + 7% ≱ 25%.

“73,111” events in “1,000,000” have A=Yes with B=No. The number “73,111” conforms to “1,000,000*0.5*sin2(45o/2)”. The “0.5” factor applies to A=Yes and the factor “sin2(45o/2) = 0.146…” applies to the subsequent detection for B=No. After that, “73,056” events have B=Yes with C=No. And then, “249,986” events have A=Yes with C=No.

286 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY We do a statistical evaluation to explain away the minor error in the numbers. We conclude that Bell’s inequality is not satisfied because: QTY(YNC) + QTY(AYN) ≱ QTY (YBN) for electrons 73,111 + 73,056 = 146,167 ≱ 249,986 In the laboratory, Bell’s inequality was not satisfied in photons, electrons, protons or any other subatomic particles. Properties Determined at the Instant of Detection. We expected Bell’s inequality to be unsatisfied because of the “sin2(/2)” rule, which was derived from the requirement the spin of the electron becomes parallel or else anti-parallel with the axis of measurement. Because Bell’s Inequality is unsatisfied, we conclude there are no hidden variables or other memory mechanisms by which the direction of spin of an electron was determined when emitted and simply carried as information to the detectors. It means the direction of spin is determined at the moment of detection. A typical emission is one particle, not two entangled particles as in the EPR experiment. When this one particle is detected, perhaps by your eye, or perhaps by a scientific device that we were monitoring, we were unsure if that particle traversed the distance from its emission time and place to the detection time and place as a particle with distinct particle properties (for example, a spin axis direction), or not. When the particle travels it is a wave, and a wave should not be a carrier of particle properties, but subatomic particles are a bit mysterious. Before we had the EPR experiment results, it was thought that perhaps waves did carry particle properties. And that is because, when detecting one particle, we did not have a means to know if the spin was established at the time of emission or at the time of detection. Now, because of entanglement, we know. Instantaneous Information Travel. The second conclusion is that information traveled a macroscopic distance instantaneously. There is a macroscopic distance between the two detectors, and those two detectors each detect the spin direction simultaneously. The direction of spin is determined only at the instant of detection, and that means the

287 APPENDIX B - SPOOKY ACTION AT A DISTANCE readings on the two detectors are mysteriously coordinated one to the other. It means information traveled from one detector to the other at instantaneous speed, not as fast as the speed-of-light but faster, faster to the ultimate speed: instantaneous. For the typical situation of one emitted particle (and not two entangled particles), spooky action at a distance is a “collapse of the wave function”, in which the wave disappears from everywhere the particle is not (relative to an observing particle), when the particle is detected. The challenge is to explain spooky action at a distance with a mathematical model. New Theory. An electron and the observed photon from that electron are both one-in-the-same particle, per the proposed Theory of Special Relativity with Non-Finite Numbers. Entangled particles, too, are one particle: A whole light cone for two entangled photons is one-in-thesame as the electrons at its center, relative to an observing particle, until an observation event (or lack of observation event) of a photon is made by an observing particle. More rigor is needed, but the one particle theory has promise in explaining the EPR experiment results. It is through the EPR experiment that quantum mechanics is brought to a macroscopic scale, to where we can physically see how strange quantum mechanics is. Therefore, naturally, we want to use the EPR experiment to see with our own eyes any proposed new theory.

288 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

289 APPENDIX C – DISCOVERING AN ABSTRACTION

Appendix C – Discovering an Abstraction About the Author. The author, Paul C Daiber, is a mechanical engineer who designs and services combustion turbine engines. Outside of professional employment, the engineer chose to solve this puzzle: Define infinity by finding an inertial frame of reference in which a blue photon has a faster speed than a red photon. An engineering spec/procedure was written for developing new theory: 1) Abstractions. How we understand our world. •

Definitions. A person categorizes patterns in the physical, real, natural world by creating a definition for each identified pattern. Each definition is an abstraction. An abstraction is defined in words or other symbols for people to use.

•

Ours. Abstractions as definitions of categories are for our use. We define abstractions as we find useful or otherwise as we want. No external entity gives us our abstractions or defines our abstractions.

•

Application and Ambiguity. Definitions of abstractions have some degree of ambiguity. The definitions are only valid to the extent the definitions are consistently applied to the real world by the definition’s prescribed rules.

•

Names. Abstractions are each named. Each is accompanied by a counter-set of abstractions, each of which also get a name.

•

Start Simple. Begin the development of definitions of new abstractions by first finding patterns that are obvious. Develop more definitions by finding less obvious patterns.

290 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2) Discovery. A discovery is outside what can logically be derived from the abstractions, used as axioms, that we already have. •

Research. Become familiar with what are assumed to be the inherently valid abstractions by which we know our world.

•

Guess a new definition by investigating an observation, or variations, extrapolations, combinations, tangents, violations, or alternatives.

•

Analysis. Build a bridge of logic from old definitions to the guess or from the guess to old definitions by evolving the guess to conform to logic.

•

Finish. Write it up to communicate the discovered abstraction to other people. Definitions of abstractions can be communicated and comprehended only by physical, real, natural world example experiences other people have with that abstraction, through the use of the prescribed rules of the definition.

The method for discovering an abstraction was inspired from a passage in Morris Kline’s book A History of Mathematical Thought, Volume 1 (Oxford University Press, 1972) in which he paraphrased Descartes who said logic could not be used to investigate unknown fundamental truths. It seemed Descartes thought that thoughts from outside of logic were needed when searching for something the rules of logic were not designed to find. Elsewhere in that chapter, Morris Kline said mathematicians at the beginning of the age of algebra developed theory without discipline or rigor. In contrast to that era four hundred years ago, the algebra of today is full of discipline and rigor. If we were to rethink algebra enough to include infinity in applied mathematics, then, it seemed, we had to remove ourselves from today’s discipline and rigor. Being undisciplined, we grab what we want. We violate an axiomatic rule to create something illogical, because we really want it. (The value of grabbing what is wanted was perhaps learned from Star Trek in the 1960’s: Captain Kirk got what he wanted by leading his crew to do what he wanted, regardless of rules and regulations imposed

291 APPENDIX C – DISCOVERING AN ABSTRACTION on him. The intelligent, unemotional and exclusively logical first officer, Spock, could not lead by violating rules. Therefore, Spock could only be a support person to his captain.) To solve the puzzle of the blue and red photons, the existing theories of physics had to be learned. Maxwell’s Equations was learned in engineering school using Halliday and Resnick’s textbook. Schrodinger’s Equation had to be self-taught. Fortunately, Eisberg’s and Resnick’s textbook was on someone’s bookshelf at work, and Gasiorowicz’s textbook was found in a used-book store. Real treasure came from the author’s dad’s college textbooks, from the mid-1950’s, which were intercepted on their way to the trash when the author was home from college. In those books was Methods of Theoretical Physics Part I by Morse and Feshbach, McGraw-Hill Book Company, Inc., 1953. Unlike what was found on the internet, this old book’s development of the Dirac Equation was written for engineers and easily learned. Next, the Dirac Equation and Maxwell’s Equations had to be placed into one algebra. The most effective algebra had been assumed to be the geometric algebra in Chapter 6 of Hildebrand’s textbook Advanced Calculus for Applications from college. But geometric algebra didn’t work well. Quaternions were found better because of the ease of writing identities and tracking gauge invariance, and because the electric and magnetic fields could be combined into one invariant by having complex components. Reading math textbooks became an enjoyable hobby. That might sound strange, but the author was recently with two old college roommates (Gator and Sonny) and they went straight to the math section when the three of them entered the Half Price Books bookstore in the (Walnut Creek, CA) square. During the talking in Starbucks that evening, it appeared that, in general: People seek answers to the mysteriousness of the world around them by learning the rational and logical techniques of math. Infinity was one such mystery. What was the actual infinity? Extensive reading did not find an answer. One of the best books on infinity was Aristotle’s book Physics. He summarized what was suspected about infinity at that time, twentyfour hundred years ago. A generation or two prior to Aristotle, the Pythagoreans had proven the irrationality of the square-root-of-two, and that proof forced the Pythagoreans to admit that infinity, as something

292 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY different from finite natural numbers, had mathematical relevance. The Pythagoreans disliked infinity because they attributed to infinity the unwanted properties of “mysterious, vague, and remote” (these three words are a remembered quote from a book that cannot be found), and, in general, the Pythagoreans disliked the contamination of something proven to be irrational in their system of rational deduction. Archimedes, too, disliked infinity (per his story The Sand Reckoner) as did most of the great men of science in ancient Greece, Aristotle included. The general dislike of a positive actual infinity in ancient Greece grew on the author like a piece of good advice from a parent. The feeling was that an actual infinity existed in our mathematics as a temporary substitute for a theory presently unknown or unavailable to us. Early attempts to learn Cantor’s theory of infinite sets were unsuccessful because Cantor’s real numbers seemed to not include irrational numbers. Other people had likely seen this issue, with the evidence being this quote: “Cantor took N1 to be the power of the continuum. This question, however, remains open, and for the present we see no trace of a path to its solution.” (from The World of Mathematics Volume Three by Newman, 1956, Simon and Schuster, page 1599 in the article Infinity by Hans Hahn.) That book opened the door to seeking alternatives to what Cantor had specified to be the real number continuum. One early attempt (by the author) gave an irrational number a positive infinity of place-value digits after the decimal point, and then unwritten and unspecified place-value digits after that. It was close to what became local-real numbers, but at the time it led nowhere. In the last two years of the project, the author’s son’s college pure mathematics textbook, written by Chartrand and others, was found in a box in the basement, and Cantor’s theory could finally be learned as a full and complete theory. Once the bigger picture of what Cantor intended was learned, then his theory could be adapted into what was becoming the solution to the blue/red photon puzzle. A second lesson from Cantor was that he grabbed at a definition for infinity, and he then tried to build a bridge of logic to it. He had the right idea with regard to process. Long before Cantor’s theory was understood, at a time when there really wasn’t much hope the project would be successful, an idea for infinity came from the phrase “forever and a day”. The reciprocal-ofzero was the forever. To add a one to it, the hyperbolic trig functions were used: “cosh(1/0) = 1 + 1/0” and “sinh(1/0) = 1/0”. From that guess

293 APPENDIX C – DISCOVERING AN ABSTRACTION Maxwell’s Equations were derived from the Dirac Equation. It was the first big break. The “1/0” meant the mathematics was invalid, but, but, there was truth down inside it, truth somewhere deeply hidden. The truth became certain when the correct electromagnetic field force density was calculated using the complex conjugate. A burden of responsibility was felt, because, if the bridge of logic did not get built, then this discovery would likely become lost and forgotten, and that was because anyone who cared would not get to hear about it because any mathematics dependent on a division by zero cannot be published. Division by zero is famous as a tool of deception and is called “pseudo-mathematics”. If the division by zero could not be removed, then, most likely, the discovery would not be stumbled upon again for many, many years, perhaps for as long as it had already lay hidden, from when the Dirac Equation was discovered and understood (1930) to then (2010). The formal prohibition against division by zero is in the definition of rational numbers, and that meant, maybe, “1/0” and “0/0” could be changed into the wanted infinity by investigating irrational numbers. “2” was ripped apart every way possible. After some years the focus changed to “” (zeta) in “1 = 1 + 5” and “1 = 2 + 3”, which were found by equating lengths thirty degrees apart on the equilateral triangle spiral. A rational approximation to “” was calculated for each of the two equations. Was the portion not-yet-calculated the same with regards to the geometric construction of the triangle spiral? It took many years to develop that question, and even longer to answer it. It was guessed the two “”’s could only be precise to an actual infinity quantity of place-value digits. Their difference was called a real number zero. The unknown place-value digits starting at the actual infinity count meant the real number zero was different from an integer zero, and that meant it could be in the denominator. The division reciprocal of the real number zero became the infinity that applied to the hyperbolic-angle for motion at the speed-of-light. This was another moment of celebration. With division by the integer zero gone, the theory seemed complete enough to actually tell someone. A downloadable early version of this book (titled: Infinity Applied to Special Relativity) was placed on the internet, to make it available to the public in case the project was never finished. The book

294 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY told a somewhat logical story from beginning to end but it lacked key essentials, for example, it lacked the exotic Lorentz Transformation needed to model motion at the speed-of-light. It wasn’t advertised and no copies sold, and tallies showed only a person or two reviewed the first several free pages. The real number zero became the “local-zero” when Aristotle’s potential infinity, a finite number, replaced the positive actual infinity in the count of place-value digits. With no actual infinity in the numbers, there was a worry the proposed system of numbers had deviated too far from what was generally accepted as numbers. At the very end of the project a new proposed axiom substantiated the removal of the positive actual infinity, and the bridge of logic was completed. Per the method, we first guessed at the final definition of the abstraction (which was the combination of Maxwell’s Equations into the Dirac Equation using division by zero), and then the ideas evolved until the core idea was found (which was the proposed new reciprocal-of-zero axiom). To communicate the discovery logically, per the method, reverse the activities (with the proposed new reciprocal-of-zero axiom explained first, and the derivation of Maxwell’s Equations from the Dirac Equation last). The reversed activities is needed for a logically derived argument. But, the reversed activities in the communication erased any trace of the process of discovery that was actually followed. To not lose that excitement, with its guesses, its stress, the finding of critical books, and its little victories, this “About the Author” was written. Early versions for this math book had little more than an initial guess at what infinity could be. A guess cannot be the logical foundation of a math book, regardless of how strongly a person claims the guess is correct. Discipline came from Paul’s brother Andy J Daiber, another engineer. Andy was/is a hobby mathematician, but Andy was not someone like the author who searched for playgrounds in numbers. Rather, Andy was someone who appreciated the serious core values pure mathematicians have. (Andy sent the book by Mary Tiles, the Teach Yourself book on group theory, Paul Cohen’s Set Theory and the Continuum Hypothesis, and over the years he sent maybe a dozen other pure mathematics books to the author.) The author is grateful to Andy for reading this book every several years, when the book was incomplete and illogical. Andy mercilessly shot down every idea presented without substantiation. Our human tendency is to think our claims are correct, and to defend our claims when subjected to criticism. In mathematics,

295 APPENDIX C – DISCOVERING AN ABSTRACTION only logic can be a defense. When digging deeper to find fundamental evidence to prove one’s claims, we stumble upon fantastic discoveries that are unexpectedly interesting and useful. That generalization of an interactive process summarizes the human side of the development of the proposed Theory of Non-Finite Numbers. The attitude, or the sense of values, that fundamentally led to the need to write this book came from something said by a Citizenship in the Community merit badge counselor, Mr. O’Brien, paraphrased as: Don’t bother reading something unless you intend to do something with that knowledge. Now, forty years later and no longer a child, this rule for living is changed to: Success in life is not measured by how much knowledge you take to the grave with you, but rather by what good you did with that knowledge. I hope some good comes out of this effort to insert an infinity into applied mathematics. What I really want is for your curiosity to have become ignited, and for you to further grow theory for non-finite numbers. And, I hope you enjoyed reading this book. \\//,

The Storybook In its first draft, the storybook Alien Invasion Math Story was a collection of essays and calculations written during the hunt for a useful infinity. The added storyline with its exaggerated emotional extremes portrayed the confusion, frustration, misdirection and all the stress felt in the hunt. Inspiration came from the movie Race for the Double Helix (BBC), in which Crick and Watson mixed incomplete bits of data (from Franklin) with staggered inspiration and persevered past criticism and little failures to win the race. Their moment of success came with a revelation. Me, sitting at a desk or standing in a store watching my wife shop while thinking through proofs is not nearly as exciting as Crick and Watson in their quaint apartments and labs, so my storyline is fiction. But, at least the math and the essays on human will are real. And, the revelation at the climax, well, you’ll just have to read it. \\//,

296 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Glossary Actual Infinity. Aristotle’s completed infinity. An infinity that is larger than finite. Aleph Null. Cantor’s version of the completed/actual infinity to which the Continuum Hypotheses is assigned. Algebra. A set of rules by which logic can be exercised. Algebra Field. Formed by two groups and a distributive property. A set of elements and the criteria by which operations on those elements are performed. All-Number. An expression formed from quantity numbers, label-numbers, the unspecified-speed-parameter, and the unspecified-label-number. Anti-Matter. The negative of matter. If matter is a something, then anti-matter is the hole created by the removal of that something from nothing. Applied Mathematics. The adaptation of mathematics to mathematical models of physics. Argument. The independent variable on which a function operates. Axiom. A fundamental assumption that is accepted as a basis for subsequent derivations. Axiomatic Set Theory. The mathematical theory of a set of axioms from which algebra is derived. ZFC axiomatic set theory is one form and is most popular. Axiomatic set theory was developed in the early 1900’s. Biot-Savart Law. The mathematical model of the electromagnetic field of a moving point particle. Both Plus and Minus But Also Neither Plus Nor Minus Separately. (N) The plus or minus sign is unknown and unknowable. It is in contrast to the plus or minus sign () because the implication of the plus or minus sign is it is known or knowable. Cantor’s Continuum Hypothesis. The axiom of axiomatic set theory from which the real numbers are derived. It forces the existence of an actual infinity. Cantor’s Theory of Infinite Sets. A theory of pure mathematics in which an actual infinity is assumed to exist between finite numbers and the reciprocal of the integer zero. Cardinality. The quantity of members in a set. A number is cardinal if it is the count of members in a set. This definition is part of Cantor’s theory of infinite sets. Cartesian Grid. The typical x-y plot. Descartes used the Cartesian grid to plot roots of polynomials. Cause and Effect. The name for related events that are sequential in time. Collapse of the Wave Function. When an observation of one particle by another particle occurs, the perception of the observed particle changes from being a wave that is spread through space and time to being a particle that has one location in space and time. This

297 GLOSSARY AND INDEX transition from being everywhere to being in one place is informally called the collapse of the wave function. Complex-Conjugate. An alternate representation of a complex or hypercomplex number. The complex-label-number “i” is made negative (and the order of factors and terms is reversed but only after the quaternion simple-label-number factors in each term are reduced to only one quaternion simple-label-number). Component. The factor of term in an expression that is not a direction indicator. A component is often measurable. Compound-Label-Numbers. Label-Numbers that do have a factor for gauge invariance. For a single-term summation-form all-number expression of an invariant, the compoundlabel-number is the same as a simple-label-number. Conjugate. An alternative all-number expression. multiplication of two all-number expressions.

The conjugate form is used in

Conservation Law. A mathematical expression that specifies how a material is created or destroyed. Continuum. The property by which spacing is not finite, but, rather, is smaller than finite. Contravariant. The alternative coordinate system representation for use with nonrectilinear or non-unit-magnitude geometric-unit-vectors. Cosmological Model. Our logical model of the physical universe (plethora). Count. A finite natural number. Countable. A set that has a one-to-one member correspondence with the natural numbers, in Cantor’s theory of infinite sets. Crude Non-Number Algebra. The name given to the proposed algebra for the reciprocal of the integer zero based on the proposed new reciprocal-of-zero axiom. de Broglie Relations. Total energy equals the modified Planck’s constant times the frequency. Total momentum equals the modified Planck’s constant times the wavenumber. Dedekind Cut. A cut in the number-line for which the high-side and low-side numbers at the cut, and numbers between them, are all “essentially” the same number. Per Dedekind: “From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts.” Degradations. The break-down in symmetry for higher orders of hypercomplexity. Denumerable. Same as Countable. Descartes. Mathematician and philosopher who marks the beginning of the modern era of intellectual thought. Dirac Equation. The relativistic mathematical model for the dynamics of an electron.

298 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Dirac Spinor Solution. Any of the four Dirac Spinor Solutions presented in this book. Distributed Material. A material that exists as a distribution in space and time such that a gradient exists. Distributed material may also be called a “field”, but “field” is so general a term it should be used cautiously. Distributed Material Theory. The theory (in this book) for which continuum operators, such as the gradient differential operator, may be applied. E-M-Compound-Label-Numbers. The label-numbers used for a six-term invariant. Enabler Functions. Variables inserted into an equation to accomplish a purpose. Energy-Momentum. The invariant that combines energy (as the time term) with momentum (as the space terms). Engineering-Calculation Algebra. Algebra designed to be as efficient as possible in the task of making a calculation. Entropy. The amount of disorder in the universe. The amount of entropy in the universe increases with time. EPR Experiment. The EPR experiment verified, through the use of Bell’s Inequality, that the collapse of the wave function occurs at the moment of observation and not at the moment of emission of the particle that is observed such that the properties of the particle do not exist prior to the observation. As a secondary result there is the experimental verification of “Spooky Action at a Distance”. Euclid’s Textbook. A textbook for geometry (and, to some extent, numbers) written at the beginning of the Hellenistic age (after Alexander’s conquests) that survived as the mathematics textbook until mathematics was reformulated by the work of Descartes and other contemporaries in the 1500’s using the Arabic numerals previously introduced to Europe by Fibonacci. Euler’s Constant. The difference between the sum of reciprocal positive integers and the natural logarithm of the largest positive integer in the sum plus one. Euler’s Equation. Relates the exponential function to the trigonometric functions. Exotic Lorentz Transformation. A Lorentz Transformation that uses a hyperbolic-angle that is not a real (or rational) number. The exotic Lorentz Transformations take advantage of the label-numbers in the all-number algebra. Exponential Function. The single valued equivalent of a base of “e” to an exponent. The exponential function has a polynomial expansion to infinity as its definition. Factor. A factor in a mathematical expression is separated by multiplication and division signs, such that factors are multiplied. Feynman. A theoretical physicist who contributed extensively to QED and to the theory of electrons and photons in general. Feynman’s fame is perhaps due to his ability to present concepts through visualizations and explanations that were easily comprehendible by his intended audience.

299 GLOSSARY AND INDEX Final-Result. The third step of the Process from Descartes of a local-real number algebra operation applies the condition that the result be a local-real number (with “Lmax” maximum count of known or knowable place-value digits before or after the decimal point). The final-result applies to what can be measured in geometric space, with examples being the components of invariants and speed. Finished-Calculation. The second step of the Process from Descartes of a local-real number algebra operation uses probability theory to find the “finished-calculation”. Finite. A finite number is a number from the set of natural numbers (defined by starting at one and repeatedly and unboundedly adding one), or a finite number is a number that is a result of a binary operation (addition, subtraction, multiplication or division) performed using a natural number or other finite number, per the algebra field for rational numbers. A finite (irrational) number is bounded high and low by two rational numbers. Gauge. A reference against which a perspective is made. Geometric-Unit-Vector. One of the four geometric entities that represent mathematically a direction in space or the direction of time. Governing Equation. A mathematical model for a theory of physics. Gradient Operator. The time and space differential operator. Group. A set and the operations performed on that set, as a basis for unambiguous derivations using that set and those operations. Handedness. The fingers curl with the geometric feature and the thumb points a direction for the geometric feature. If it is the right thumb, then the handedness is righthandedness. If it is the left thumb, then the handedness is left-handedness. In geometry the handedness is the correlation between the nomenclature of x, y, z relating to the fingers passing through positive x to positive y and the thumb pointing to positive z. In physics matter can have a geometrically right-hand or a left-hand spin but, more typically, a reverse of handedness refers to a reverse of parity to the extent that matter becomes anti-matter with the example being a right-hand glove (matter) has the same appearance as a left-hand glove that is turned inside out (anti-matter). Hypatia. Woman teacher at the end of the Hellenic / Hellenistic era. She contributed to the mathematics of conics and had other contributions to science and mathematics. Her murder marks the end of the era of the thousand years of Greek intellectual thought. Hypercomplex-Conjugate. An alternate representation of a hypercomplex number. The quaternion simple-label-numbers are made negative and the order of factors (and of terms) is reversed. Hypercomplex-Plane. Similar to the Complex-Plane, but with one of the two dimensions imaginary rather than real. An illustration of time-space on the space-space of a sheet of paper. What is a circular rotation in time-space is a hyperbolic rotation in space-space and so hyperbolic-angles are illustrated in the hypercomplex-plane using circular angles and that substitution causes a distortion in the illustration of a hypercomplex-plane. Hypothesis. An educated guess.

300 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Identity. A mathematical statement that is true as a proven theorem and is stated generically without reference to a mathematical model of physics. Identity Elements. The zero for addition and the one for multiplication. Imprecision term “” (xi). Either a local-zero or else a local-infinity with the selection unknown and unknowable. It is added to a rational truncated number to form a localreal number. Inertial Reference Frame. A constant speed reference frame. The word “inertial” is used because a reference frame that is accelerating because it is falling in a gravitational field may be approximated as a constant speed reference frame. Infinitesimal. The division reciprocal of infinity. Infinitely small. Infinity. Not Finished. No Finality. Unbounded. No limit. Infinity Relation Equations. The first pass function equations using a real number (or positive real number) zero in the argument. Infinity Summation Equation. A sum of reciprocal natural numbers that extends in the count of terms beyond the largest natural number. Invariant. A mathematical expression for something that is physically real, such that the expression cannot change when the physically real something is observed from a different vantage. Irrational Number. A number proven to not be rational. Kinetic Energy. The energy due to particle movement. Known or Knowable. Known or else knowable in that it may have a definite value. Label-Numbers. Numbers of unit magnitude (or zero magnitude) that represent direction rather than quantity. Length Contraction. The reduction in length when the object is moving relative to an observer. Local-Infinity. The division reciprocal of a local-zero. Local-Real Number. The truncated number portion has the precision of the number limited to a finite maximum count of place-value digits after the decimal point in base two. The same limit is before the decimal point for the truncated number portion. A local-zero or else a local-infinity is added to the truncated number to form a local-real number. Local-Zero. “d = b – b” for the value of each place-value digit after a count of “Lmax” after the decimal point (in base two). Lorentz Transformation. The method for changing component and compound-labelnumbers (or geometric-unit-vectors) for a different inertial reference frame. Macroscopic. The scale of objects for which quantum effects are negligible.

301 GLOSSARY AND INDEX Many Worlds Hypothesis. In traditional quantum mechanics there is only one reality with that reality being the result of particle to particle observations. It seems reasonable to assume that each observation is not unique in that whenever an observation occurs there is a split in reality such that another universe is created in which the opposite observation occurred. The repeated formation of these other universes forms the many worlds of the Many Worlds Hypothesis. Mathematical Model. A logical model for something physical. Typically, a mathematical model is used to make a prediction of a measurement or other definite observation. Equation relating quantitative values with measurement units that can be used to predict the results of an experiment or other observation in the real, natural, physical world. Matrix Equation. An equation that uses a matrix and column vectors. Matrix Isomorph. A matrix that has the same mathematical properties as a label-number. A matrix isomorph is used to find the mathematical properties of a label-number. A matrix equivalent of a label-number such that the set of matrices corresponding to a set of label-numbers has the same behavior in an algebra field. Matter. The stuff of the universe. Matter-Wave. A wave associated with a particle per the theories of Schrödinger’s Equation and the Dirac Equation. The alternative existence of a particle when the particle is not being observed. The matter-wave interference pattern with itself forms the constructive interference group. The particle location is most likely at the peak of the group. Maxwell’s Equations. The mathematical model for the electric field, the magnetic field, and electric current. The set of first order differential equations that mathematically model the electric field and the magnetic field formed by a static or moving electric charge density field. Maxwell’s Wave Equation. mathematically modeled.

The equation by which electromagnetic radiation is

Natural Numbers. The numbers formed by starting at the number one and adding one repeatedly and unboundedly. The set of natural numbers “N = {1, 2, 3, …}” is the set of numbers used for counting. Newton’s Second Law. Force equals mass times acceleration, or, as Newton said it, force equals the time derivative of momentum. Non-Linear. Has products of independent variables. Number-Line. A geometric model of the real numbers. Numerator and Denominator. The fraction 5/7 has 5 as the numerator and has 7 as the denominator. Observer. In Special Relativity the properties of a particles are specified relative to the observer particle that the particle interacts with. The observer particle is personified as a person, as us with our measurement equipment.

302 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Octonions. Similar to the complex number factor but pertain to the seven directions of ultra-space. Particle. An existence with location (and time), momentum (and energy), and angular momentum. A particle may extend over a region of space, but, typically, a particle is approximated as a point particle. Pascal’s Triangle. A pyramid of rows formed by adding the two numbers immediately above, after starting with one: 1, 1 1, 1 2 1, 1 3 3 1, 1 4 6 4 1, etc. Pauli Spin Matrices. The Pauli Spin Matrices are 2x2 matrices used by Dirac in the development of the Dirac Equation. Place-Value Digits. Place-value digits are the integer numbers used before or after the decimal point. The value of the integer number is based on its placement relative to the decimal point, per a power series notation equivalent to place-value digit notation. Polynomial. An algebraic summation in which the independent variable has an integer exponent in each term with the integer exponent typically non-negative. Potential. A quantification of a sum such that the gradient of the potential is a vector field. Potential Energy. The energy due to location in a potential field. Examples are a gravity field, a pressure field, an electric voltage field, and a magnetic field. Potential Infinity. A finite number that is either so large that its value is not relevant, or else a finite number that increases instantly and unboundedly. Defined per Aristotle Process from Descartes. The three-step process that begins and ends with geometry and has all-number algebra for the analysis in the middle. It was pioneered by Descartes in what he called analytic geometry. The process of translating a geometric problem into the abstract for analysis and then translating the result back into geometry as a prediction for a measurement. Product. The result of factors. Property. A partial definition of something. Pseudo-Mathematics. The hidden use of the reciprocal-of-zero to prove true something that is not true. Pseudo-Vector. A vector in a plane such that is depicts a rotation in the plane and not a direction perpendicular to the plane. Pure Mathematics. A system of axioms and proofs. Pure mathematics is in contrast to applied mathematics because in applied mathematics the axioms must be applicable to models of our physical, real, natural world. Pythagorean Theorem. The sum of the squares of the lengths of the two perpendicular sides of a right triangle equals the square of the length of the hypotenuse side. Quantity. A quantity is different from a count because a quantity may be an actual infinity in value. A count is limited to being finite.

303 GLOSSARY AND INDEX Quaternions. Similar to the complex number factor but pertain to the three directions of space. Multi-valued square root of negative one values first proposed by Hamilton in 1843. Rational Numbers. The numbers formed as ratios of finite natural numbers. Real Numbers. The numbers that are not imaginary or complex (per Descartes), or else, the numbers that have the infinite quantity aleph one over an interval (per Cantor). Reciprocal-of-Zero Axiom. Proposed in this book. A new proposed axiom for numbers that has its basis in the proof of irrationality of a logarithm. The purpose of this axiom is to bring axiomatic set theory into applied mathematics. Reciprocal of the Integer Zero. The reciprocal of the integer zero exists as an abstract concept for the largest magnitude number for both positive numbers and negative numbers. The non-number reciprocal of the integer zero is the never-ending end of number magnitudes. Recursive. A recursive process has repeated steps. For example, the natural numbers are created through the recursive process of repeatedly adding one after starting at the number one. Remnant-Product. The real portion of the product of three four-component time-space invariants. Reverse Parity. The complete reversal of a particle, as if the particle were a glove and the glove was inside-out. A particle that is reverse parity of matter is anti-matter. Rest Mass. The hyperbolic-radius of the energy-momentum invariant. Rest mass is thought of as the stuff of our macroscopic world, but there is no stuff of rest mass. Rather, we can think of rest mass as the fields inside a subatomic particle. Root. The root of an equation is a variable value for which the equation is satisfied. The term “square root of two” is the positive root of the equation “x2 – 2 = 0”. Scalar. Having only one term. Schrödinger’s Cat. Until we open the box, we do not know if Schrödinger’s Cat is dead or alive: Schrödinger’s Cat is a cat in a box with a glass flask of poison and with a radioactive atomic nucleus. Inside the box is a radioactive nucleus. If the nucleus decays, then the emitted particle breaks a container of poison and the cat dies. An observer outside the box does not know if the nucleus has emitted a particle to break the flask and kill the cat, or hasn’t. Relative to the observer the cat is both dead and alive as a state that is both and so is neither dead nor alive separately. The decay both happened and did not happen until the decay is observed by the box being opened. The state of the cat becomes known relative to the observer when the box is opened. Schrödinger’s Equation. The non-relativistic predecessor to the Dirac Equation as a model for the dynamics of an electron using a wave as the intermediate observed form of the electron. It is a second order ordinary differential equation and so is quite different from the first order partial differential equation set by Dirac.

304 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Sedonions. Similar to the complex number factor but pertain to the fifteen directions of ultra-ultra space. Simple-Label-Numbers. Label-Numbers that do not have a factor for gauge invariance. Singular-Label-Number. A label-number of zero magnitude. A label-number that results in zero when multiplied by its hypercomplex-conjugate. Singularity. A division by zero. In physics, it is where a division by zero occurs in a theory for a phenomenon, with the popular example being the center of a black hole of General Relativity. Six Dots. Repeats as the zeros repeat after the decimal point of an integer. The items are of a quantity that cannot be counted, and, therefore, cannot have individual properties and are addressed as a quantity in bulk. The quantity is not finite and is assumed to equal the non-number reciprocal of the integer zero. Space-Like Invariant. A four-term (or two-term) invariant that has only a space term for the case of zero speed. Space-Negative. The operator by which the Lorentz Transformation is the inverse matrix of the normal Lorentz Transformation, and, to compensate, the three-space labelnumbers are negative. Normally used for the gradient operator and normally used by convention for anti-matter. An explicitly written marker that identifies an invariant or other mathematical entity as requiring an inverted matrix for the Lorentz Transformation and as requiring a negative on the quaternion compound-label-numbers (that is, on the space label-numbers and not on the time label-number). Special Theory of Relativity. Einstein’s theory for there being no preferred inertial (constant speed) reference frame with the condition that the speed-of-light be the same for every inertial reference frame. Einstein’s Special Theory of Relativity is based on the premise that there is no preferred speed of an observer and the speed-of-light is the same for all observers. Speed-of-Light. The speed of light over long distances in a vacuum. It is a unit conversion factor from time measurement units to space measurement units. Spin. The property of a particle by which it has angular momentum. The spin of a particle is in units of Planck’s constant. Spin may be positive or negative. Spooky Action at a Distance. Einstein’s name for the coordinated polarization (or spin) of two macroscopically separated entangled photons (or electrons) of the EPR experiment. This is Einstein’s name for the particle properties of two entangled properties to become specified at two locations that are a macroscopic distance apart. Sum. The result of terms. Term. A term in a mathematical expression is separated by plus and minus signs, such that terms are added. Theory. In physics a theory is a logical or otherwise mathematical model that is generally accepted as valid for predicting measurements within set limits. In mathematics a theory is a branch or subset of mathematics

305 GLOSSARY AND INDEX Theory-Development-Algebra. Algebra designed to explicitly represent in symbols the subtle aspects of the physics being modeled and designed explicitly to be as abstract as possible. Theory of Non-Finite Numbers. Proposed in this book. The theory of numbers based on the proposed new reciprocal-of-zero axiom in which numbers have an unknown and unknowable portion. Theory of Special Relativity with Non-Finite Numbers. Proposed in this book. The application of the proposed Theory of Non-Finite Numbers to the existing Special Theory of Relativity. Three Dots. The three dots represent a count using the finite natural numbers. Each and every item being counted is individually addressed in the count. Time Dilation. The reduction in the rate of a clock’s mechanism when the clock is moving relative to an observer. Time-Like Invariant. A four-term (or two-term) invariant that has only a time term for the case of zero speed. Time-Space. Also called “space-time”. It refers to the union of time with space as a fourth dimension. Trigonometric Function. exponential function.

Sine and Cosine and other functions, as derived from the

Triple-Vector-Product. The imaginary portion of the product of three four-component time-space invariants. Truncated Numbers. Rational numbers of limited place-value digits. Uncountable. A set that has a quantity of members that cannot be counted because the quantity is greater than the quantity of members in the set of natural numbers, per Cantor’s theory of infinite sets. Unknown and Unknowable. The value cannot be known because the value does not exist relative to the observer. The analogy is Schrödinger’s Cat. Unspecified-Label-Number. A simple-label-number that is unknown and unknowable but is restricted to being one of the three simple-label-numbers and not a combination of them. The unspecified-label-number “” (kappa) is unknown and unknowable and is restricted to being exclusively one of the simple-label-numbers “qx”, “qy”, or “qz”. Unspecified-Speed-Parameter. A hyperbolic-angle that is unknown and unknowable. It differs from an independent variable “x” as an unknown because “x” is unknown but is knowable. The unspecified parameter “” specific to use in compound-label-numbers as a hyperbolic-angle to ensure Relativistic gauge invariance (that is, to ensure there is no preferred inertial reference frame). Vector. Having multiple terms. Voltage. The potential for the electric field. The voltage as a vector-space field includes the potential for the magnetic field.

306 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Wave Group. A wave group is a constructive interference location of a set of individual waves. The wave group moves at a speed that may be different from the speed of the crests of the individual waves, if the waves themselves have different wave crest speeds. World Line. The path of trajectory of a particle or other object on the hypercomplexplane or the four-dimensional version of the hypercomplex-plane. Or, in general, the path of a particle through four-dimensional space-time with that world line a continuum of time and space coordinates. ZFC Set Theory. The comprehensive set theory used in mathematics, evolved from Cantor’s theory of infinite sets and other beginnings. Uses specific axioms as its bases. A field of pure mathematics

307 GLOSSARY AND INDEX

Index aleph null, x-xi, 197-203, 296 Ampere, 92-3 Android, 71 angular momentum, 55, 84-6, 88, 160, 169, 184, 245-7, 278 anti-associative, ix, 7-14, 258-9, 275 anti-commutative, ix, 7-14, 176, 227, 258-60 anti-matter, 64-71, 75, 81, 84, 88, 130, 156, 162-9, 180, 181-5, 252, 255, 296 Archimedes, 292 area differential operator, 146-7 Aristotle, 291 axial vector, 19-20, 29 axiom of infinity, x, 201 axiomatic set theory, x, 200-1, 296 beautiful, 107, 136, 156, 244 Bell’s Inequality, 277, 279-86, 298 Bianchi, Eugenio, 253 Biot-Savart Law, 116-8, 124, 155, 296 black hole, 251, 253, 304 Bronstein, Matvei (Planck length), 253 Cantor, ix, x, 195-203, 248, 251, 253, 292, 296 Cantor’s Continuum Hypothesis, ix, 187, 196, 200-3 cardinality, 198, 296 Cohen, 201, 294 collapse of the wave function, 71, 88, 251-2, 287, 296 complex numbers, 4, 6, 8, 11-2, 14, 23-4, 29, 33-34, 106, 164, 169, 178, 234, 260, 291. 297 continuum, xii, 72, 195, 198, 204, 292, 297, 306 contravariant, 18, 19, 23, 27-8, 78, 297 cross-product, 3, 6, 15-17, 26, 29, 89, 99, 136 crude non-number algebra, 191, 202, 249, 250, 251, 297 de Broglie, 160, 217, 297 Dedekind, 197-8, 203, 297 denumerable / countable, 197, 198, 297 Descartes, 1, 8, 253, 290, 297 determinant 20 differential geometry, 78 dot-product, 3, 15-17, 26, 29, 76, 89, 99, 109 Einstein, 37, 46, 58, 89, 196, 251, 277, 304 E-M-compound-label-numbers, 96

308 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY enabler functions,162, 165, 230, 298 Entropy, 88, 251, 298 EPR Experiment, 88, 254, 277-87, 298, 304 Evert, Hugh, 252 exponential function, 6, 21, 24, 42, 136, 191, 206-9, 298 Faraday, 91-2, 156 Feynman, 64, 246, 298 final-result, 1, 204, 206, 209, 247, 298 finished-calculation, 1, 204, 206, 209, 247, 299 finite, 187, 299 force, 2-3, 67-9, 122, 132, 159, 238, 245, 252, 301 Galileo, 36 Gauss, 91-2 General Relativity, 68, 78, 251, 253, 304 Gödel, 200 Hamilton, 4, 257, 303 Hawking / Bekenstein, 253 Heaviside, 4, 89, 118 Hurwitz, 273 Hypatia, 36, 299 imprecision term “” (xi), ix, 193, 206-7, 209, 229, 300 irrational, x-xi, 8, 187-8, 192, 197-8, 203, 249, 291-3, 300 Kline, Morris, 290 length contraction, 47-8, 300 local-real numbers, xii, 187, 195, 203, 244, 249, 300 local-zero, xii, 193-5, 204-5, 294, 300 local-infinity, xii, 193-5, 204-5, 294, 300 Lorentz Transformation ix, 33, 36, 44, 108, 175, 300 Lorenz Condition, 98, 100, 103, 155 magnetic moment, 246 matrix isomorph, 5, 11-4, 20, 31, 260, 301 non-Relativistic, 39, 42, 74, 118, 157, 304 Pascal’s Triangle, 19, 23, 29, 144, 302 Pauli Spin Matrices, xiii, xv, 5, 31, 165, 260, 302 Planck’s constant, 160, 253, 298, 302 polar vector, 19, 23, 29 Poynting Vector, 120, 128, 135 Process from Descartes, 1

309 GLOSSARY AND INDEX pseudo-vector, 19 QED, 246, 298 quaternion hypercomplex numbers, 4-31, 257-75, 303 rational numbers, ix, 8, 188, 303 real numbers, x, 8, 197-203 reciprocal-of-zero axiom, x, 189, 294, 303 remnant-product, 17-9, 100-2, 303 Rovelli, Carlo, 254 Schrödinger, xii, 169, 194, 301, 303, 305 six dots, xii, 187-8, 192, 194, 304 space-like, 48-55, 58, 88, 304 space-negative, 72-83, 110, 146, 157, 161-3, 168, 228, 304 spin, 55, 84-6, 129, 169, 184, 245-7, 253, 277-87, 304 spiral waves, 127, 167 Spooky Action at a Distance, 58, 277-87, 298, 304 super-potential, 105-7 tau particle, 124 Theory of Non-Finite Numbers, 187, 305 Theory of Special Relativity with Non-Finite Numbers, 209, 244, 305 Tiles, Mary, 201 time dilation, 46, 305 triple-vector-product, 17-9, 23, 28, 100-1, 156, 305 truncated numbers, xii, 192-195, 248-9, 305 uncountable sets, 198-200, 305 unspecified-label-number, 95, 305 unspecified-speed-parameter, 38-9, 72, 88, 205-6, 305 vector-front multiplication, 227, 262 vector identities, 101, 103, 152 vector-space, 267-8 volume, 18, 20, 23, 121, 148-50 Wolchover, Natalie, 201 World-volume, 18, 151 ZFC, 201

310 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY

Back Cover Algebra without an actual infinity is proposed for applied mathematics. Maxwell’s Equations unite with the Dirac Equation to combine electron dynamics with photon dynamics because, by use of the algebra, an electron projects itself as a photon. Electron/photon double existence derives from Schrödinger’s Cat because each place-value digit of a real number beyond a finite maximum in count is unknown and unknowable, analogous to the cat being both alive and dead inside its unopened box. The algebra is derived from a proposed axiom that replaces Cantor’s Continuum Hypothesis. Empirically derived energy density and the Poynting Vector unite in the force density invariant as one mathematical model. It suggests quantities in our geometric world actually do have finite imprecision, and that the new algebra applies to more modern theories of physics. Visualizations and exercises help comprehension. The mathematics is simple enough to be understood by a high school student who has taken first year level college math and physics classes (and is familiar with trigonometry and logarithms, complex numbers, matrix multiplication, geometric-unit-vectors, and partial differential equations). Visualizations and examples help the reader comprehend each subtle feature in the algebra. Each chapter has exercises so that the reader can check their comprehension.