A Geometrical Description of Space-Time [2, 3 ed.]

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titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

Umberto Bellini

idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

TITLE D : A SPACE2TIME IN PROPAGATION _____________________________________________________ Book D I : The propagation of space 2 time Book D II : The experience of space 2 time

idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

BOOK D I : THE PROPAGATION OF SPACE2TIME

2 Chapter 1 : A travelling pulse 2 Chapter 2 : The propagation of space2 time

2 Chapter 3 : A luminal space-time only 2 Chapter 4 : The propagation rate

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idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

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idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

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idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

INDEX 1.1

Generalities

1.2

An appreciated present

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idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

1.1

GENERALITIES

THE SYSTEM Let us consider a pulse emitted during a time interval [ t A , t B ] from a source at r = 0 and travelling along the r axis at speed v . At moment t 0, the system can be sketched as in figure :

f(x) t = t0 v

0

rT

rF

r

r F = v (t 0 2 t A ) r T = v (t 0 2 t B )

The duration of the time interval [ t A , t B ] will be called duration tpulse . The length of the interval [ r T , r F ] will be called base r pulse. It is straightforward to check that duration and base are related through the relation : r pulse = v tpulse.

THE TRANSIT OF THE PULSE At moment t 0 we have that : 2 At r F < r , the pulse is not yet passed 2 At r F = r , the pulse starts its passing 2 In the interval ( r T ; r F ) the pulse is passing 2 At point r = r T the pulse has just passed 2 In the interval r < r T the pulse has already passed

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idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

1.2

AN APPRECIATED PRESENT

THE PRESENT Years ago a friend of mine made me an original present. It was a transparent plastic bag containing a transparent liquid in metastable equilibrium. Inside the bag there was a thin metal coin with a relief in the centre. If you pressed the relief, making the coin clicking, a small shock pulse departed from the coin. The propagation of that shock through the liquid made it to crystallize. The change of state was mildly exothermic so that you could feel the warmth holding the bag in your hands. Once the experience was over, if you put the bag in boiling water, the crystallized substance would return to the liquid state, ready to be re-used.

THE CRYSTALLIZATION OF THE LIQUID Let us consider the shock pulse as it propagates through the substance inside the plastic bag. One can distinguish three regions : 2 Where the shock pulse has not yet passed, the substance is in the liquid state 2 Where the shock pulse is passing, the substance is crystallizing. In such a region the substance is neither properly a liquid nor a solid 2 Where the shock pulse has already passed, the substance is in the solid state As one can see : 2 The state of the substance is well defined before and after the pulse 2 The transition from one state to the other one is not instantaneous but is a process covering all the region between the front edge and the tail edge of the pulse 2 After the pulse has passed, the state of the substance is no longer as before

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titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

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idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

INDEX 2.1

Two basic assumptions

2.2

Space2time propagation

2.3

Position in function of time

2.4

Global reference frames

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titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

2.1

TWO BASIC ASSUMPTIONS

In this text, two assumptions will be introduced :

FIRST ASSUMPTION Space2time is mathematically described by a space H ¢ 2 C . Space H is propagated by an expanding pulse as described at Chapter 1.

SECOND ASSUMPTION There was an event from which the expanding pulse formed and departed. That event will be called < originating event +O ORIG+ = .

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titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

2.2

SPACE2TIME PROPAGATION

WHAT IS PROPAGATED Assumed +O ORIG + as reference event, every space-time point is located through the pair of complex quantities ( ³ , ³ ) defined as follows : (³,³ )* H¢2C

³c r2

t

³c r+

t

2 r

2 r

The propagating pulse extends the space H from the null value of the norm n c |³| + |³| at the originating event to greater and greater values of n . Graphical representations of that propagation will be provided in Appendix A II < Graphical representations = .

PROPAGATING THROUGH WHAT ? It makes no sense to wonder through what space2time is in propagation. This is because space2time is the only reality we can know and nothing can be said of what there is outside of it, if any. To be strictly, even the words < what =, < exists = and < outside = couldn9 t be used because they have meaning only inside space2time.

THE PULSE PROFILE It makes no sense to speak about the profile of the propagating pulse. This is for the good reason that all the space2time quantities are being formed under the expanding pulse.

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titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

2.3

POSITION IN FUNCTION OF TIME

The expanding pulse propagates space2time from the null value n = 0 to greater and greater values of n . In consequence of that, s is in function of n : s is formed when n is formed. In ( t , r ) coordinates you have : r = r ( t ) .

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titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

2.4

GLOBAL REFERENCE FRAMES

GLOBAL REFERENCE FRAME Definition In the discussion that follows it is useful to introduce the notion of < global reference frame = . A global reference frame { O ORIG } is a free floating reference frame defined as follows : 2 It assumes as a reference event the originating event + OORIG + 2 Its origin is where the originating event happened

Remark It will be seen in Book F II < The natural motion = that any free reference frame is entitled to state : < My origin is where the originating event happened = . So, every reference frame assuming + OORIG + as reference event is a global reference frame, no matter its origin.

CONNECTION BETWEEN {OORIG } AND {O} Connection Let {O} be a reference frame assuming +O+ as reference event and { O ORIG } a global reference frame having the following properties: 2 It assumes the same origin as {O} 2 It assumes the same unit of length as {O} 2 Its axes are parallel with the {O} one9 s

9=

t = 9 + t ORIG

By construction, the coordinate transformations between them are :

2 t ORIG 9=

= 9

Notation Distances and time intervals will be denoted by characters in Courier New when referred to a global reference frame and in Bookman Old Style when referred to an ordinary reference frame : 2 { O ORIG } : t, r 2 {O} : t , r

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titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

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idJob: 1330709

titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

INDEX 3.1

Space2time is only luminal

3.2

Reference frames

3.3

The maximum distance r MAX

3.4

No tachyonic particles, except entanglers

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titolo: A Geometrical Description of Space-Time - 3rd edition - Volume 2

3.1

SPACE2TIME IS ONLY LUMINAL

LUMINAL EITHER TACHYONIC SPACE2TIME As written in Bok BI