The Origin of Gravity From First Principles 1536195669, 9781536195668

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PHYSICS RESEARCH AND TECHNOLOGY

THE ORIGIN OF GRAVITY FROM FIRST PRINCIPLES

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PHYSICS RESEARCH AND TECHNOLOGY

THE ORIGIN OF GRAVITY FROM FIRST PRINCIPLES

VOLODYMYR KRASNOHOLOVETS EDITOR

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Library of Congress Cataloging-in-Publication Data Names: Krasnoholovets, Volodymyr, editor. Title: The origin of gravity from first principles / Volodymyr Krasnoholovets, PhD., editor, Department of Theoretical Physics, Institute of Physics, Kyiv, Ukraine. Description: New York : Nova Science Publishers, [2021] | Series: Physics research and technology | Includes bibliographical references and index. | Identifiers: LCCN 2021022288 (print) | LCCN 2021022289 (ebook) | ISBN 9781536195668 (hardcover) | ISBN 9781536196917 (adobe pdf) Subjects: LCSH: Gravitation. | Quantum gravity. | Gravity. Classification: LCC QC178 .O745 2021 (print) | LCC QC178 (ebook) | DDC 531/.14--dc23 LC record available at https://lccn.loc.gov/2021022288 LC ebook record available at https://lccn.loc.gov/2021022289

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface

vii

Chapter 1

Quantum Field Theoretical Origin of Gravity Iberê Kuntz

Chapter 2

Low-Energy Quantum Gravity and Cosmology Michael A. Ivanov

23

Chapter 3

Only Gravity Thomas C. Andersen

57

Chapter 4

Sub-Quantum Gravity. The Condensate Vortex Model Peter A. Jackson

71

Chapter 5

Gravity in Space Particle Dualism Theory Sky Darmos

121

Chapter 6

Quantum Gravity Hidden in Newton Gravity and How to Unify It with Quantum Mechanics Espen Gaarder Haug

133

Chapter 7

The Origin of Gravity and Its Effects: According to the Subquantum Kinetics Paradigm Paul A. LaViolette

217

1

vi Chapter 8

Contents Derivation of Gravity from First Submicroscopic Principles Volodymyr Krasnoholovets

281

Editor’s Contact Information

333

Index

335

PREFACE Newton’s initial theory of the gravitational field, which is action at a distance theory, is now substituted by Einstein’s geometry of spacetime. However, these views work only at a macroscopic scale. When we are approaching the de Broglie wavelength, all principles of macroscopic physics stop working as the laws of the quantum world begin to guide the physical systems studied... But what is happening at the smallest scale? Does quantum gravity solve the problem? Unfortunately, the answer is negative because the most developed approaches, loop quantum gravity and string theory, rather are searching for the quantization rules of general relativity but not for the source of gravity as such. Usually the most reasonable approaches to quantum gravity use Planck size entities as building blocks for space, however, a general understanding of how these entities are ordered and what their properties are has not yet been developed. Yet, how can we reach understanding the phenomenon of gravity starting from first principles? When this book was planned, I contacted about a hundred of world leading physicists studying gravity. I asked them to contribute to a future book with the title The origin of gravity from first principles. Six researchers replied that they had been overburdened with duties and hence did not have time for this project. Eight researchers replied they were interested in the project and would be happy to contribute. Chapters below disclose the problem of the origin of gravity from the viewpoint of the researchers who have agreed to try to answer such an insidious question. Some approaches are based on the field equations and ideas of general relativity, but others suggest their own procedures. Chapter 1 considers the quantization of matter fields in a non-dynamical curved space-time, which is able to produce ultraviolet divergences, such that

viii

Volodymyr Krasnoholovets

curvature invariants are induced. The gravitational dynamics is derived from the quantum fluctuations of matter particles and the coupling of matter, which stands for the origin of the gravitational field. Chapter 2 examines a model of low-energy quantum gravity as the conjecture about the existence of the background of super-strong interacting gravitons, such that gravity is considered as the screening effect. Chapter 3 proposes that physicists consider studying “N=4 General Relativity” as a toy model. This ‘Only Gravity’ toy model uses Einstein’s field equations on their own in the hope that ignoring complicated interactions of gravity with other fields and physical theories may help us understand more about quantum gravity. Chapter 4 suggests a new vortex model of gravity that is treated across all dimensional orders. The smallest is a sub-matter ‘Higgs Condensate’ with the vacuum energy scale likely below the Planck length. This tiniest ‘granularity’ is the ‘stuff of’ electromagnetic waves and condensed matter vortex pair (e) fermions at ‘dark matter’ scale. The motion of vortices produces a radial pressure gradient proportional to the ‘mass’, as generated by the vortex orbital momentum. Chapter 5 overviews the space particle dualism theory presenting a new approach to quantum gravity, which allows one to change fundamentally the way of calculate gravity. One of its most radical aspects of the theory suggested is its total rejection of the equivalence principle for all gravitating matter. Chapter 6 considers the Planck scale as directly linked to gravity, which possesses a Lorentz symmetry. The relativistic wave equation, the relativistic energy momentum relation, and Minkowski space are represented by simpler equations owing to the understanding of mass at a deeper level. Namely, standard mass (kg) is represented via a collision ratio: One kg has the appropriate number of internal collisions per second (the Compton frequency). This allows gravity and quantum mechanics to be unified under the same theory. Chapter 7 represents the phenomenon of gravitation in the novel physics methodology of subquantum kinetics. The gravitation in this model is shown to spawn physically realistic particle-like structures having mass and charge, which generate gravity potential fields capable of exerting forces on neighboring particles. Subquantum kinetics predicts supercritical conditions in gravity potential wells that allow spontaneous zero-point energy fluctuations to grow and continuously materialize subatomic particles. The gravity fields of

Preface

ix

subquantum kinetics are also shown to account for the various general relativistic effects. Chapter 8 introduces a submicroscopic theory of real physical space, consisting of a mathematical lattice of primary topological balls, or the tessellattice. A cell in the tessellattice has the size equal to Planck’s length. A particle is treated as a deformed cell and a degree of a volumetric fractal deformation of the cell is related to the notion of mass. Due to the interaction of a moving particle with ongoing cells, a cloud of spatial excitations named inertons appears around the particle, which is mapped to the quantum mechanical particle’s wave -function. Inertons behave like standing spherical waves and establish a peculiar landscape proportional to 1/r, which results in Newton’s gravity. So, dear Reader, you are welcome to get acquainted with the achievements of the authors in detail. The general picture is variegated, but nevertheless, there is an actual orientation towards novelty and experimental confirmation.

Volodymyr Krasnoholovets Editor

In: The Origin of Gravity from First Principles ISBN: 978-1-53619-566-8 c 2021 Nova Science Publishers, Inc. Editor: Volodymyr Krasnoholovets

Chapter 1

Q UANTUM F IELD T HEORETICAL O RIGIN OF G RAVITY Iberˆe Kuntz∗ Dipartimento di Fisica e Astronomia, Universit`a di Bologna, Bologna, Italy I.N.F.N., Sezione di Bologna, IS - FLAG, Bologna, Italy

Abstract The quantization of matter fields in a non-dynamical curved spacetime always produces UV divergences that are proportional to curvature invariants. Even if one sets the gravitational strength to zero at the tree level, quantum matter fields unavoidably give rise to a dynamical spacetime, which is the cornerstone of general relativity. Inducing gravity this way ameliorates some of the problems associated with the quantization of gravity as the emergent gravitational interaction is inherently classical. In this chapter, we shall review some of the main avenues to induced gravity and present an approach based on effective field theory arguments. We also discuss the coupling of matter to the induced gravitational field in light of the Schwinger-Keldysh formalism, which is necessary to make sense of time-dependent gravitational evolutions.

Keywords: induced gravity, semi-classical gravity, quantum field theory in curved spacetime, effective field theory ∗

Corresponding Author’s Email: [email protected].

2

1.

Iberˆe Kuntz

INTRODUCTION

The question of the origin of the gravitational force is intimately related to how the universe came into being. According to the Big Bang theory, the universe we know today is believed to have started from a period where it was extremely hot and dense and had the size of a proton. Whether the Big Bang marked the origin of everything remains debatable, nonetheless, the standard model of cosmology is generally accepted as the theory for the initial instants after the Big Bang singularity. During this period, matter and interactions were created via symmetry breaking, leading to the standard model of particle physics described by the gauge group U (1) × SU (2) × SU (3) [1, 2, 3, 4]. Although this mechanism is well understood and accepted as the origin of standard model particles and their corresponding interactions, it is much less clear how this process culminated in the creation of the gravitational force. One possibility that can very well fit into this picture is the creation of the gravitational dynamics from the quantum fluctuations of matter particles [5]. In this sense, one can think of gravity as a classical condensate field whose origin is attributed to a symmetry-breaking mechanism due to matter fields. This interpretation of induced gravity not only explains the origin of the gravitational field but also avoids many technical and epistemological issues regarding the quantum nature of spacetime. In fact, should gravity be intrinsically classical, no renormalizability problem would ever appear in the covariant quantization program and the standard model of particle physics could very well be trusted even for Planckian energies. Moreover, the time problem would be non-existent in the canonical approach. Inducing gravity from matter fields is thus a very appealing alternative to quantum gravity. In this chapter, we shall give a detailed overview of induced gravity, particularly from the point of view of the effective field theory. This chapter is organized as follows. In Sec. 2, we review the formalism of the Euclidean effective action in curved spaces and we adopt Schwinger’s proper time method in Sec. 3 to calculate the one-loop divergences of matter fields in curved spaces. In Sec. 4, we review some of the main approaches to deal with the one-loop divergences in order to interpret the emergent curvature invariants as gravity. Sec. 5 is devoted to the application of the effective action to the study of induced gravity from an effective field theory viewpoint. In Sec. 6, we show how the Lorentzian effective action can be deduced from its Euclidean counterpart, in particular for the obtention of in-in amplitudes that are important in scenarios

Quantum Field Theoretical Origin of Gravity

3

with time evolution. We discuss prospects and give some conclusions in Sec. 6.

2.

E FFECTIVE ACTION F IELD METHOD

AND

THE

BACKGROUND

The effective action plays a central role in quantum field theory [6, 7]. It is indeed the object that contains all information relative to scattering amplitudes between asymptotic states, which ultimately comprises everything one looks for in collider physics. With slight variations in the boundary conditions, one is also able to obtain the backreaction of quantum fields onto the mean fields via the effective equations of motion obtained from the effective action (see Sec. 6). Therefore, finding the effective action is equivalent to solving a quantum field theory. In this section, we shall review the effective action formalism for a generic field theory. To make our notation less cluttered, we shall adopt DeWitt’s condensed notation in which small indices (e.g., i, j, k, . . .) run over all discrete indices, hereby denoted by capital indices (e.g., I, J, K, . . .), and the continuum spacetime x. One can thus imagine the correspondence i = (I, x) between small and capital indices. Fields shall be collectively denoted by ϕi = ϕI (x). Einstein’s summation convention for small indices thus implies an additional integration over spacetime: Z X ϕi ϕi = d4 x ϕI (x)ϕI (x). (1) I

The functional generator of one-particle irreducible (1PI) diagrams is given, as usual, by the path integral: Z exp(−W [J]) = Dφ exp(−S[φ] − φi Ji ), (2)

where S[φ] denotes the classical action for the collection of fields φi and Ji correspondingly denotes their external current. Note that we adopted the Euclidean signature in Eq. (2). The Lorentzian effective action can be obtained by analytical continuation from the Euclidean space. This will be particularly important in Sec. 6 when obtaining the effective action for the in-in mean field. The effective action is defined by the Legendre transform of the functional W [J] with respect to the mean field ϕi ≡ h0|φi|0i =

δW [J] , δJi

(3)

4

Iberˆe Kuntz

namely Γ[ϕ] = W [J[ϕ]] − ϕi Ji [ϕ].

(4)

Variation of Eq. (4) yields the effective equations of motion δΓ[ϕ] = −Ji , δϕi

(5)

which generalizes the classical equations of motion to include quantum fluctuations of the fields. From Eqs. (2), (4) and (5), one obtains:   Z i i δΓ[ϕ] exp(−Γ[ϕ]) = Dφ exp −S[ϕ] + (φ − ϕ ) . (6) δϕi Noticeably, Eq. (6) is an infinite-dimensional integro-differential equation. Needless to say, solving it for the effective action Γ[ϕ] is utterly difficult in general. One useful method consists in using the background field method [8, 9], in which one makes the following change of variables in the path integral: φi → ϕi + φi .

(7)

In the new variables, one can then obtain a semiclassical expansion Γ[ϕ] = S[ϕ] + Γ1-loop [ϕ] + Γ2-loop [ϕ] + · · · ,

(8)

by expanding the right-hand side of (6) in powers of φi . It is precisely the contribution from Γ1-loop [ϕ] that shall give origin to the gravitational sector. Such a correction reads Γ1-loop [ϕ] =

1 Tr log(F (∇)), 2

(9)

where F (∇)δ(x, y) =

δ2 S . δϕI (x)δϕJ (y)

(10)

The precise form of the operator F (∇) depends on the theory, but for the sake of this chapter, we shall assume R F (∇) =  + Pˆ − ˆ1 − m2 ˆ1, 6

(11)

Quantum Field Theoretical Origin of Gravity

5

which covers all theories with canonical actions. The box symbol denotes the d’Alembert operator in curved spaces  = g µν ∇µ ∇ν , the matrix ˆ1 = (δ IJ ) is the identity matrix in the space of fields, Pˆ is a generic potential matrix and we singled out the Ricci scalar factor R6 and the mass m2 from Pˆ for convenience. The main difficulty is then to calculate the one-loop contribution, which is the subject of the next section.

3.

SCHWINGER ’ S P ROPER T IME METHOD

In the presence of non-trivial mean fields or when the background spacetime is curved, the calculation of propagators, namely Green’s function for the operator F (∇), becomes more subtle. This is partly due to the difficulty in defining the Fourier transform. In these cases, Schwinger’s proper time method comes in handy [10], allowing one to write the propagator in terms of the heat kernel K(s) = esF (∇) : Z G ≡ −F (∇)−1 =

∞

dsK(s).

(12)

0

The second equality in Eq. (12) is an identity that can be trivially obtained by direct integration for a negative definite operator F (∇) 1 . The one-loop contribution to the effective action can thus be written Z 1 ∞ 1 Tr log (F (∇)) = − ds Tr K(s). (13) 2 2 0 The problem of calculating one-loop corrections reduces to finding the heat kernel K(s), which satisfies the heat equation: ∂K ij (s) ∂s

= F (∇)K ij (s)

(14)

with boundary condition given by K ij (0) = δ IJ δ(x, y).

(15)

Here K ij (s) = K IJ (s|x, y) denotes the kernel of K(s), namely the matrix representation of the functional K(s). Eq. (14) admits a solution in terms of an 1 The negative sign of F (∇) might be spoiled depending on the theory’s potential, but this is not a problem when the potential is treated perturbatively along with the other curvature invariants.

6

Iberˆe Kuntz

asymptotic expansion for s → 0: ∞ X ∇1/2 (x, y) 1/2 − σ(x,y) −sm2 2s sn an (x, y), K(s|x, y) = g (y)e (4πs)d/2 n=0

(16)

where σ(x, y) is the Synge world function and ∇(x, y) is the Pauli-Van VleckMorette determinant. Because Eq. (13) is expressed in terms of the trace of the heat kernel Z Tr K(s) = d4 x tr K(s|x, x), (17)

one only needs the coincidence limit x → y of the so called HAMIDEW parameters an (x, y) [11, 8]. It turns out that such parameters are local functions of the curvature. The first few HAMIDEW parameters are given by [12, 13, 14] a0 (x, x) = ˆ1, a1 (x, x) = Pˆ ,

(18) (19)

1 1 (Rµνρσ Rµνρσ − Rµν Rµν )ˆ1 + Rµν Rµν 180 12 1 1 1 Rˆ1, + Pˆ2 + Pˆ + 2 6 180

a2 (x, x) =

(20)

where Rµνρσ is the Riemann tensor, Rµν is the Ricci tensor, R is the Ricci scalar and Rµν denotes the fiber bundle curvature (i.e., the gauge field strength). From Eqs. (13), (16) and (17), we then find the divergent part of the one-loop effective action to be: Z d/2 X √ 1-loop Γdiv [ϕ] = αn dd x g tr an (x, x), (21) n=0

where we adopted dimensional regularization ω → d/2 to obtain   1 1 (−m2 )d/2−n αn ≡ − ψ (d/2 − n + 1) − d/2 − ω (d/2 − n)! 2(4π)d/2

(22)

and ψ(x) is the logarithmic derivative of the Euler gamma function. In light of Eqs. (18)–(20), the quite remarkable result (21) shows that the divergent part of the one-loop effective action is a linear combination of curvature invariants. Note that the results (18)–(20) and (21) are completely general. They are valid for any theory whose operator F (∇) is of the form (11), which includes all standard cases of canonical fields. Irrespective of the number and the spin of fields

Quantum Field Theoretical Origin of Gravity

7

present in the theory, some of the UV divergences will always be proportional to spacetime curvatures. The particular information on the existing field species in the theory appears only in the factors of proportionality in front of the curvatures. Taking the trace and integrating Eqs. (18)–(20), allows a more explicit dependence on the field content: Z √ (23) d4 x g tr a0 (x, x) = k0 , Z √ d4 x g tr a1 (x, x) = k1 R − m2 , (24) Z 1 √ d4 x g tr a2 (x, x) = k2 Cµνρσ C µνρσ + k4 R2 − m2 k1 R + m4 , (25) 2 where ki denotes the factors of proportionality and Cµνρσ is the Weyl tensor. Table 1 summarizes the values of ki for different types of fields. As an example, consider a massive scalar field in a four-dimensional curved space: Z √ 1 S[φ] = d4 x g φ( + m2 )φ. (26) 2

The application of Eq. (21) for this particular case gives:  Z 1 1 26 m2 1-loop 4 √ 4 Γdiv [ϕ] = − d x g 10 m + R 2(4π)2  3  7 2 1 2 149 38 C + R + E − R , 10 2 90 15

(27) (28)

where  = 2 − ω and we have now written the result in terms of the Weyl tensor Cµνρσ and the Euler density E = Rµνρσ Rµνρσ − 4Rµν Rµν + R2 .

(29)

The important point is that the divergences to first order in the curvature are proportional to a constant and a linear term in R that would resemble the EinsteinHilbert action with a cosmological constant if they were not divergent. To interpret such terms as gravitational interaction, one needs to get rid of such divergences. In the following, we shall give an overview of the main approaches to make sense of these divergent couplings.

8

Iberˆe Kuntz Table 1. Values of the coefficients ki for each type of particle’s species. Here ξ is the non-minimal coupling coefficient of scalars to gravity. The net value is calculated by multiplying each value in the table by the number of fields present in each category k0

k1

k2

k4

Scalar (minimal)

1

1/6

1/120

1/72

Scalar (conformal)

1

0

1/120

0

Scalar (arbitrary)

1

1/6 − ξ

1/120

(1 − 6ξ)2 /72

Weyl spinor

2

−1/6

−1/40

0

Dirac spinor

4

−1/3

−1/20

0

Vector (1 ⊕ 0)

4

−1/3

7/60

1/36

Spin 1 (massive)

3

−1/2

13/120

1/72

Spin 1 (massless)

2

−2/3

1/10

0

Chiral supermultiplet

0

1/2

1/24

1/36

Vector supermultiplet (massless)

0

1/2

1/24

0

Vector supermultiplet (massless)

0

0

5/24

1/24

4.

R EVIEW ON THE MAIN A PPROACHES G RAVITY

TO I NDUCED

At this point, there are several alternatives to make sense of the gravitational couplings appearing in the effective action [5, 15, 16, 17, 18, 19, 20, 21] (see [22] for a review). The original proposal of Sakharov imposes the dominance of the one-loop corrections along with an explicit Planckian cutoff [5]. In this case, setting the tree-level couplings to zero, one obtains 1 1 = − k1 Λ 2 , G 2π

(30)

Quantum Field Theoretical Origin of Gravity

9

for k1 ≈ −1. Note that we switched the regularization method to an explicit UV cutoff Λ to make the comparison with early works more transparent. The result in dimensional regularization can be recovered by replacing log(Λ/µ) → 1/ and setting the other divergences to zero. From Eq. (30), one then obtains Newton’s constant as a function of the particle spectrum in the theory. The other gravitational couplings are similarly obtained: 1 k0 Λ4 , with k0 ≈ 0, 64π 2  2 1 Λ c2 = k4 log ≈ 1, 2 32π µ2  2 1 Λ c3 = k4 log ≈ 1. 2 32π µ2

Λc = −

(31) (32) (33)

In Sakharov’s approach, one imposes Λc ≈ 0 and c2 , c3 ≈ 1. It is quite remarkable that one can express all gravitational couplings in terms of the particle spectrum corresponding to the classical action. Notice, however, that the gravitational couplings were made finite by explicitly cutting off the theory at the Planck scale Λ ∼ Mp . This is rather unnatural and it makes it impossible to take the continuum limit. Another approach, originally attributed to Pauli, demands finite contributions from one-loop corrections. Pauli’s method thus requires the accidental cancellations of contributions from different matter fields, an idea quite similar to supersymmetry. For these cancellations to take place, one needs to enforce k0 = m2 = m4 = 0,

(34)

which guarantees that the cosmological constant is finite. In this case, it is then given by  2 1 m 4 Λc = Λ0 + m log . (35) 2 64π µ2 Similarly, one needs k 1 = k 1 m2 = 0

(36)

to obtain a finite Newton’s constant, namely  2 1 1 1 m 2 = − k1 m log . G G0 2π µ2

(37)

10

Iberˆe Kuntz

Finally, one also needs to impose the conditions k2 = k4 = 0 to make the quadratic curvature contributions finite, leading to:  2 1 m c3 = (c3 )0 − k2 log , 2 32π µ2  2 1 m c4 = (c4 )0 − k4 log . 2 32π µ2

(38)

(39) (40)

The most important feature of Pauli’s approach is indeed the finiteness of the gravitational couplings. Notice that the UV regulator dropped out in Eqs. (35), (37), (39) and (40), thus one can take the continuum limit normally, which makes Pauli’s approach much more natural than Sakharov’s. Nonetheless, the cancellations imposed in Eqs. (34), (36) and (38) are extremely expensive and too constraining. They are unlikely to yield anything useful. There are many other attempts to make sense of the gravitational couplings by combining the above approaches. Frolov and Fursaev, for example, combined the absence of tree-level couplings of Sakharov’s approach with the oneloop finiteness of Pauli’s [15, 16, 17, 18, 19, 20, 21]. Their model thus enjoys the benefits from both worlds, but it still requires very restricted constraints due to Pauli’s finiteness conditions. In [22], it was also considered an approach based on the standard renormalization. In this scenario, invariance constraints were imposed to make the gravitational couplings constant when the theory goes through a phase transition or a spontaneous symmetry breaking, namely δk0 = δm2 = δm4 = 0, 2

(41)

δk1 = δ(k1 m ) = 0,

(42)

δk2 = δk4 = 0.

(43)

One can interpret these constraints as cancellations between mass changes in the bosonic and fermionic sectors. When such constraints are satisfied, one obtains a relation between the variation of the gravitational couplings and the variations

Quantum Field Theoretical Origin of Gravity

11

in the particle spectrum and masses:   2  m 1 4 δ m log , δΛc = 2 64π µ2     2  1 1 m δ = − δ k1 m2 log , G 2π µ2   2  m 1 δ k2 log , δc3 = − 2 32π µ2   2  1 m δc4 = − δ k4 log . 2 32π µ2

(44) (45) (46) (47)

These constraints are much weaker than Pauli’s, but they are still very difficult to be satisfied. In the following, we shall propose a way to induce gravity without the need of imposing any constraint whatsoever. This is possible with the modern understanding of renormalization using effective field theories.

5.

E FFECTIVE F IELD T HEORY: G RAVITY

L ET T HERE B E

We shall here take a slightly different route to the renormalization approach by adopting an effective field theory viewpoint [23, 24]. As Eq. (21) shows, the one-loop divergences require counter-terms to second-order in the curvature so that they can be canceled. However, no curvature invariant was originally present in the bare action, which indicates that such cancellation of divergences cannot take place and that the theory is not renormalizable. The calculation of divergences at two-loops also reveals the need for cubic curvature terms for renormalization, sustaining the non-renormalizability of the theory. This turns out to be a general pattern, thus the removal of UV divergences at the m-loop order requires curvature invariants of order m + 1. None of these curvature invariants were ever-present in the bare action particularly because we started with a theory without gravity. Canceling the divergences to all orders would then require the inclusion of infinitely many curvature invariants to act as counter-terms. After renormalization, one would thus end up with infinitely many coupling constants, which appear as coefficients of the counter-terms, to be fixed by experiments. Performing infinite experiments is impossible, which makes the theory lose predictability.

12

Iberˆe Kuntz

In modern quantum field theory, however, this does not mean much. It is just a reflection of our naive assumption regarding the validity of physical models in arbitrary scales. The systematical treatment of non-renormalizable theories is performed in the effective field theory formalism, which is nothing else than a small generalization of renormalization. The only actual difference comes with a choice of the typical energy scale E of the underlying processes and a corresponding accuracy n, where n is the power of (E/Λ)n and Λ is a cutoff scale. For a given energy scale E  Λ and n, higher-order contributions are suppressed, reducing the aforementioned infinitely many curvature invariants to a finite set of terms. In the effective field theory spirit, we renormalize the one-loop divergences in Eq. (21) by introducing the counter-terms up to second order in the curvature, which is equivalent to n = 4 as each factor of curvature contributes with two spacetime derivatives. Up to one-loop order, the counter-terms take the form Z  √  (48) ∆Γ1-loop = d4 x −g b0 + b1 R + b2 R2 + b3 Rµν Rµν , where bi = bi (Λ) are formally divergent quantities that depend on the cutoff Λ. From Eqs. (21)–(25) and (48), we thus obtain the renormalized action at one-loop: Z  √  1-loop (49) Γren [gµν ] = d4 x −g c0 + c1 R + c2 R2 + c3 Rµν Rµν , with

c0 = b 0 − c1 = b 1 − c2 = b 2 + c3 = b 3 +

 4  2  1 Λ m4 Λ 2 2 −m Λ + log , 2 32π 2 2 m2   2  1 Λ 2 2 k1 Λ − k1 m log , 2π m2  2 1 Λ k2 log , 32π 2 m2  2 1 Λ k4 log . 2 32π m2

(50) (51) (52) (53)

Here ci are renormalized coupling constants that must be fixed by experiments.

Quantum Field Theoretical Origin of Gravity

13

The latest measurements of such coupling constants give c0 ≡ Λc ' 10−48 GeV 4 , 1 ' 2.9 × 1036 GeV 2 , c1 ≡ 16πG

(54) (55)

where we have identified c0 = Λc as the cosmological constant and c1 = (16πG)−1 as the inverse of Newton’s constant. The coupling constants due to quadratic curvatures are practically unconstrained as they satisfy [25, 26] c2 , c3 . 1061 ,

(56)

which reflects the smallness of quantum effects. Because of renormalization, the coupling constants acquire a dependence on the renormalization scale, which is described by the renormalization group: µ

δci = βi , δµ

(57)

where the beta functions βi are obtained from the coefficients of the divergence 1/, which can also be extracted from Table 1 for all particle species. They are explicitly given by m4 , 32π 2 k 1 m2 β1 = − , π k2 β2 = − , 16π 2 k4 β3 = − . 16π 2 β0 =

(58) (59) (60) (61)

The important thing to notice is that the action has acquired terms containing derivatives of the metric and such terms resemble exactly the Einstein-Hilbert action at leading order. At next-to-leading order, one obtains quadratic curvature invariants which become important at energies close (but yet much smaller than) the Planck scale. What initially was just a fixed background has now gained dynamics. No matter what background one starts with, it must become dynamical to accommodate quantum matter fields. Contrary to the approaches outlined in Sec. 4, we should stress that in the effective field theory one need

14

Iberˆe Kuntz

not impose any constraint in the particle spectrum of the theory. Yet, we obtain an effective action with finite couplings that must be fixed by experimental data. The particle spectrum only dictates the running of the gravitational couplings, which need not be constrained by any ad hoc assumption.

6.

T HE E FFECTIVE ACTION K ELDYSH F ORMALISM

IN THE

SCHWINGER -

As we have seen so far, the idea of generating gravity from quantum matter fields relies on the calculation of the effective action Γ[ϕ; gµν ] for the mean field ϕi defined on a curved spacetime whose metric is gµν . The metric gµν that we started with was not dynamical, it was just used to describe the background where matter fields were defined, thus, initially, gµν could not be interpreted as gravity. In particular, at the onset, there is no field equation in which gµν is a solution of nor there are terms in the classical action corresponding to its dynamics. We then showed that even if we start with no classical dynamics for gµν , in obtaining the effective action Γ[ϕ; gµν ] for matter fields one naturally generates curvature invariants in the action, thus giving dynamics to the background gµν . Nonetheless, the usual formalism for the effective action, as employed in the previous sections, is intended to calculate scattering amplitudes, which requires in-out amplitudes h0out|φi |0in i by the LSZ reduction formula. In-out amplitudes are not observables by themselves as they are generally complex even for selfadjoint operators. Moreover, their evolution is not causal as it is expected from a scattering process. In this type of experiment, one is indeed interested in the S-matrix, whose elements are real and can be observed in accelerators. On the other hand, classical dynamics are very much different from scattering experiments. In gravity, for example, one would like to know the time evolution of gµν itself or of matter fields over gµν . If ϕi = h0out|φi |0in i is complex and does not evolve causally, then its evolution obviously does not correspond to a physical motion. In the previous sections, we have intentionally omitted the terms corresponding to the background matter fields because we were particularly interested in showing the appearance of the gravitational couplings. However, when the action for the background fields ϕi are taken into account, one has to face the problems of complex correlation functions and a causal evolutions for ϕi . At the level of the equations of motion, these problems

Quantum Field Theoretical Origin of Gravity

15

are reflected into non-causal issues for the evolution of the metric gµν and it would yield complex metric solutions. The vast number of works that have arguably reproduced general relativity as an emergent interaction from quantum matter fields have largely ignored this fact. Even though one can reproduce the correct curvature terms for the gravitational sector, the couplings of the dynamical metric to background matter fields lead to fundamental issues for the spacetime dynamics. We must remember that the coupling to matter plays a role of utmost importance, thus these issues cannot be ignored when studying the appearance of gravity from quantum matter fields. To reproduce the classical dynamics of gravity coupled to matter correctly, one must employ the Schwinger-Keldysh approach for obtaining in-in amplitudes such as ϕi = h0in |φi|0in i. Because in-in amplitudes are evaluated using the same state, they turn out to be real for self-adjoint operators. Futhermore, they do not require information from the future as in-out amplitudes do. In-in correlation functions do not result directly from the standard Feynman path integral calculation. However, it can be carried out using the somewhat different Schwinger-Keldysh formalism (or closed-time path integral) which needs in-out amplitudes only at mid-steps [27, 28]. This is done by doubling every single degree of freedom φ, which are then usually denoted by φ+ and φ− . The field φ+ is produced by an external source J+ and is in charge of the transition between |0in i and an intermediate state |Σα i, which belongs to a future Cauchy surface Σ. On the other hand, the field φ− is generated by J− and is responsible for the transition from |Σαi back to |0in i. If we assume that {|Σαi} forms a complete set of states, then the functional generator of connected in-in amplitudes is derived by summing over all possible intermediate states |Σαi, leading to X ei W [J+ ,J− ] = h0in |ΣαiJ− hΣα|0in iJ+ . (62) α

If, in addition, we suppose that {|Σαi} are eigenstates of φ on Σ, then we can rewrite Eq. (62) in terms of Feynman path integrals: Z i i W [J+ ,J− ] (63) e = Dφ+ Dφ− e ~ {S[φ+ ]+S[φ− ]+J+ φ+ −J− φ− } , where the integration fields are subjected to vacuum boundary conditions in the remote past (corresponding to the state |0in i) and φ+ = φ− on Σ. The many in-in amplitudes are calculated by functionally differentiating W [J+ , J− ] with

16

Iberˆe Kuntz

respect to the sources and then setting J+ = J− = 0 in the end. Since there are now two kinds of fields, namely φ+ and φ− , and two corresponding kinds of sources, there will be two types of vertices and four types of propagators entailed in Feynman diagrams, that is ~δ ~δ i W [J+ ,J− ] 0 e , (64) Gab (x, x ) = sign(a) i δJa(x) sign(b) i δJb(x0 ) J+ =J− =0 where

( +1 sign(a) = −1

for a = + for a = − .

(65)

The diagonal elements of Gab are associated with the Feynman and antiFeynman propagators, G++ (x, x0) = h0in | T φ(x) φ(x0) |0in i , G−− (x, x0) = h0in | T¯ φ(x) φ(x0) |0in i ,

(66) (67)

where T and T¯ are the time-ordered and anti-time-ordered operators, respectively. The off-diagonal elements, on the other hand, correspond to Wightman correlation functions: G+− (x, x0 ) = h0in | φ(x0 ) φ(x) |0ini , 0

0

G−+ (x, x ) = h0in | φ(x) φ(x ) |0ini .

(68) (69)

Despite the new vertices and propagators, the Feynman rules for in-in amplitudes are identical to the standard ones for in-out point functions. From the above considerations, we can see that the calculation of the effective action in the in-in formalism involves the path integral from |0in i to |Σαi and another path integral corresponding the inverse process from |Σα i back to |0in i. Needless to say, such a calculation is much more involved than the standard one for in-out amplitudes. Fortunately, an easier method for the calculation of the in-in effective action exists. Indeed, it was shown in [29] that one can calculate the Euclidean effective action as usual and in the end, after the variation of the action, impose retarded boundary conditions by replacing Euclidean Green’s functions by the retarded ones: GE (x, x0 ) →

1 ret G (x, x0), i

(70)

Quantum Field Theoretical Origin of Gravity

17

where Gret (x, x0) = G++ − G+− .

(71)

This procedure results from an analytic continuation, via the Wick rotation, from the Euclidean effective action to the Lorentzian case. For in-out amplitudes, such an analytic continuation results in the usual Feynman boundary conditions, whereas retarded boundary conditions naturally appear in the case of in-in amplitudes. Such procedure shows that both cases of physical interest, namely scattering amplitudes as well as time-dependent evolutions, can be obtained from the Euclidean effective action. The only caveat is the requirement of asymptotic flatness or trivial boundary conditions in the asymptotic region. After quantization and renormalization, the complete effective action contains both the curvature invariants and the background matter fields: Γ[ϕ; gµν ] = Γg [gµν ] + ΓM [ϕ; gµν ],

(72)

where the induced gravitational action is given by Eq. (49), namely Γg [gµν ] = Γ1-loop ren [gµν ],

(73)

and ΓM is the effective action for the background matter fields. The equations of motion for the induced gravitational field is thus obtained from the action (49) with the additional prescription described above, replacing GE by Gret: δΓM δΓg = − µν . (74) δg µν δg GE → 1 Gret i

The equations of motion for the background matter fields are similarly obtained as δΓM = 0. (75) δϕi GE → 1 Gret i

Notice that the evolutions of both the induced metric and the background matter fields are causal due to the retarded Green’s function, which appears naturally as a consequence of the Schwinger-Keldysh formalism. For induced gravity, we must not overlook the importance of the SchwingerKeldysh formalism in obtaining the reality and causality properties of in-in mean fields ϕi = h0in | φi |0in i. We recall that it is a fact of Nature that the gravitational and matter fields are both real and evolve causally. These properties can also be confirmed explicitly order by order in a ~ expansion by using

18

Iberˆe Kuntz

the effective equations obtained from the effective action in the in-in formalism Γ[ϕ+ , ϕ− ], which is defined by the Legendre transform of the in-in generating functional W [J+ , J− ] with respect to the sources J± . We stress the importance of the reality of the background field for the interpretation of gµν as a physical metric field. Causality is also equally crucial and it results from the fact that the Schwinger-Keldysh formalism singles out the retarded Green’s function Gret = G++ − G+− as the correct propagator to be used in all calculations. For more details on the Schwinger-Keldysh approach, including the proof of the reality and causality of the background field, please see Refs. [28, 29].

C ONCLUSION The quantization of fields in curved spacetime naturally generates UV divergences that turn out to be proportional to spacetime curvature invariants. This brings the possibility of interpreting gravity as an emergent phenomenon that results from UV physics, whereas the gravitational interaction is set to zero at the tree level. To give a proper account of the dynamics of gravity, one has to deal with the UV divergences that appear along the way. As we pointed out, there are many different avenues one can take to sort this out, such as demanding oneloop dominance or one-loop finiteness. In this chapter, we have rather adopted the point of view of effective field theories to induce the gravitational action. The effective field theory approach is much more natural and less constraining. In fact, the contributions to the gravitational couplings, which at leading order comprise the cosmological constant Λc and Newton’s constant G, from the matter couplings show up only in the beta functions for Λc and G. Hence, Λc and G are affected only by their dependence on the renormalization scale µ obtained from the renormalization group equation. Their actual values at a certain scale µ must be determined by the matching procedure, i.e., by demanding consistency with Newton’s law or from the current experimental data. Apart from that, no other requirement is necessary. We should emphasize that, despite the quantization of matter fields, the gravitational interaction remains classical. From the effective field theory viewpoint, the curvature invariants appear as counter-terms that correct only the coupling of matter with the background and background self-couplings, i.e., the vertices of Feynman diagrams. On the other hand, there is no external line corresponding to the gravitational interaction. This indeed reflects the initial absence of gravitational degrees of freedom in the classical action. The effective

Quantum Field Theoretical Origin of Gravity

19

nature of such an approach thus means that no quantizable degree of freedom exists in the gravitational sector, notwithstanding the resurgence of a dynamical background. An intrinsically classical gravitational field has interesting properties. It would not face the usual issues encountered by many of the approaches to quantum gravity, such as the problem of time and the definition of observables. The effective field theory, nonetheless, brings the possibility of an emergent classical gravitational field that is valid only below the Planck scale. As one probes Planckian energies, all the infinitely many curvature invariants become relevant, and such an approach breaks down. Whether this picture of an induced gravitational field remains valid in the deep UV is not clear. However, this problem is shared among all different interpretations of induced gravity.

R EFERENCES [1] Yang, C. N. and Mills, R. L., Phys. Rev. 96 (1954), 191-195 doi:10.1103/PhysRev.96.191. [2] Glashow, S. L., Nucl. Phys. 22 (1961), 579-588 doi:10.1016/00295582(61)90469-2. [3] Weinberg, S., Phys. Rev. Lett. 19 (1967), 1264-1266 doi:10.1103/PhysRev Lett.19.1264. [4] Salam, A., Conf. Proc. C 680519 (1968), 367-377 doi:10.1142/978981279 5915 0034. [5] Sakharov, A. D., Usp. Fiz. Nauk 161 (1991) no.5, 64-66 doi:10.1070/PU19 91v034n05ABEH002498. [6] Buchbinder, I. L., Odintsov, S. D. and Shapiro, I. L., “Effective action in quantum gravity,” Bristol (IOP, London, 1992). [7] Toms, D. J., “The Schwinger Action Principle and Effective Action,” Cambridge University Press (2007), doi:10.1017/CBO9780511585913. [8] DeWitt, B. S., “Dynamical theory of groups and fields,” Conf. Proc. C 630701 (1964), 585-820. [9] Abbott, L. F., Acta Phys. Polon. B 13 (1982), 33 CERN-TH-3113.

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[10] Schwinger, J. S., Phys. Rev. 82 (1951), 664-679 doi:10.1103/PhysRev.82. 664. [11] Gibbons, G. W., “Quantum Field Theory In Curved Space-time,” MPIPAE-ASTRO-145. [12] Barvinsky, A. O. and Vilkovisky, G. A., Phys. Rept. 119 (1985), 1-74 doi:10.1016/0370-1573(85)90148-6. [13] Avramidi, I. G., Lect. Notes Phys. Monogr. 64 (2000), 1-149 doi:10.1007/3-540-46523-5. [14] Vassilevich, D. V., Phys. Rept. 388 (2003), 279-360 doi:10.1016/j.physrep. 2003.09.002 [arXiv:hep-th/0306138 [hep-th]]. [15] Frolov, V. P., Fursaev, D. V. and Zelnikov, A. I., Nucl. Phys. B Proc. Suppl. 57 (1997), 192-196 doi:10.1016/S0920-5632(97)00373-3. [16] Frolov, V. P. and Fursaev, D. V., Phys. Rev. D 61 (2000), 024007 doi:10.1103/PhysRevD.61.024007 [arXiv:gr-qc/9907046 [gr-qc]]. [17] Frolov, V. P. and Fursaev, D. V., Phys. Rev. D 58 (1998), 124009 doi:10.1103/PhysRevD.58.124009 [arXiv:hep-th/9806078 [hep-th]]. [18] Frolov, V. P. and Fursaev, D. V., Class. Quant. Grav. 15 (1998), 2041-2074 doi:10.1088/0264-9381/15/8/001 [arXiv:hep-th/9802010 [hep-th]]. [19] Frolov, V. P. and Fursaev, D. V., J. Astrophys Astron 20, 121-129 (1999), doi.org/10.1007/BF02702347 [arXiv:hep-th/9705207 [hep-th]]. [20] Frolov, V. P. and Fursaev, D. V., Phys. Rev. D 56 (1997), 2212-2225 doi:10.1103/PhysRevD.56.2212 [arXiv:hep-th/9703178 [hep-th]]. [21] Frolov, V. P., Fursaev, D. V. and Zelnikov, A. I., Nucl. Phys. B 486 (1997), 339-352 doi:10.1016/S0550-3213(96)00678-5 [arXiv:hepth/9607104 [hep-th]]. [22] Visser, M., Mod. Phys. Lett. A 17 (2002), 977-992 doi:10.1142/S0217732 302006886 [arXiv:gr-qc/0204062 [gr-qc]]. [23] Donoghue, J. F., Phys. Rev. D 50 (1994), 3874-3888 doi:10.1103/PhysRev D.50.3874 [arXiv:gr-qc/9405057 [gr-qc]].

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[24] Burgess, C. P., Living Rev. Rel. 7 (2004), 5-56 doi:10.12942/lrr-2004-5 [arXiv:gr-qc/0311082 [gr-qc]]. [25] Hoyle, C. D., Kapner, D. J., Heckel, B. R., Adelberger, E. G., Gundlach, J. H., Schmidt, U. and Swanson, H. E., Phys. Rev. D 70 (2004), 042004 doi:10.1103/PhysRevD.70.042004 [arXiv:hep-ph/0405262 [hep-ph]]. [26] Stelle, K. S., Gen. Rel. Grav. 9 (1978), 353-371 doi:10.1007/BF00760427. [27] Keldysh, L. V., Zh. Eksp. Teor. Fiz. 47 (1964), 1515-1527. [28] Jordan, R. D., Phys. Rev. D 33 (1986), 444-454 doi:10.1103/PhysRevD.33. 444. [29] Barvinsky, A. O. and Vilkovisky, G. A., Nucl. Phys. B 282 (1987), 163-188 doi:10.1016/0550-3213(87)90681-X.

In: The Origin of Gravity from First Principles ISBN: 978-1-53619-566-8 c 2021 Nova Science Publishers, Inc. Editor: Volodymyr Krasnoholovets

Chapter 2

L OW-E NERGY Q UANTUM G RAVITY AND C OSMOLOGY Michael A. Ivanov∗ Physics Department, Belarus State University of Informatics and Radioelectronics, Minsk, Belarus

Abstract The model of low-energy quantum gravity by the author is based on a conjecture about the existence of the background of super-strong interacting gravitons, gravity is considered as a screening effect. The Newton constant G and the Hubble constant H are computable variables in this model. In this chapter, main effects of the model are discussed, some cosmological consequences are confronted with observations. Galaxy number counts/redshift and counts/magnitude relations are being considered. It is shown that this model can fit observations with the theoretical luminosity distance without dark energy. The Hubble parameter of this model is a linear function of the redshift, that is consistent with existing observations. The outcomes of numerical modeling of the influence of the additional deceleration of bodies, and some possibilities to verify the model are described.

PACS: 98.80.Es, 04.50.Kd, 04.60.Bc Keywords: quantum gravity, cosmology ∗

Corresponding Author’s Email: [email protected].

24

1.

Michael A. Ivanov

INTRODUCTION

There are very different approaches to unify general relativity with quantum mechanics or with the standard model of particle physics (SM), but there are almost no theoretical predictions which may be verified by experiments or observations. Known predictions, if the ones are possible, concern mainly Planckscale physics and geometry, for example, foamy space-time in loop quantum gravity. This poorness of theoretical predictions of existing models and the absence of manifestations of quantum gravity accepted by the scientific community make the situation around quantum gravity very vague: theorists are not sure in the validity of used approaches, experimentalists and observers do not know what to search to help them. Taking into account logical difficulties of existing approaches, the main sought by H. Nicolai [1] about the situation is that we have no other choice but to try to create a future consistent theory out of purely theoretical basics. It seems that one of the possible ways is choosing a symmetry group which may lead us further as it was by the creation of the SM. But the SM’s symmetries were established due to big experimental efforts. From another side, the SM’s continuous symmetries may result from underlying discrete symmetries if the fundamental fermions are the two-component composite particles [2, 3]. Unfortunately, even availability of the consistent model of quantum gravity talking us about Planck-scale physics cannot help to understand why micro particles prefer not to move along geodesics by small energies which are very far from the Planck scale. Perhaps, we should search and introduce some more non-evident ideas to come nearer to the unknown quantum nature of gravity. In the model of low-energy quantum gravity [4, 5, 6], gravitation is considered as the screening effect in the sea of super-strong interacting gravitons. The Newton constant G and the Hubble constant H are computable in the model as functions of the background temperature. There is no need of any expansion of the universe and dark energy in the model to fit corresponding cosmological observations. The two-parametric theoretical luminosity distance of the model is caused by forehead and non-forehead collisions of photons with gravitons. The additional deceleration of massive bodies has the same nature as the redshift of remote objects in the model: these effects are caused by collisions with gravitons, but we should take into account both forehead and backhead collisions with gravitons in case of massive bodies [7]. Some consequences of the model are described in this chapter.

Low-Energy Quantum Gravity and Cosmology

2.

MAIN F EATURES

OF THE

25

MODEL

I would like to describe here some important features of my model of lowenergy quantum gravity [4, 5, 6]. It is supposed in it that the background of super-strong interacting gravitons exists with the same temperature T as CMB. In the sea of gravitons, a pressure force of single gravitons and a repulsive force due to scattered gravitons are approximately equal for any pair of usual bodies. However they are three order greater than the Newtonian force between bodies. It leads immediately to the very surprising conclusion: Einstein’s equivalence principle would be roughly violated for black holes, because this repulsive force is equal to zero for them. The ratio of gravitational to inertial masses of a black hole is equal to 1215.4. For a binary system of a black hole and a usual body, the third Newtonian law will also be violated. If single gravitons of running flux couple in pairs which are destructed in collisions, then we have for the Newton constant G : G≡

4 D2 c(kT )6 · · I2 , 3 π 3 ¯h3

(1)

where I2 = 2.3184 · 10−6 . It follows from this expression that by T = 2.7K the new constant D should have the value: D = 0.795 · 10−27 m2 /eV 2 . The inverse-square law of classical gravity describes the main quantum effect of this model. The possibility to calculate G makes the model underlying for general relativity. To have the condition of big distances: σ(E, <  >)  4πr 2 , where σ(E, <  >) is the cross-section of interaction of gravitons with an average energy <  > with a particle having an energy of E, r is a distance between massive particles, be fulfilled, it is necessary to accept an ”atomic structure” of matter, i.e., gravitons cannot interact with big bodies in the aggregate, they may interact only with ”small particles” of matter - for example, with atoms. For photons, there are two small effects in the sea of super-strong interacting gravitons: average energy losses of a photon due to forehead collisions with gravitons and an additional relaxation of a photonic flux due to non-forehead collisions of photons with gravitons. The first effect leads to the geometrical distance/redshift relation: r(z) = ln(1 + z) · c/H0 ,

(2)

where H0 is the Hubble constant. The both effects lead to the luminosity dis-

26

Michael A. Ivanov

tance/redshift relation: DL (z) = c/H0 · ln(1 + z) · (1 + z)(1+b)/2 ,

(3)

where the ”constant” b belongs to the range 0 - 2.137 (b = 2.137 for a very soft radiation, and b → 0 for a very hard one). In the general case it should depend on a rest-frame spectrum and on the redshift. Because of this, the Hubble diagram should be a multi-value function of the redshift: for a given z, b may have different values for different kinds of sources. The average time delay of photons due to multiple interactions with gravitons of the background is computed in my paper [8]. The two variants of evaluation of the lifetime of a virtual photon are considered: 1) on a basis of the uncertainties relation (it is a common place in physics of particles) and 2) using a conjecture about constancy of the proper lifetime of a virtual photon. In the first case any Lorentz violation is negligible: the ratio of the average time delay of photons to their propagation time is equal approximately to 10−28 ; in the second one (with a new parameter of the model), the time-lag is proportional √ free√ to the difference E01 − E02 , where E01 , E02 are initial energies of photons, and more energetic photons should arrive later, also as in the first case. The effect of graviton pairing is taken into account. The Hubble constant may be computed in the model, too: H0 =

1 45 σT 4 I42 1/2 D · ¯ · (σT 4 ) = (G ) , 2π 64π 5 c3 I2

(4)

where ¯ is an average graviton energy, I4 = 24.866. We have for its value: H0 = 2.14 · 10−18 s−1 = 66.875 km · s−1 · M pc−1 . The additional deceleration w of massive bodies has the same nature as the redshift of remote objects in the model: these effects are caused by collisions with gravitons, but we should take into account both forehead and backhead collisions with gravitons in the case of massive bodies [7]. The deceleration w is equal to: w = −w0 · 4η 2 · (1 − η 2 )0.5 , (5)

where w0 ≡ H0 c = 6.419 · 10−10 m/s2 , if we use the theoretical value of H0 in the model; η ≡ V /c, V is a body’s velocity relative to the graviton background. For small velocities we have: w ' −w0 · 4η 2.

(6)

Low-Energy Quantum Gravity and Cosmology

3.

MODIFIED DYNAMICS BACKGROUND

IN THE

27

G RAVITON

Some results of numerical modeling of a motion of bodies in the central field by the influence of this additional deceleration are described in this section [10].

Figure 1. A star orbit in a galaxy with M = 1010 · M by u = 5 · 105 m/s and r(0) = 1 kpc; t ' 30 Gyr, single loops interflow, the change of the distance to the center ∆r/r(0) = −0.034. In the Newtonian approach, if u is a more massive body’s velocity relative to the background, M is its mass, and V = v + u is the velocity of the small body relative to the graviton background, we will have now the following equation of motion of the small body: ¨r = −G

M r 4w0 · + 2 (u · u − | v + u | · (v + u)), r2 r c

where r is a radius-vector of the small body.

(7)

28

Michael A. Ivanov

To model the motion in the central field, I have slightly modified the program in C++ written for our work [9] to work in 3 dimensions using Eq. 7.

Figure 2. A star orbit in a galaxy with M = 1010 · M by u = 5 · 105 m/s and r(0) = 100 kpc; t ' 300 Gyr, the first unclosed external loop corresponds to 29.2 Gyr. Let us consider the initial conditions by which a material point trajectory in the classical case is circular, i.e., v(0) = (G · M/r(0))0.5, and v(0) ⊥ r(0), T is a period of motion in the classical case of a circular trajectory by the given initial distance to the center. To evaluate a stability of planetary orbits in the solar system in a presence of the anomalous deceleration w, we can use the following trick: to increase w by hand to see a very small change of the orbit’s radius, and to re-calculate a value of the resulting effect. In a case of the Earth-like circular orbit, i.e., by M = M , r(0) = 1 AU, given u = 4 · 105 m/s and that three vectors r, v, u lie in one plane, we get by the replacement: w → 104 · w for one classical period T : ∆r/r(0) = −1.08 · 10−8 yr−1 by ∆t = 10−10 · T. It

Low-Energy Quantum Gravity and Cosmology

29

means that by the anomalous deceleration w we should have now: ∆r/r(0) = −1.08 · 10−12 yr−1 . For the case when u is perpendicular to r, v we have: ∆r/r(0) = −7.2 · 10−13 yr−1 . The Earth orbit will be stable enough to have not contradictions with the estimated age of it in the solar system.

Figure 3. The deviation z(t) (solid) of a star orbit in a galaxy (with M = 1010 · M by u = 5 · 105 m/s and r(0) = 10 kpc) from the classical plane (x, y) for the case of v(0) = 1.2 · (G · M/r(0))0.5; T = 0.781 Gyr, the graph of 10−4 · y(t) (dotted) is shown for the comparison. Results of modeling a star orbit in a galaxy in the similar way are shown in Figures 1 and 2 for M = 1010 · M , u = 5 · 105 m/s by r(0) = 1 kpc (Fig. 0 1) and r(0) = 100 kpc (Fig. 2). The ratio r¨w(0) is equal to 2.2 and 0.00022 respectively. By r(0) = 1 kpc the relative change of the distance to the center is ∆r/r(0) = −0.034 during the time interval of ' 30 Gyr. By r(0) = 1 kpc the first unclosed external loop in Fig. 2 corresponds to 29.2 Gyr. We see that at all scales closed orbits do not exist in the model: bodies inspiral to the center of attraction, but for the Earth-like orbits this effect is very small. When u is perpendicular to r, v, another effect takes place: the motion of

30

Michael A. Ivanov

Figure 4. The same graphs as in Fig. 3, but for the case of v(0) = (G · M/r(0))0.5; T = 0.781 Gyr, 10−4 · y (dotted), z (solid). the body in the central field is not planar. The deviation z(t) of a star orbit in a galaxy (with M = 1010 · M by u = 5 · 105 m/s and r(0) = 10 kpc) from the classical plane (x, y) is shown in Figures 3 and 4. For the case of v(0) = (G · M/r(0))0.5 (the classical orbit would be circular), deviations from the classical plane (x, y) occur in one side off this plane, with returns to it (Fig. 4). In the case of the Earth-like circular orbit, the maximal deviation from the classical plane is lesser of 1 mm by u = 4 · 105 m/s. If v(0) 6= (G · M/r(0))0.5, deviations from the classical plane (x, y) occur in both sides off this plane (Fig. 3, v(0) = 1.2 · (G · M/r(0))0.5), and the ones may be interpreted as a slow revolution of a quasi-classical planar orbit around some axis in this plane. The described results show two peculiarities of modified dynamics in the model: an absence of closed orbits and a possibility of the non-planar motion of massive bodies in the central field due to the anomalous deceleration by the graviton background. These effects are negligible for the Earth-like orbits and, perhaps, too small to be observable during an acceptable time interval in galaxies. But the interaction of photons with the background leads to the observable

31

Low-Energy Quantum Gravity and Cosmology effects which can be essential for our understanding of the universe.

4.

C OSMOLOGICAL C ONSEQUENCES

OF THE

MODEL

Small additional effects of this model have essential cosmological consequences. In the model, redshifts of remote objects and the dimming of supernovae 1a may be interpreted without any expansion of the Universe and without dark energy. Some of these consequences are discussed and confronted with galaxy number counts, supernovae 1a, long GRBs, and QSOs observations in this section. It is shown that the two-parametric theoretical luminosity distance of the model fits observations with high confidence levels, if all data sets are corrected for no time dilation. These two parameters are computable in the model.

4.1.

Galaxy Number Counts

Figure 5. The graph of the function f2 (z) (solid) of this model. The typical error bar and data point are taken from paper by Loh and Spillar [13].

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Michael A. Ivanov

In this subsection, I consider galaxy number counts/redshift and counts/magnitude relations on a basis of this model [11]. I assume here that a space is flat and the Universe is not expanding. 4.1.1.

The Galaxy Number Counts-Redshift Relation

Total galaxy number counts dN (r) for a volume element dV = dΩr 2 dr is equal to: dN (r) = ng dV = ng dΩr 2 dr, where ng is the galaxy number density (it is constant in the no-evolution scenario), dΩ is a solid angle element. Using the function r(z) of this model, we can re-write galaxy number counts as a function of the redshift z: dN (z) = ng dΩ(H0/c)−3

ln2 (1 + z) dz. 1+z

(8)

Let us introduce a function (see [12]) f2 (z) ≡

(H0 /c)3 dN (z) ; ng dΩz 2 dz

then we have for it in this model: f2 (z) =

ln2 (1 + z) . z 2 (1 + z)

(9)

A graph of this function is shown in Fig. 5; the typical error bar and data point are added here from paper by Loh and Spillar [13]. There is not a visible contradiction with observations. There is not any free parameter in the model to fit this curve; it is a very rigid case. It is impossible to count a total galaxy number for big redshifts so as very faint galaxies are not observable. For objects with a fixed luminosity, it is easy to find how their magnitude m changes with a redshift. So as dm(z) under a constant Rluminosity is equal to: dm(z) = 5d(lgDL(z)), we have for ∆m(z1 , z2 ) ≡ zz12 dm(z) : ∆m(z1 , z2 ) = 5lg(f1(z2 )/f1 (z1 )). The graph of this function is shown in Fig. 6 for z1 = 0.001; 0.01; 0.1.

(10)

Low-Energy Quantum Gravity and Cosmology

33

Figure 6. Magnitude changes ∆m as a function of the redshift difference z2 −z1 in this model for z1 = 0.001 (solid); 0.01 (dot); 0.1 (dash). 4.1.2.

Taking into Account the Galaxy Luminosity Function

Galaxies have different luminosities L, and we can write ng as an integral: R ng = dng (L), where dng (L) = η(L)dL, η(L) is the galaxy luminosity function. I shall use here the Schechter luminosity function [14]: L α L L ) exp(− )d( ) (11) L∗ L∗ L∗ with the parameters φ∗ , L∗ , α. So as we have by a definition of the luminosity distance DL (z) that a light flux I is equal to: I = 4πDL2 (z) , and a visible magL nitude m of an object is m = −2.5 lg I + C, where C is a constant, then m is equal to: η(L)dL = φ∗ (

m = −2.5 lg I + 5 lg DL (z) + (C − 4π). We can write for L : L =A·

2 (z) DL , κm

(12) (13)

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Michael A. Ivanov

Figure 7. The relative difference (f3 (m) − a(m))/a(m) as a function of the magnitude m for α = −2.43 by 10−2 < A1 < 102 (solid), A1 = 104 (dash), A1 = 105 (dot), A1 = 106 (dadot). where κ = 100.4 , A = const. For a thin layer with z = const we have: dL = where

∂L · dm, ∂m

D2 (z) ∂L = −mκ · A Lm = −mκL. ∂m κ

(14)

Then dng (m, z) = −(φ∗ κ) · l α(m, z) exp(−l(m, z)) · (m · l(m, z))dm,

(15)

where (−dm) corresponds to decreasing m by growing L when z = const, and l(m, z) ≡

L(m, z) . L∗

Low-Energy Quantum Gravity and Cosmology

35

Let us introduce a function f3 (m, z) with a differential df3 (m, z) ≡

dN (m, z) dΩ(−dm).

(16)

We have for this differential in the model:

Figure 8. Number counts f4 (m, z) (dot) and f5 (m1 , m2 ) (solid) (logarithmic scale) as a function of the redshift by A1 = 105 for α = −2.43, m1 = 10 and different values of m = m2 : 15, 20, 25, 30; m = 10 (only f4 (m, z)).

df3 (m, z) = (

φ∗ κ ln2 (1 + z) α+1 ) · m · l (m, z) · exp(−l(m, z)) · dz, a3 (1 + z)

(17)

where a = H0 /c. An integral on z gives the galaxy number counts/magnitude relation: f3 (m) = (

φ∗ κ )·m· a3

Z

0

zmax

l α+1 (m, z) · exp(−l(m, z)) ·

ln2 (1 + z) dz; (18) (1 + z)

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Michael A. Ivanov

I use here an upper limit zmax = 10. To compare this function with observations by Yasuda et al. [15], let us choose the normalizing factor from the condition: f3 (16) = a(16), where a(m) ≡ Aλ · 100.6(m−16)

(19)

is the function assuming ”Euclidean” geometry and giving the best fit to observations [15], Aλ = const depends on the spectral band. In this case, we have two free parameters - α and L∗ - to fit observations, and the latter one is connected with a constant A1 ≡ a2AL∗ if l(m, z) = A1

f12 (z) . κm

Figure 9. The theoretical Hubble diagram µ0 (z) of this model (solid); Supernovae 1a observational data (580 points of the SCP Union 2.1 compilation) are taken from [18] and corrected for no time dilation. If we use the magnitude scale in which m = 0 for Vega then C = 2.5 lg IV ega , and we get for A1 by H0 = 2.14 · 10−18 s−1 (it is a theoretical

Low-Energy Quantum Gravity and Cosmology

37

estimate of H0 in this model): A1 ' 5 · 1017 ·

L , L∗

(20)

where L is the Sun luminosity; the following values are used: LV ega = 50L , the distance to Vega rV ega = 26 LY.

Figure 10. Values of k(z) (580 points) and < k(z) >, < k(z) > +σk , < k(z) > −σk (lines) for the SCP Union 2.1 compilation. Without the factor m, the function f3 (m) by exp(−l(m, z) → 1 would be close to a(m) by α = −2.5. Matching values of α shows that f3 (m) is the closest to a(m) in the range 10 < m < 20 by α = −2.43. The ratio f3 (m)−a(m) is shown in Fig. 7 for different values of A1 by this value of α a(m) (to turn aside the problem with divergencies of this function by small L for negative values of α, all computations are performed here for z > 0.001). All such the curves conflow by A1 ≤ 102 (or 5·1015 < L∗ ), i.e., observations of the galaxy number counts/magnitude relation are non-sensitive to A1 in this range. For fainter magnitudes 20 < m < 30, the behavior of all curves is identical: they go below of the ratio value 1 with the same slope. If we compare this

38

Michael A. Ivanov

figure with Figs. 6,10,12 from [15], we see that the considered model provides a no-worse fit to observations than the function a(m) if the same K-corrections are added (perhaps, even the better one if one takes into account positions of observational points in Figs. 6,10,12 from [15] by m < 16 and m > 16) for the range 102 < A1 < 107 that corresponds to 5 · 1015 > L∗ > 5 · 1010 . Observations of N (z) for different magnitudes are a lot more informative. If we define a function f4 (m, z) as f4 (m, z) ≡ (

a3 df3 (m, z) )· , φ∗ κ dz

(21)

this function is equal in the model to: f4 (m, z) = m · l α+1 (m, z) · exp(−l(m, z)) ·

ln2 (1 + z) . (1 + z)

(22)

Galaxy number counts in the range m1 < m < m2 are proportional to the function: Z m2 f5 (m1 , m2 ) ≡ f4 (m, z)dm = (23) =

Z

m1

m2

m1

m · l α+1 (m, z) · exp(−l(m, z)) ·

ln2 (1 + z) dm. (1 + z)

Graphs of both f4 (m, z) and f5 (m1 , m2 ) are shown in Fig. 8 by α = −2.43, A1 = 105 ; they are very similar between themselves. We see that even the observational fact that a number of visible galaxies by z ∼ 10 is very small allows us to restrict a value of the parameter A1 much stronger than observations of N (m). Quasar number counts are considered in [11], too.

4.2. 4.2.1.

Fitting Observations with the Theoretical Luminosity Distance The Hubble Diagram of This Model

In this model, the luminosity distance is given by Eq. 3. The theoretical value of relaxation factor b for a soft radiation is b = 2.137. Let us begin with this value of b, considering the Hubble constant as a single free parameter to fit observations [16]. All observational data should be corrected for no time dilation as: µ(z) → µ(z) + 2.5 · lg(1 + z) in this model without expansion.

Low-Energy Quantum Gravity and Cosmology

39

Figure 11. The theoretical Hubble diagram µ0 (z) of this model with b = 2.365 (solid); Supernovae 1a observational data (31 binned points of the JLA compilation) are taken from Tables F.1 and F.2 of [19] and corrected for no time dilation. Two big compilations of SN 1a observations are used here: the SCP Union 2.1 compilation (580 supernovae) [18] and the JLA compilation (740 supernovae) [19]. These compilations may be used to evaluate the Hubble constant in this approach. Using the definition of distance modulus: µ(z) = 5lgDL(z)(M pc) + 25, we get from Eq. 3 for the theoretical distance modulus µ0 (z): µ0 (z) = 5lgf1(z) + k, where f1 (z) ≡ ln(1 + z) · (1 + z)(1+b)/2, and the constant k is equal to: k ≡ 5lg(c/H0) + 25. If the model fits observations, then we shall have for k(z): k(z) = µ(z) − 5lgf1(z),

(24)

40

Michael A. Ivanov

Figure 12. Values of k(z) (31 binned points) and < k(z) >, < k(z) > +σk , < k(z) > −σk (lines) for the JLA compilation. where µ(z) is an observational value of distance modulus. The weighted average value of k(z) : P k(zi )/σi2 , (25) < k(z) >= P 1/σi2

where σi2 is a dispersion of µ(zi ), will be the best estimate of k. Here, σi2 is defined as: σi2 = σi2 stat + σi2 sys . The average value of the Hubble constant may be found as: c · 105 . 10/5 · M pc For a standard deviation of the Hubble constant we have: < H0 >=

(26)

ln10· < H0 > · σk , (27) 5 where σk2 is a weighted dispersion of k, which is calculated with the same weights as < k(z) > . σ0 =

Low-Energy Quantum Gravity and Cosmology

41

Figure 13. The theoretical Hubble diagram µ0 (z) of this model (solid); long GRBs observational data (109 points) are taken from Tables 1,2 of [20] and corrected for no time dilation. The theoretical Hubble diagram µ0 (z) of this model with < k(z) > which is calculated using the SCP Union 2.1 compilation [18] is shown in Fig. 9 together with observational points corrected for no time dilation. Values of k(z) (580 points) and < k(z) >, < k(z) > +σk , < k(z) > −σk (lines) are shown in Fig. 10. For this compilation we have: < k > ±σk = 43.216 ± 0.194. Calculating the χ2 value as: χ2 =

X (k(zi)− < H0 >)2

σi2

,

(28)

we get χ2 = 239.635. By 579 degrees of freedom of this data set, it means that the hypothesis that k(z) = const cannot be rejected with 100% C.L. Using Eqs. 25, 26, we get for the Hubble constant from the fitting: < H0 > ±σ0 = (2.211 ± 0.198) · 10−18 s−1 = (68.223 ± 6.097)

km . s · M pc

42

Michael A. Ivanov

Figure 14. Values of k(z) (109 points) and < k(z) >, < k(z) > +σk , < k(z) > −σk (lines) for long GRBs. The theoretical value of the Hubble constant in the model: H0 = 2.14 · 10−18 s−1 = 66.875 km · s−1 · M pc−1 belongs to this range. The traditional dimension km · s−1 · M pc−1 is not connected here with any expansion. To repeat the above calculations for the JLA compilation, I have used 31 binned points from Tables F.1 and F.2 of [19] (diagonal elements of the correlation matrix in Table F.2 are dispersions of distance moduli). We have for this compilation by b = 2.137: < k > ±σk = 43.174 ± 0.049 with χ2 = 51.66. By 30 degrees of freedom of this data set, it means that the hypothesis that k(z) = const cannot be rejected only with 0.83% C.L. Varying the value of b, we find the best fitting value of this parameter: b = 2.365 with χ2 = 30.71. It means that the hypothesis that k(z) = const cannot be rejected now with 43.03% C.L. This value of b is 1.107 times greater than the theoretical one. For the Hubble constant we have in this case: < H0 > ±σ0 = (2.254 ± 0.051) · 10−18 s−1 = (69.54 ± 1.58)

km . s · M pc

Low-Energy Quantum Gravity and Cosmology

43

Results of the best fitting are shown in Figs. 11,12. If observations of long Gamma-Ray Bursts (GRBs) for small z are calibrated using SNe 1a, observational points are fitted with this theoretical Hubble diagram, too [6]. But for hard radiation of GRBs, the factor b may be smaller, and the real diagram for them may differ from the one for SNe 1a. With this limitation, the long GRBs observational data (109 points) are taken from Tables 1,2 of [20] and fitted in the same manner with b = 2.137. In this case we have: < k > ±σk = 43.262 ± 8.447 with χ2 = 70.39. By 108 degrees of freedom of this data set, it means that the hypothesis that k(z) = const cannot be rejected with 99.81% C.L. For the Hubble constant we have in this case: km . < H0 > ±σ0 = (2.162 ± 0.274) · 10−18 s−1 = (66.71 ± 8.45) s · M pc Results of the fitting are shown in Figs. 13,14.

Figure 15. The theoretical Hubble diagram µ0 (z) of this model with b = 1.11 (solid); GRB observational data with the Yonetoku calibration (44 points) are taken from Table 3 of [21] and corrected for no time dilation. A data set of 44 long Gamma-Ray Bursts was compiled with the redshift

44

Michael A. Ivanov

Figure 16. The theoretical Hubble diagram µ0 (z) of this model (solid); quasar observational data (18 binned points) [23] are corrected for no time dilation. range of [0.347; 9.4] [21], in which two empirical luminosity correlations (the Amati relation and Yonetoku relation) were used to calibrate observations. Because the GRB Hubble diagram calibrated using luminosity correlations is almost independent on the GRB spectra, as it has been shown by the authors, I use here values of µ(zi ) ± σi from columns 7 of Tables 2 and 3 of [21], based on the Band function, but with both calibrations. If this data set is fitted in the same manner with b = 2.137, we have for the Amati calibration: < k > ±σk = 43.168 ± 1.159 with χ2 = 40.585. By 43 degrees of freedom of this data set, it means that the hypothesis that k(z) = const cannot be rejected with 57.66% C.L. For the Hubble constant we have in this case:

< H0 > ±σ0 = (2.26 ± 1.206) · 10−18 s−1 = (69.732 ± 37.226)

km . s · M pc

By b = 2.137, we have for the Yonetoku calibration: < k > ±σk = 43.148 ± 1.197 with χ2 = 43.148. It means that the hypothesis that k(z) = const cannot

45

Low-Energy Quantum Gravity and Cosmology be rejected with 46.5% C.L. For the Hubble constant we have in this case: < H0 > ±σ0 = (2.281 ± 1.257) · 10−18 s−1 = (70.386 ± 38.793)

km . s · M pc

But best fitting values of b are less than 2.137 in both cases: b = 1.885 for the Amati calibration (< k > ±σk = 43.484 ± 1.15, χ2 = 39.92, with 60.57% C.L. and < H0 > ±σ0 = (1.954 ± 1.035) · 10−18 s−1 = (60.309 ± 31.932)km/s/M pc.), and b = 1.11 for the Yonetoku one (< k > ±σk = 44.439 ± 1.037, χ2 = 32.58, with 87.62% C.L. and < H0 > ±σ0 = (1.259±0.601)·10−18 s−1 = (38.841±18.546)km/s/M pc.). Namely smaller values of this parameter for bigger photon energies are expected in the model. For best fitting values of b, values of distance moduli are overestimated in both calibrations: on ∼ 0.225 for the Amati calibration, and on ∼ 1.18 for the Yonetoku calibration, if we compare values of < k > with its theoretical value of 43.259. It leads to the corresponding underestimation of the Hubble constant. Results of the best fitting for the Yonetoku calibration are shown in Fig. 15. A new method to test cosmological models was introduced, based on the Hubble diagram for quasars [22]. The authors built a data set of 1,138 quasars for this purpose. Some later, this method and the data set were used to compare different models [23]. I have used here the binned quasar data set (18 binned points) of the paper [23] to verify my model in the described above manner. This data set contains the sum of observed distance modulus and an arbitrary constant A. To find this unknown constant for the calibration of QSO observations, I have computed < k0 (z) >=< k(z) > +A and replaced < k(z) > by its value for the JLA compilation; it gave: A = 50.248. This linking means that the average values of the Hubble constant should be identical for the two data sets. Subtracting this value of A, we get from the fitting of the quasar data by b = 2.137: < k > ±σk = 43.175 ± 0.340 with χ2 = 23.378. By 17 degrees of freedom of this data set, it means that the hypothesis that k(z) = const cannot be rejected now with 13.73% C.L. For the Hubble constant we have: < H0 > ±σ0 = (2.253 ± 0.340) · 10−18 s−1 = (69.534 ± 10.873) Results of the fitting are shown in Fig. 16.

km . s · M pc

46

Michael A. Ivanov

4.2.2.

Comparison with the LCDM Cosmological Model

The luminosity distance in the concordance cosmology by w = −1 is: DL (z) = c/H0 · (1 + z)

Z

where f2 (z) ≡ (1 + z)

Rz

0

z

[(1 + x)3 ΩM + (1 − ΩM )]−0.5 dx ≡ c/H0 · f2 (z),

(29) 3 −0.5 [(1 + x) Ω + (1 − Ω )] , Ω is the normalM M M 0

Table 1. Results of fitting the Hubble diagram with the model of low-energy quantum gravity and the LCDM cosmological model. The best fitting values of b and ΩM for 44 long GRBs are marked by the bold typeface the model of low-energy quantum gravity Data set b χ2 C.L., % SCP Union 2.1 [18] 2.137 239.635 100 JLA [19] 2.365 30.71 43.03 109 long GRBs [20] 2.137 70.39 99.81 44 long GRBs [21], 2.137 40.585 57.66 the Amati calibration 1.885 39.92 60.57 44 long GRBs [21], 2.137 43.148 46.5 the Yonetoku calibration 1.11 32.58 87.62 quasars [23] 2.137 23.378 13.73 the LCDM cosmological model Data set ΩM χ2 C.L., % SCP Union 2.1 [18] 0.30 217.954 100 JLA [19] 0.30 29.548 48.90 109 long GRBs [20] 0.30 66.457 99.94 44 long GRBs [21], 0.30 40.777 56.81 the Amati calibration 0.49 40.596 57.61 44 long GRBs [21], 0.30 38.456 66.85 the Yonetoku calibration 1.0 34.556 81.72 quasars [23] 0.30 21.368 21.03

< H0 > ±σ0 68.22 ± 6.10 69.54 ± 1.58 66.71 ± 8.45 69.73 ± 37.23 60.31 ± 31.93 70.39 ± 38.79 38.84 ± 18.55 69.53 ± 10.87 < H0 > ±σ0 69.68 ± 5.94 70.08 ± 1.56 70.04 ± 8.62 68.99 ± 36.92 60.75 ± 32.44 69.59 ± 36.10 49.51 ± 24.35 69.68 ± 10.42

ized matter density. To compare the above results of fitting with results for the LCDM cosmology, let us replace f1 (z) → f2 (z) and repeat the calculations. Of course, all data sets should remain now corrected for time dilation. The re-

Low-Energy Quantum Gravity and Cosmology

47

sults of fitting are presented in Table 1; for convenience, the main above results for the model of low-energy quantum gravity are collected in the table, too. It is obvious, that confidence levels for both models do not allow to reject any of them. It is a big surprise that the Einstein–de Sitter model (Eq. 29 with ΩM = 1) cannot be rejected on a base of the full SCP Union 2.1 data set and the χ2 −criterion. We get χ2 = 428.579 and 99.9999% C.L. The cause is in a big number of small-z supernovae 1a in this set; it leads to a big number of degrees of freedom, but to small differences of χ2 for models with similar values of DL (z) in this range of z. But if one splits the data set in two subsets, for example with z ≤ 0.5 and z > 0.5, and uses the first subset to evaluate < H0 >, then using this < H0 > and the second subset to compute χ2 by much smaller number of degrees of freedom, one can reject this model with high probability (when z > 0.5, we get χ2 = 247.551 by 166 observations and 0.004% C.L.). Results for the model of low-energy quantum gravity and the LCDM cosmological model are not essentially changed by the splitting. But the Einstein–de Sitter model with ΩM = 1 bests the LCDM cosmological model with any amount of dark energy for the 44 long GRBs data set with the Yonetoku calibration.

4.3.

The Hubble Parameter of This Model

If the geometrical distance is described by Eq. 2, for a remote region of the universe we may introduce the Hubble parameter H(z) in the following manner: dz = H(z) ·

dr , c

to imitate the local Hubble law. Taking a derivative H(z) : H(z) = H0 · (1 + z). It means that in the model:

H(z) = H0 . (1 + z)

(30) dr dz ,

we get in this model for (31) (32)

The last formula gives us a possibility to evaluate the Hubble constant using observed values of the Hubble parameter H(z). To do it, I use here 28 points of H(z) from [24] and one point for z < 0.1 from [25]. The last point is the result

48

Michael A. Ivanov

Figure 17. The ratio H(z)/(1 + z) ± σ and the weighted value of the Hubble constant < H0 > ±σ0 (horizontal lines). Observed values of the Hubble parameter H(z) are taken from Table 1 of [24] and one point for z < 0.1 is taken from [25]. of HST measurement of the Hubble constant obtained from observations of 256 low-z supernovae 1a. Here I refer this point to the average redshift z = 0.05. Observed values of the ratio H(z)/(1 + z) with ±σ error bars are shown in Fig. 17 (points). The weighted average value of the Hubble constant is calculated by the formula: P H(zi ) 2 1+z /σi < H0 >= P i 2 . (33) 1/σi The weighted dispersion of the Hubble constant is found with the same weights: P H(zi ) ( 1+zi − < H0 >)2 /σi2 2 P σ0 = . (34) 1/σi2

Low-Energy Quantum Gravity and Cosmology

49

Figure 18. The ratio H(z)/(1 + z) ± σ and the weighted value of the Hubble constant < H0 > ±σ0 (horizontal lines). Observed values of the Hubble parameter H(z) are taken from [26]. Calculations give for these quantities: < H0 > ±σ0 = (64.40 ± 5.95) km s−1 M pc−1 .

(35)

The weighted average value of the Hubble constant with ±σ0 error bars are shown in Fig. 17 as horizontal lines. Calculating the χ2 value as: χ2 =

i) 2 X ( H(z 1+z − < H0 >) i

σi2

,

(36)

we get χ2 = 16.491. By 28 degrees of freedom of our data set, it means that the hypothesis described by Eq. 31 cannot be rejected with 95% C.L.

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Michael A. Ivanov

If we use another set of 21 cosmological model-independent measurements of H(z) based on the differential age method [26], we get (see Fig. 18): < H0 > ±σ0 = (63.37 ± 4.56) km s−1 M pc−1 .

(37)

The value of χ2 in this case is smaller and equal to 3.948. By 21 degrees of freedom of this new data set, it means that the hypothesis described by Eq. 31 cannot be rejected with 99.998% C.L. Some authors try in a frame of models of expanding universe to find deceleration-acceleration transition redshifts using the same data set (for example, [24]). The above conclusion that the ratio H(z)/(1 + z) remains statistically constant in the available range of redshifts is model-independent. For the considered model, it is an additional fact against dark energy as an admissible alternative to the graviton background.

4.4.

The Alcock-Paczynski Test of the Model

The Alcock-Paczynski cosmological test consists in an evaluation of the ratio of observed angular size to radial/redshift size [27]. This test has been carried out for a few cosmological models by Fulvio Melia and Martin LopezCorredoira [28]. They used model-independent data on BAO peak positions from [29] and [30]. For two mean values of z (< z >= 0.57 and < z >= 2.34), the measured angular-diameter distance dA (z) and Hubble parameter H(z) give for the observed characteristic ratio yobs (z) of this test the values: yobs (0.57) = 1.264 ± 0.056 and yobs (2.34) = 1.706 ± 0.076. In this model we have: dcom (z) = dA (z) = r(z), where dcom (z) is the cosmological comoving distance. Because the Universe is static here, the ratio y(z) for this model is defined as: r(z) · H(z) 1 r(z) y(z) = = = (1 + ) · ln(1 + z), (38) d cz z z · dz r(z) where H(z) is defined by Eq. 31. This function without free parameters characterizes any tired light model (model 6 in [28]). We have only two observational points to fit them with this function. Calculating the χ2 value as: χ2 =

X (yobs (zi ) − y(zi ))2

σi2

,

(39)

we get χ2 = 0.189, that corresponds to the confidence level of 91% for two degrees of freedom.

Low-Energy Quantum Gravity and Cosmology

5.

51

T HE L IGHT- FROM -N OWHERE E FFECT

The additional relaxation of a photonic flux of a remote galaxy due to nonforehead collisions of photons with gravitons is accompanied with the deviation of some part of photons from the galaxy-observer direction. Given multiple collisions on their long ways, the number of initial photons scattered in such the manner rises quickly, and each of them may be scattered again and again. It should lead to the appearance of a diffuse background with a complex spectrum. A tentative detection of a diffuse cosmic optical background [33] may be connected with this light-from-nowhere effect. To evaluate how big is the ratio δ(z) of the scattered flux to the the remainder 2 (b, z) reaching the observer, we can compute the flux Φ (z) ≡ Φ(z) ≡ L/DL 0 2 L/DL(0, z), where L is the luminosity, DL (b, z) and DL (0, z) are luminosity distances by b 6= 0 and b = 0. Φ0 (z) corresponds to the absence of non-forehead collisions. Then the ratio may be defined as: δ(z) ≡ (Φ0 (z) − Φ(z))/Φ(z).

(40)

δ(z) = (1 + z)b − 1.

(41)

Using Eq. 3 we get: We have by b = 2.137: δ(1) = 3.34, δ(2) = 9.46, δ(10) = 167.06. To find the sky brightness in the optical range, for example, it is necessary to know the ratio δ(z), and, at least, the light flux of galaxies and their number counts by different redshifts.

C ONCLUSION The Newton constant G has been measured up to now with the relative standard uncertainty only ∼ 10−4 (about the long story of these measurements, see [31]). In this model, the Newton constant arises as an average value of the stochastic variable characterizing the interaction of a couple of bodies with a huge number of gravitons. Uncertainties of G and T are connected as: ∆G ∆T =6 . G T If fluctuations of the temperature of the graviton background have the same order of magnitude as the ones of the CMB temperature, then ∆G/G ≤ 6·10−4 .

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It is important that measured values of G may depend on the orientation of two bodies relatively to remote stars. Further attempts to measure G taking into account these circumstances may be interesting for the verification. In this model, the luminosity distance is a multi-value function of the redshift due to different values of the factor b for soft and hard radiation. It opens another way to verify the model by cosmological observations comparing the Hubble diagrams of sources with different spectra. But to realize it we should have the possibility to calibrate the luminosity, for example, of remote GRBs independently of the Hubble diagram of supernovae Ia. The Hubble parameter H(z) of this model is a linear function of z: H(z) = H0 · (1 + z) (as well as in the Rh = ct cosmological model [32]), that is in a big discrepancy with ΛCDM. As it was shown, this function fits available observations of H(z) very well [6, 32], and further investigations of this problem are important. The most important cosmological consequence of the model is the local quantum nature of redshifts of remote objects. At present, advanced LIGO technologies may be partly used to verify this redshift mechanism in a groundbased laser experiment [6]. One should compare spectra of laser radiation before and after passing some big distance in a high-vacuum tube. If one constructs a future version of the LIGO detector with some additional equipment, the verification of the redshift mechanism may be performed in parallel with the main task or during a calibration stage of the detector. The positive expected result of such the experiment would mean also that the universe does not expand. It seems that to open minds for the broader perception of possible manifestations of quantum gravity and ways to its future theory, we should doubt in some commonly accepted things. The very bright example is the claimed existence of dark energy that is unnecessary in the considered model. If redshifts of remote objects have the local quantum nature, the expansion of the universe becomes not necessary, and some observable effects may be interpreted as the long-awaited manifestation of quantum gravity but in the absolutely unexpected scale of energies ∼ 10−3 eV. This scale may move us much closer to the understanding of the existing chasm between general relativity and quantum mechanics. And, perhaps, it can give us chances to construct if not a bridge between them, then a new common base for both theories.

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R EFERENCES [1] Nicolai, H. (2017). Gravity’s quantum side. https://cerncourier.com/a /gravitys-quantum-side/.

Eprint:

[2] Ivanov, M. A. (1988). The system of equations describing 4 generations with the symmetry group SU (3)C × SU (2)L × U (1) (in Russian). Deposited in VINITY 19.12.1988 as VINITI No 8842B88; in English: Eprint: https://vixra.org/pdf/2006.0185v3.pdf. [3] Ivanov, M. A. (1992). Primary postulates of the standard model as consequences of the composite nature of the fundamental fermions. Nuovo Cimento, 105A: 77. Eprint: arXiv:hepth/0207210. [4] Ivanov, M. A. (2001). Possible manifestations of the graviton background. Gen. Rel. Grav., 33: 479-490; Erratum-ibid. (2003), 35: 939-940. Eprint: arXiv:astro-ph/0005084v2. [5] Ivanov, M. A. (2006). ”Gravitons as super-strong interacting particles, and low-energy quantum gravity”. In Focus on Quantum Gravity Research, 89-120, edited by D.C. Moore, NY: Nova Science. Eprint: arXiv:hep-th/0506189v3. [6] Ivanov, M. A. (2018). Selected papers on low-energy quantum gravity. Eprint: https://vixra.org/pdf/1110.0042v2.pdf. [7] Ivanov, M. A. (2019). Low-energy quantum gravity and cosmology without dark energy. Advances in Astrophysics, 4: 1-6. [8] Ivanov, M. A. (2009). Lorentz symmetry violation due to interactions of photons with the graviton background. Eprint: arXiv:0907.1032v2 [physics.gen-ph]. [9] Ivanov, M. A., Narkevich, A. S., and Shenetz, P. S. (2017). Modified dynamics due to forehead collisions of bodies with gravitons: Numerical modeling. Eprint: http://vixra.org/pdf/1706.0427v1.pdf.

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Michael A. Ivanov [10] Ivanov, M. A. (2019). Modified dynamics of massive bodies in the graviton background. Eprint: https://vixra.org/pdf/1907.0257v2.pdf. [11] Ivanov, M. A. (2006). Galaxy number counts in a presence of the graviton background. Eprint: arXiv:astro-ph/0606223v3. [12] Cunha, J. V., Lima, J. A. S., N. Pires. (2002). Deflationary cosmology: Λ(t) Observational expressions. A&A, 390: 809-815. [13] Loh, E. D. and Spillar, E. J. (1986). A Measurement of the Mass Density of the Universe. ApJ, 307: L1. [14] Schechter, P. L. (1976). An analytic expression for the luminosity function for galaxies. ApJ, 203: 297-306. [15] Yasuda, N., et al. (2001). Galaxy number counts from the sloan digital sky survey commissioning data. AJ, 122: 1104-1124. [16] Ivanov, M. A. (2016). ”Cosmological consequences of the model of low-energy quantum gravity”. In Proc. Int. Conf. Cosmology on Small Scales 2016, 179-198, edited by M. Krizek and Yu. Dumin, Prague: Institute of Mathematics CAS. Eprint: http://http://css2016.math.cas.cz/proceedingsCSS2016.pdf. [17] Riess, A. G., et al. (2004). Type Ia Supernova Discoveries at z > 1 From the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution. ApJ,, 607: 665-687. Eprint: astro-ph/0402512. [18] Suzuki, N., et al. (2012). The Hubble Space Telescope Cluster Supernova Survey: V. Improving the Dark Energy Constraints Above z > 1 and Building an Early-Type-Hosted Supernova Sample. ApJ, 746: 85. Eprint: arXiv:1105.3470v1 [astro-ph.CO]. [19] Betoule, M., et al. (2014). Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples. A&A, 568: A22. Eprint: arXiv:1401.4064v2 [astro-ph.CO]. [20] Wei, H. (2010). Observational Constraints on Cosmological Models with the Updated Long Gamma-Ray Bursts. JCAP, 08. Eprint: arXiv:1004.4951v3 [astro-ph.CO].

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[21] Lin, H.-N., Li, X., and Chang, Z. (2016). Effect of GRB spectra on the empirical luminosity correlations and the GRB Hubble diagram. MNRAS, 459: 2501-2512. Eprint: arXiv:1604.02285 [astro-ph.HE]. [22] Risaliti, G., and Lusso, E. (2015). A Hubble Diagram for Quasars. ApJ,, 815: 33; Eprint: arXiv:1505.07118 [astro-ph.CO]. [23] Lopez-Corredoira, M., Melia, F., Lusso, E., and Risaliti, G. (2017). Cosmological test with the QSO Hubble diagram. International Journal of Modern Physics D, 25, No. 05, id. 1650060. Eprint: arXiv:1602.06743 [astro-ph.CO]. [24] Farooq, O., and Ratra, B. (2013). Hubble parameter measurement constraints on the cosmological deceleration-acceleration transition redshift. ApJ letters, 766: L7. Eprint: arXiv:1301.5243. [25] Riess, A. G., et al. (2011). A 3% Solution: Determination of the Hubble Constant with the Hubble Space Telescope and Wide Field Camera 3. ApJ, 730: 119. [26] Moresco, M. (2015). Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z∼2. MNRAS Letters, 450: L16-L20. Eprint: arXiv:1503.01116v1 [astro-ph.CO]. [27] Alcock, C. and Paczynski, B. (1979). An evolution free test for non-zero cosmological constant. Nature, 281: 358-359. [28] Melia, F., and Lopez-Corredoira, M. (2017). International Journal of Modern Physics D, 26, Issue 6, id. 1750055-265. Eprint: arXiv:1503.05052v1 [astro-ph.CO]. [29] Anderson, L., et al. (2014). The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Data Releases 10 and 11 Galaxy samples. MNRAS, 441: 24-62. [30] Delubac, T., et al. (2015). Baryon acoustic oscillations in the Lyα forest of BOSS DR11 quasars. A&A, 574: A59.

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Michael A. Ivanov [31] Wu, Junfei, et al. (2019). Progress in Precise Measurements of the Gravitational Constant. Annalen Phys., 531, 5, 1900013. [32] Melia, F., and Yennapureddy, M. K. (2018). Model selection using cosmic chronometers with Gaussian Processes. JCAP, 2018, 02, 034. Eprint: arXiv:1802.02255v2 [astro-ph.CO]. [33] Lauer, T. R., et al. (2011). New Horizons Observations of the Cosmic Optical Background. Eprint: arXiv:2011.03052v2 [astroph.GA].

In: The Origin of Gravity from First Principles ISBN: 978-1-53619-566-8 Editor: Volodymyr Krasnoholovets © 2021 Nova Science Publishers, Inc.

Chapter 3

ONLY GRAVITY Thomas C. Andersen nSCIr, Ontario, Canada

ABSTRACT Gravity in its current accepted form has a lot of success: from gravitational wave detection and black hole parameterization to Einstein lensing and GPS satellite operation. The main issues with gravity in current theory concern its interaction with other fields and laws, such as quantum mechanics. In this chapter, we consider simply dropping the troublesome other fields and laws and looking at general relativity itself, as perhaps a ‘toy model’. Simplified toy theories abound in theoretical physics. These models are extremely useful. An example is N = 4 supersymmetric Yang–Mills theory. In this model alone, tens of thousands of papers have been published, some cited thousands of times. This chapter proposes that physicists consider studying “N = 4 General Relativity” as a toy model (and also as a serious contender for a fundamental underlying field). This ‘Only Gravity’ model uses Einstein’s field equations on their own in the hope that ignoring complicated interactions of gravity with other fields (electromagnetism, etc) and physical theories (quantum mechanics, QFT, etc) may paradoxically help us understand more about quantum gravity, electromagnetism and nuclear physics. The conclusion is that there is a tremendous amount of unexplored phenomena in classical general relativity.



Corresponding Author’s Email: [email protected].

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Keywords: quantum gravity, toy models, general relativity, quantum mechanics, emergent electromagnetism, emergent quantum mechanics

INTRODUCTION If gravity is to have an origin, one approach would be to build it from a familiar or proposed field, so that in the end the gravity we see today is some emergent phenomena on an underlying field, ideally a single underlying field or process (Verlinde 2016; Merali 2013). But it’s not simple (Carlip 2014)! In this chapter we consider that this single underlying field is gravity itself! We point out that the Einstein field equations are firstly not about gravitational attraction, but rather mostly a soliton – wave interaction theory that happens to have – almost as a side effect – Newtonian gravitational attraction. Consider a model called for this chapter ‘Only Gravity’ that uses Einstein’s field equations (EFE): 1 2

𝑅μν − 𝑔μν R + Λ𝑔μν =

8πG 𝑇 𝑐 4 μν

(1)

and nothing more. This drastic simplification allows the investigator to ignore electromagnetism, quantum effects, and other forces. While it may seem that throwing out ‘most’ of physics will in this case result in a rather boring, simple environment, we will show that on the contrary many interesting phenomena occur in Only Gravity, with deep parallels to real world fields and particles. The EFEs (1) are, unlike Newtons law of gravitation, not manifestly a recipe for determining the gravitational force bewteen two objects. Instead one has to use one of several derivations, for example (Katz 1968). The usual way to determine what non-linear equations might behave like is to linearize them, which results in (Lorentz guage): ∇2 ̅̅̅̅̅ ℎμν = 16π𝑇μν

(2)

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with “2” taken to be the  – d’Alembertian operator. This equation looks more like a 2nd order wave equation than a prescription for gravity. Indeed – if one works out the energy flux for waves of high frequency and small (h ~ 10-22) dimensionless wave amplitude we find a flux of 1033watts/m2 (see later in the chapter), while the gravitational attraction of a large amount of energy is very tiny, so small as to be safely ignored for most proceses. Had in some other hypothetical world, the EFEs (1) been hypothesized as a new non linear wave equation, with big ‘G’ hidden or a slightly different value, one can see how it is that even in general relativity Newtonian gravity can (mostly) be ignored.

The Role of 𝚲 For simplicity we will ignore Λ here (O’Raifeartaigh 2018). If in the real world Λ is small but non-zero, it will not affect objects smaller than a galaxy, and this chapter concerns much smaller objects. Λ like effects can be put into the stress energy tensor in pure Only Gravity.

THE STRESS ENERGY TENSOR Einstein described the EFEs (1) as ... similar to a building, one wing of which is made of fine marble (left part of the equation), but the other wing of which is built of low grade wood (right side of equation). The phenomenological representation of matter is, in fact, only a crude substitute for a representation which would correspond to all known properties of matter. (Albert Einstein 1936)

Is equation (1) needed in full for Only Gravity? After all the stress energy tensor Tμν describes the matter and electromagnetic fields, etc existing in a spacetime. A more pure approach would be to use the vacuum field equations – which is what Only Gravity should use: 1 2

𝑅μν − 𝑔μν R = 0.

(3)

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Quite a few solutions to the vacuum equations such as the Schwarzschild solution (Schwarzschild 1916) and gravitational waves exist. Yet the stress energy tensor can be reintroduced in a ‘pure’ vacuum solution: for example, imagine many black holes distributed in a pressure-less dust, in a Friedmann– Lemaître–Robertson–Walker (FLRW) universe. This effective Tμν would of course also include other pure vacuum field solutions like gravitational waves, and thus we would be back to equation (1) – but with the stress energy tensor now completely built out of the vacuum field equation solutions (e.g., black holes are solitons) and waves.

EXAMPLES We will consider how studying Only Gravity might help with our understanding of a few areas where general relativity is an important component and/or a place where our understanding is weak in microscopic physics. The idea is that solutions and explorations into Only Gravity within the huge unexplored parameter space that Only Gravity allows will give us insight into our real universe, perhaps even unifying some aspects. Each example topic has a section called ‘questions’ where we outline some of the many avenues which are largely unexplored today.

Example: Cosmology The role of gravity in the standard Λ𝐶DM model of cosmology is well understood, but of course there are problems with the understanding of inflation, dark matter, and dark energy, the sum of which has led to serious doubts about the entire Λ𝐶DM concept(Bullock and Boylan-Kolchin 2017; Lelli et al.,.g. 2017). Within Only Gravity, a universe comparable to our current universe (from a general relativity standpoint) can likely be constructed using a black hole ‘dust’ and the FLRW metric.

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Example: Cosmology – Early Epoch Questions 

   

Is there a way to distribute primordial black holes and other vacuum EFE solutions such as gravitational waves at some unknown earlier epoch, such that coalescing, and radiation scattering evolves to a universe that looks (at least gravitationally) like our current one? What would be the distribution of black holes in such a universe in the current epoch? What would be the spectrum and strength of stochastic gravitational radiation be in the current epoch? What does the vacuum look like in Only Gravity at various epochs? Is inflation possible in an Only Gravity universe?

Investigating these questions in a model with far fewer free parameters than, for example Λ𝐶DM, could provide insights into the cosmological role of other processes and fields such as electromagnetism and quantum mechanics.

Example: Particle and Nuclear Physics Einstein was one of a long line of researchers considering how general relativity might affect particle physics (A Einstein 1919), but Only Gravity allows more freedom, as other interfering fields and quantum mechanics are to be ignored. One starting point is the observation that black holes are sometimes described as fundamental particles - after all they have just a few parameters, like elementary particles.

The Electron If we look at the electron, we have no issue constructing a black hole of its mass in Only Gravity, with a tiny Schwarzschild radius of  rs = 2Gme /c 2 = 1.4x10−57 m. We can seek to improve the model by assigning an angular momentum ℏ/2𝑚𝑒 c, invoking the Kerr solution. Doing this of course results in a ‘highly illegal’ naked ring singularity of radius ℏ/2me c = 1.93x10−13 m. We

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note that singularities are much more of a problem in accepted physical theory, as for example electromagnetism and quantum field theory don’t behave well at a singularity. That is why we are using Only Gravity – to explore general relativity without having to follow the rules and limitations other fields. Consider this ‘Kerr electron’. It has the same mass and spin as the one in our own universe, yet it has no charge. Charged electron models similar to this (Kerr Newman) have been studied (Arcos and Pereira 2004; Burinskii 2012). One pleasant surprise with our uncharged ‘Kerr electron’ is that the ratio of the two lengths we have for this solution – the ratio of the Schwarzschild radius to the size of the ring singularity is almost exactly the same as the ratio of the electromagnetic to gravitational force for two electrons ~1044. This ratio becomes an exact match if 4α is added in by hand. ℏ/(2me c) 2 e /c )

size ratio = 4α (2Gm

=

αℏc Gm2e

= 4.17x1042 =

EM Force Grav Force

(4)

We think this connection might be more than a curious accident. How can we generate a ratio that is close to that of the electromagnetic to gravitational force using only non electromagnetic (just mass and spin) of the electron? One might consider the addition of 4α to get to an exact agreement is cheating, but it’s just a small unitless correction factor of ~1/30, compared to the 42 orders of magnitude in the raw ratio. Ignoring 𝛼 still shows a remarable coincidence between the uncharged Kerr solution and the actual ratio of electromagnetic to gravitational forces. Electromagetism? If the large, Compton sized ring has an effective area around its circumference of its length times the Schwarzschild radius rs and we compare this area to the Schwarzschild area πrs2, we can see that the above ratio (4) is a ratio of cross sections. Perhaps electromagnetism can be built out of an interaction mechanism within pure general relativity. For more information see (Andersen 2017).

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Nuclear Physics If we turn to nucleon and nuclear sized constructions, let us rashly assume that someone could construct a model of a proton, neutron or nucleus like object from some hypothetical Only Gravity soliton(s). Call these heavier solutions Only Gravity Solitons (OGS). A nuclear-like particle or nucleus made of OGS would fairly obviously radiate gravitational waves, due to internal motions of one OGS relative to another. If one works through a simple calculation using the Eddington gravitational radiation (Eddington 1922) formula, the gravitational radiation levels are quite small - in the eV per universe age time scale for a nucleus. So even in a classical nucleonic system, the radiation of gravitational waves (at high frequencies) would amount to such a small amount of energy that it would not affect nuclear structure. It is interesting to note that when an OGS takes on higher masses and smaller dimensions (analogous to real short-lived particles such as heavy pions) gravitational radiation in our toy model increase to the point of perhaps causing these OGS assemblages to radiate energy away quickly, forming an interesting parallel to our real world.

Example: Particle and Nuclear Physics: Questions   



What properties does this ‘Kerr electron’ share with our real electron? What are the experimental consequences in our real world if atomic nuclei internal motions generate gravitational waves? Are there other possible constructions of particle like solutions of the EFEs such as trefoils or similar? What properties do these knot-like solutions have? Are singularities really less of a problem in a pure Only Gravity universe?

Example: Gravitational Waves The experimental detection of gravitational waves is one of the major triumphs of the past few decades. Only Gravity allows us the freedom to wander

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into places uncharted. An analogy can be made with electromagnetism: every time a new telescope in an as yet unexplored wavelength regime (radio, UV, IR, etc.) is constructed, discoveries are made. Why should this be different for gravitational waves? The interaction of gravitational waves with astrophysical Kerr black holes has been well studied (Brito, Cardoso, and Pani 2015), and shows that a gravitational wave of the right frequency and amplitude can extract a substantial (> 10%) amount of a black hole’s energy in a single superradiant interaction showing that geometric objects can have large interactions with gravitational waves.

Example: Gravitational Waves: Questions    

How do gravitational waves work at Compton frequencies? Can we detect gravitational waves at atomic or nuclear frequencies? Is it possible to build a Teukolsky and Press (Press and Teukolsky 1972) black hole ‘bomb’ using a ‘gas’ of active gravitational objects? How would gravitational waves interact with singularities, is the cross section extremely large or in some sense ‘perfect’?

Example: Quantum Mechanics Quantum mechanics deals with the small. In order to see what gravitation itself has to say about small scale effects we should look at small things - like our Kerr electron or gravitational waves at Compton frequencies. The formula for gravitational wave energy flux F is straightforward and simple: from (Kokkotas 2002) (h is the dimensionless strain here) 2 2 ergs f h ) ( −22 ) 1kHz 10 cm2 sec

F =  3 (

(5)

A Compton frequency gravitational wave with a strain at the LIGO limit (say h~10−22 ) implies a gravitational wave flux of 1033 watts/m2! It would seem that even tiny gravitational waves can carry enormous amounts of energy around without being detected by electromagnetic means. This shows that

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general relativity has the bandwidth in terms of both information and energy carrying capacity to deliver physical effects at atomic and particle physics length scales. These physical effects may in fact be a good model for the em, strong and weak nuclear forces.

The Quantum Electron As another example, consider again the Kerr ring from the previous section with no charge and the experimental mass and spin of an electron. The naked singularity produced by the Kerr metric has a tension and a fundamental frequency. As with zitterbewegung, this singularity is rotating at the speed of light. The equation for tension in a rotating loop L of rope of mass M, which can be found in elementary physics textbooks is: 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 = 𝑀𝐿ω2 /(2π)2

(6)

For a ring singularity the rotational velocity is the speed of light, so we have then a direct relationship between ω and L: ω  =  2π𝑐/𝐿. Thus, the tension in a Kerr singularity is: 2

T=

2(𝑚𝑐 2 ) ℎ𝑐

,

𝑇𝐾𝑒𝑟𝑟 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 = 0.067N

(7)

The tension in a Kerr singularity is finite, equals the mass per unit length (in geometric units), and quadratic in mass (for different particles). Keep in mind that larger masses correspond to smaller rings. The calculation is accurate without using a relativistic expression, as it uses net values for mass per unit length and energy, etc. Note that this tension has the same characteristics as a cosmic strings, whereas for example Brandenberger states: (Brandenberger 2014) “Since cosmic strings are relativistic objects, a straight string is described by one number, namely its mass per unit length µ which also equals its tension...” The fundamental frequency of a tensioned string is given by

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Thomas C. Andersen 𝑇

𝑓=

√𝑚/𝐿 2𝐿

(8)

𝑇

or since √𝑚/𝐿 = 𝑐 in our case and using the electron mass and spin, 𝑓=

𝑐 2𝐿

=

𝑚𝑒 𝑐 2 ℎ

= 1.24𝑥1020 𝐻𝑧

(9)

The Compton frequency is the de Broglie fundamental frequency and is 1/2 the zitterbewegung frequency. See for example Wignall (Wignall 1993) on the relationship between de Broglie frequency and matter waves. Also see (Bohm and Hiley 1982; Shuler 2015; Wignall 1985). Regarding the rotating ring singularity as a tensioned spinning entity allows one to see that the singularity will not be fixed in a circular shape, but rather it will be deformed by any gravitational waves impinging on it. There is lots of evidence to suggest that singularities in general relativity have a ‘perfect’ connection to the gravitational field, and exchange radiation strongly (Nakao and Morisawa 2005), and thus this uncharged Kerr electron will have a very sharp resonance at its Compton frequency, which leads to a possible mechanism to build quantum behavior out of nothing more than general relativity.

Example: Quantum Mechanics: Questions Can the quantum behavior of an electron be modelled by the general relativistic behavior of a Kerr ring with the right mass and angular momentum? Without assuming an explicit model, does assigning a general relativity defined bandwidth limitation to quantum mechanics suggest a solution to the quantum measurement problem?

CONCLUSION This chapter begs the reader to consider the origin of gravity as gravity itself! An explanation is in order. By considering general relativity as the only field, other fields – for instance electromagnetism – are to be built of general

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relativity, and thus plain old weak gravitation, as holds us to the earth is but a side effect of the machinations of the vibrant, active microscopic effects that general relativity is capable of generating all on its own. Put another way, if someone were to postulate the general theory of relativity today as a microscopic solitonic wave theory, it might take a while before constants were arranged and calculations done to even determine that general relativity could in fact have a small accretive (Newtonian) component. Thus the origin of gravity is in a small side effect of the EFEs (1). Studying Einstein’s field equations on their own using a toy model approach, which we term ‘Only Gravity’ presents a research opportunity on its own, independent of its viability as an actual theory. Only Gravity employs a clear theoretical model with a huge unexplored parameter space. We may learn about our own universe by investigating a model universe made exclusively of Einstein’s aether (Albert Einstein 1920). Or indeed gravity may be all we need.

REFERENCES Andersen, Thomas C., 2017. “A Kerr Connection to the Electric Constant Thomas,” no. 1: 1–2. https://doi.org/doi: 10.13140/RG.2.2.23834.47 041. Arcos, H. I., and Pereira, J. G., 2004. “Kerr-Newman Solution as a Dirac Particle.” General Relativity and Gravitation 36 (11): 2441–64. https://doi.org/10.1023/B:GERG.0000046832.71368.a5. Bohm, D. J., and Hiley, B. J., 1982. “The de Broglie Pilot Wave Theory and the Further Development of New Insights Arising out of It.” Foundations of Physics 12 (10): 1001–16. https://doi.org/10.1007/BF0 1889273. Brandenberger, Robert H. 2014. “Searching for Cosmic Strings in New Observational Windows.” Nuclear Physics B - Proceedings Supplements 246–247: 45–57. https://doi.org/10.1016/j.nuclphysbps. 2013.10.064. Brito, Richard, Cardoso, Vitor, and Pani, Paolo, 2015. Superradiance. General Relativity and Quantum Cosmology; High Energy Astrophysical Phenomena; High Energy Physics - Phenomenology; High Energy Physics - Theory; Fluid Dynamics. Vol. 906. Lecture Notes in Physics. Cham:

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Springer International Publishing. https://doi.org/10.1007/978-3-31919000-6. Bullock, James S., and Boylan-Kolchin, Michael, 2017. “ Small-Scale Challenges to the Λ CDM Paradigm .” Annual Review of Astronomy and Astrophysics 55 (1): 343–87. https://doi.org/10.1146/annurev-astro091916-055313. Burinskii, A., 2012. “Gravity beyond Quantum Theory: Electron as a Closed Heterotic String.” Physics of Particles and Nuclei 43 (5): 697–99. https://doi.org/10.1134/S1063779612050085. Carlip, Steven, 2014. “Challenges for Emergent Gravity.” Studies in History and Philosophy of Science Part B - Studies in History and Philosophy of Modern Physics 46 (1): 200–208. https://doi.org/10.1016/j.shpsb. 2012.11.002. Eddington, Arthur S., 1922. “The Propagation of Gravitational Waves.” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 102 (716): 268–82. Einstein, A., 1919. “Eintstein Do Gravitational Fields Play an Essential Role in the Structure of the Elementary Particles of Matter?.Pdf.” Sitzungsberichte Der Preussischen Akademie Der Wissenschaften, (Math. Phys.), 349–56. Einstein, Albert, 1920. “Einstein : Ether and Relativity.” University of Leiden Address, no. May 1920: 1–6. ———. 1936. “Physics and Reality.” Journal of the Franklin Institute 221 (3): 349–82. Katz, Amnon, 1968. “Derivation of Newton’s Law of Gravitation from General Relativity.” Journal of Mathematical Physics 9 (7): 983–85. https://doi.org/10.1063/1.1664691. Kokkotas, Kostas D., 2002. “Gravitational Wave Physics.” Encyclopedia of Physical Science and Technology 7. Lelli, Federico, McGaugh, Stacy S., Schombert, James M., and. Pawlowski, Marcel S., 2017. “One Law to Rule Them All: The Radial Acceleration Relation of Galaxies.” The Astrophysical Journal 836 (2): 152. https://doi.org/10.3847/1538-4357/836/2/152. Merali, Zeeya, 2013. “Theoretical Physics: The Origins of Space and Time.” Nature 500 (7464): 516–19. https://doi.org/10.1038/500516a.

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Nakao, Ken-ichi, and Morisawa Yoshiyuki, 2005. “Gravitational Radiation from a Cylindrical Naked Singularity” 124007: 1–9. https://doi.org/ 10.1103/PhysRevD.71.124007. O’Raifeartaigh, Cormac, 2018. “Investigating the Legend of Einstein’s ‘Biggest Blunder.’” Physics Today October (October). https://doi.org/ 10.1063/PT.6.3.20181030a. Press, William H., and Teukolsky, Saul A., 1972. “Floating Orbits, Superradiant Scattering and the Black-Hole Bomb.” Nature 238 (5361): 211–12. https://doi.org/10.1038/238211a0. Schwarzschild, Karl, 1916. “Über Das Gravitationsfeld Eines Massenpunktes Nach Der Einsteinschen Theorie.” [“About the gravitational field of a mass point according to Einstein's theory.”] Sitzungsberichte Der Königlich Preußischen Akademie Der Wissenschaften (Berlin, January, 189–96. Shuler, Robert L., 2015. “Common Pedagogical Issues with de Broglie Waves: Moving Double Slits, Composite Mass, and Clock Synchronization.” Physics Research International 2015: 1–11. https://doi.org/10.1155/ 2015/895134. Verlinde, Erik P., 2016. “Emergent Gravity and the Dark Universe,” November. https://doi.org/10.21468/SciPostPhys.2.3.016. Wignall, J. W. G., 1985. “De Broglie Waves and the Nature of Mass.” Foundations of Physics 15 (2): 207–27. https://doi.org/10.1007/BF00 735293. ———, 1993. “A Unified Approach to Classical and Quantum Physics.” In Essays on the Formal Aspects of Electromagnetic Theory, 357–98.

In: The Origin of Gravity from First Principles ISBN: 978-1-53619-566-8 Editor: Volodymyr Krasnoholovets © 2021 Nova Science Publishers, Inc.

Chapter 4

SUB-QUANTUM GRAVITY. THE CONDENSATE VORTEX MODEL Peter A. Jackson Canterbury, Kent, UK

ABSTRACT We describe a new vortex model of gravity not identified in old vortex theory, applying consistently across all dimensional orders. The smallest is a sub-matter ‘Higgs Condensate’ (HC) ‘dark’ or vacuum energy scale likely below the Planck length 10-35 m. to min 10-93m. This tiniest ‘granularity’ is the ‘stuff of’ electromagnetic (EM) waves and condensed matter vortex pair (e+/-) fermions at ‘dark matter’ scale. Larger ‘visible’ scale vortices include weather systems and more massive orbital systems including galaxies, where a cyclic process of accretion, re-ionization and active nucleus (AGN) excretion is suggested. Ultimately, the sequence extends to a consistent cyclic cosmology able to reproduce the peculiar cosmic microwave background (CMB) anisotropies. At observable scales, the motion of vortices produces a radial pressure gradient proportional to the ‘mass’, as generated by the vortex orbital momentum. We may then treat ‘matter’ as the orbital energy of the small scale components of the larger scale vortices. A mid-scale example is in the particles of air orbiting to form anticyclones. The resultant particle pressure/density gradient forming the vortex produces a standing ‘force’ towards the centre, reducing radially at the equivalent 

Corresponding Author’s Email: [email protected].

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Peter A. Jackson of Newton’s Inverse Square Law but with significant anisotropies from such as the Lense-Thirring ‘frame drag’ effect. We also derive the relationship between gravity and centripetal force. The radial pressure distribution affects other matter, causing unequal pressure on each side; as in a standing gas-pressure gradient. Vortex formation is also visible at intermediate scales, by any motions forming shear planes between medium rest states. Vortex effects are additive. All massive bodies then have the combined potentials of each component added to that of its macro rotation. In free space, the smallest vortices will form Majorana ‘dipole’ pairs, having both clockwise and anti-clockwise polar rotations, as do all spheroids and toroids, depending which pole faces the observer. The momenta pair is then equivalent to ‘electron’ (-) AND positron (+) orbital Chirality subject only to orientation with respect to the observer. Earth’s ionosphere and all spiral and disc galaxies have a similar dipole morphology. The pressure differentials are proportional to accelerated motion of a continuous medium. The Bernoulli equations and Euler’s ‘vortex force’ apply where “Accelerated motion gives lower pressure”, which the Stokes-Navier equations can’t resolve. The relative coefficient v2/c2 and principle of least action modify Euler but apply throughout up to certain velocities.

Keywords: gravity, fermions, plasma, Vortex theory, vortices, Higgs condensate, cyclic, DFM, electromagnetism, action-at-a-distance, vacuum energy, Bernoulli, centripetal force, Bow-Shock

ACRONYMS AGN CSL DFM EM GR IRF ISM (IPM) LHC LT MHD OB QM SMP SR

Active Galactic Nucleus Constant speed of light Discrete Field Model Electromagnetic General Theory of Relativity Inertial Reference Frame Interstellar/-(planetary) medium Large Hadron Collider Lorentz Transformation magneto-hydrodynamic Optical Breakdown Quantum Mechanics Solar massive particle Special Theory of Relativity

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1. INTRODUCTION Vortex theory, in all its forms, was displaced well over 100 years ago by the present main ruling paradigms of Quantum Mechanics (QM) and by the Special and General theories of Relativity (SR/GR). Yet, the new light of advancing exploration and experimentation shows that many questions remain unanswered, including the physical cause of ‘quantum’ gravity (QG). Vortex theory is revisited in that new light. The original ‘Aether’s’ role as a modulator directly dictating local light propagation speed ‘c’ is not invoked. A ‘granularity’ below the scale of condensed fermion (e+/-) pairs is suggested, at shear planes (relative motion) and at the interface of all differing media, so at ‘refractive planes’. In free space ‘Majorana fermion’ dipoles would be formed, having + and – charge at opposite poles of spheroid or toroidal rotations. Phenomena such as static electricity, surface plasmons and the ubiquitous ‘astrophysical shocks’ of all ‘moving’ bodies in space are formed. Plasma ‘shocks’ are a real manifestation of Maxwell’s Near/Far field transition zones (TZ’s). The dense ‘2-fluid plasma’ of Earth’s Bow Shock was repeatedly crossed & sampled by the multiple ‘Cluster Mission’ probes [1]. The findings were consistent with space radio telemetry transitions at shock-position ‘TZ’ crossing interactions. Our bow-shock altitude changes with solar activity but reaches well beyond the magnetopause, to some 80 – 100,000 km. The Majorana fermion plasma (also called “Majoron”) is found to be an ever more consistent dark matter candidate (Profumo 2012) [2]. Not so glamorous as more ‘exotic’ particles we now know that these ‘e+/- pairs’ populate space in far higher densities than previously assumed, as found by PAMELA and fit the energy spectrum confirmed by Fermi (Ahlers 2009) [3]. Humble condensed electrons or e+/- ‘vortex pairs’ are then able to provide a majority of the ‘missing’ gravitational potential, aided by the increasing ‘dust’ fractions also found. The Planck length ℓp, of ~1.6 x 10-35 m is well below the experimentally based classic electron radius ~10-22 m. The latest ‘Elementary Length’ of Wolfram [4] is significantly smaller than both, at 10-93m. Free fermion plasma has the highest electromagnetic (EM) coupling co-efficient (absorption/re-

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emission) known, covering all wavelengths. Below fermion size is the gamma wave coupling limit of condensed matter but there is no known reason to suppose that smaller scale states cannot exist. Wide evidence suggests that vacuum energy does exist, even without the higher ‘dark energy’ invoked for expansion theory. Data show sub-matter energy must be taken into account along with dark matter in the local 1Mpc region [5]. A ‘gauge step’ down in size is hereby hypothesized, where our sub-matter scale condensate is the ‘stuff of’ the waves. The old question; ‘what is waving?’ or rotating in ‘particles’ is then addressed. Wilczek (2008) [6] coined the term ‘Higgs Condensate’, as did Jackson-Minkowski (2019) [7] as the ‘HC’, as a variation on ‘Zero Point’ fields. Classical Logic has problems with unavoidable paradox, but a solution emerges: The binary ‘Law of the Excluded Middle’ of Boolean integer maths was founded on the ‘indivisible atom’ of Democratus. But atoms DO have components. Such ‘higher order’ states show that the “middle” between 0 & 1 is NOT ‘excluded’ by nature (physical reality) but has a non-zero distribution. A Gaussian “Law of the Reducing Middle” is then proposed, as a ‘sine curve’, with the effect of removing the ‘ultimate paradox’ in all logical systems, or at least reducing paradox to higher orders. Jackson (2019) [8] The ‘action at a distance’ problems of both EM and Gravity, emerging from Faraday and Maxwell EM can be naturally overcome with a smaller scale ‘fluid’ medium transferring the 3D energy of motions, as in waves through water. Light is now also known to have that same 3D ‘helicity’ as well as ellipticity of polarity. As in water the particle motion resolves to (Chiral ‘handed’) ellipticised helicity. ‘Motion’ always a ‘local’ and ‘relative’ concept induces an additional ‘degree of freedom’, namely the density gradient discussed below. Space probe navigation problems have confirmed that both Newtonian and Einsteinian gravity formulations are only 1st order approximations. In addition to the local ‘Lense-Thirring’ and other effects ‘anomalous accelerations’ are common. Artificial intelligence (AI) is now fitted to all deep space probes to make real-time navigational observations and course corrections. Many theories to explain the ‘Pioneer’ ‘Voyager’, ‘Flyby’ and other such anomalies have been proposed but none convincing or adopted. The anomalous effects do however naturally emerge from the vortex model proposed here.

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The methodology constructs an initial heuristic model to consistently explain the physical causes underlying all experienced ‘gravitational’ effects. Precise and complete mathematical modelling may then be developed and presented.

2. PAST VORTEX TYPE THEORIES Championed by William Thompson (Lord Kelvin) and consistent with Helmholtz’s ‘ring’ vortex masses, suggesting a smaller scale medium, most leading physicists have promoted similar hypotheses. Much initial work referred to the ‘atom’ but evolved because ‘reducibility’ of atoms became ever more evident. James Clark Maxwell called the aether a ‘bed for em fields’ and proposed a luminiferous medium of space filled with tiny aethereal vortices, with transverse elasticity, pressing against each other with centrifugal force constricting dilation. The entrained ‘aether-drag’ of George Stokes was consistent with interferometer findings but with no ‘speed modulation’ mechanism. The Lense-Thirring effect came to late to support it. The 1937 Majorana fermion, conceived as ‘its own antiparticle’ is now found to be consistent with modern laser optics. The paper presents how it can simply be represented by a single common rotating dipolar body. We here identify that all free shear plane and ‘wing tip’ vortices are paired and propagated by motion, with toroidal, dynamic, rotational vectors to be found also smoke rings. Like Isaac Newton (1717) [9], Leonard Euler (1760) [10] used a static ether with reducing density around matter following the Inverse Square Law. However neither could find a reason for such a density gradient to exist. Newton also suggests an incompressible aether with a single ‘absolute’ rest frame, later shown to be inconsistent by the ‘peculiar velocities’ of space. Others such as Fatio, Le Sage, Kelvin, & Lorentz posited that gravity is the result of tiny particles or waves moving rapidly in all directions, throughout the universe, and even ‘towards’ matter. Later conceptions are String Theory (founded in Euler’s motions), the 5th ‘curled up’ or ‘compact space’ dimension of Kaluza-Klien (KK) theory, i.e., Wesson (2006) [11], and subsequent string and ‘brane’ theories of higher order

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spaces. KK was proposed as a unified field theory of gravitation and electromagnetism, using a ‘cylinder’ model or a circle with 10-30m. radius. “Compactification” as proposed in KK theory did not normally produce group actions on chiral fermions. Subsequent work led to Yang-Mills and proposed ‘supersymmetry’ with work by Jordan and others leading to Brans–Dicke scalar–tensor theory (1961) [12]. All are precursors of our vortex particle model and antecedents to when data from space probes started to flow. No coherent theory emerged with the resolving power to explain observations such as the constant speed of light and gravity. When QM, SR and GR became the ruling paradigms along with the standard model and string theory -Vortex theories in general and Brans-Dicke, t, in particular (the closest to a physical reality) fell out of fashion and largely into neglect.

3. VORTEX SPEED/DENSITY RELATION One option for a vortex theory giving a density gradient around bodies of condensed matter had not been proposed. Paul Euler and son Leonard, then friends (also students and tutors), the Bernoulli family, developed the relationship between ‘speed’ and pressure. It didn’t appear to occur to anyone at the time that such a cause of the static pressure gradient might provide the missing process underlying the inverse square law. Romani (1975) [13], Podlaha & Sjoding (1984) [14] and others suggested gravity as aether density gradient but all lacking in any cause. Arminjon (2004) [15] reviewed Euler’s view of Newton gravity as Archimedes’ thrust in a fluid ether, linking uniform motion and gravity and suggesting an extension of Newton’s 2nd Law. Arminjon found classical mechanics violated and concluded both a “macro-ether” and perfectly fluid deformable “micro-ether” or perhaps ‘elastic fluid’ reference frames consistent with Einstein’s Special Relativity (SR). Newton also proposed ‘flows’ as Bernoulli quantified. We find the ‘macro/micro’ view consistent with the only ether model overcoming the conflict with SR in the case of differentially moving detectors all finding ‘c’, the ‘Discrete Field Model (DFM). The DFM solved the Kantor/B&B reflected signal speed anomaly (Jackson 2012) [16] by condensed

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77

pairs ‘coupling with’ EM waves carried by the ‘micro’ vacuum energy ‘Higgs’ condensate. In the DFM, electron or Majorana fermion ‘fine structure’ localises speed c by absorption and re-emission (atomic scattering). Scattering boundaries to kinetic systems are as Einstein’s 1952 [17] conceptual modification to the 1905 ‘interpretation’ of SR. In his 1952 paper Einstein conceived spatial boundaries to his “infinitely many spaces in motion within spaces,” which he wrote hadn’t been ‘part of scientific thinking.” (p. 139) The new conception was undeveloped at his death and did not update the old doctrine. We find that his ‘bounded spaces’ logic is correct, using the physical formulation of inertial systems ‘k’ which follows arithmetical rules of ‘bracketed’ functions, shown to render all logical systems consistent [18]. Einstein’s ‘52 ‘nested’ inertial systems, imply Pauli ‘exclusion’ to IRF’s as spatially limited states of motion k’ with respect to (wrt) a local background rest state k. Similar rationales were independently found by Beckmann (1987) [19] C.C. Su (2001) [20], and by (ex NASA) E.H. Dowdye (2007) [21] using astrophysical data. Such a ‘spaces in motion within spaces’ hierarchy is as found in the ‘peculiar velocities’ of all astrophysical systems, all bounded by plasma ‘scattering surfaces’. Scott & Smoot’s attempted rationalisation (2007, see 2010) [22] won a Nobel, but the link to Einstein’s 1952 re-conception wasn’t identified. Looking around us at objects in motion we find only that ‘hierarchy’ of nested kinetic states, with speeds only wrt the local background state, datum, each state k limited to the matter with that state of motion. A better rationale for comets emerges. Not ‘dirty snowballs’ but just ‘fast asteroids’. Coma are ‘shocks’ appearing above a certain local speed. Medium density and particle velocity (solar wind, etc.) affect that speed, as in ionospheric ‘bow shocks’ & magnetotails. Halley’s comet orbits at ~54.8km/sec. Asteroids, moving at ~18km/sec, only form comas in an ionosphere. The Ulysses probe found ionosphere particles in the tail of comet McNaught, and ‘magnetic reconnection’ (Neugebauer 2007) [23]. As on spacecraft re-entry there’s an acceleration due to Earth’s peculiar local velocity and also a particle density increase. Gas emissions from bodies likely do contribute to Coma, but we’ve yet to find anything resembling a dirty snowball. It must be stressed that ‘scattering surfaces’ as boundary TZ’s, changing speed to c in all local systems, remove the only substantive problem with

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‘aether’. ‘Closing’ speeds with approaching light have been assumed as c for all moving observers, but the revised understanding is of a speed changed to local c at first e+/- interaction, so overcoming that issue. Our condensate model thereby critically differs from old aether models. Put simply; ‘Closing’ speeds prior to reaching a detector is not detectable & may be c+/-v, but is instantly modulated to c on first contact. Our ‘processor’ then only receives the new wavelength, so can only calculate NEW ‘speed c’. Logic returns, because all observers/detectors, at whatever speed, will calculate ‘c’. Wavefunction ‘collapse’ is before ‘measurement’, as logic demands (i.e., Kim 2021) [24]. The critical role of EM coupling down to and at ‘vortex pair’ scale then emerges. Aether can then carry EM fluctuations without directly dictating c. (Electron Compton radius or ‘Thomson scattering length’ q

1 1−

(∆v)2 c2

>1

(123)

∆p∆x > ~

(124)

∆p∆x ≥ ~

(125)

and not to For more than hundred years standard physics has tricked itself into believing that the Heisenberg uncertainty principle is also valid for ≥ ~ because scholars have simply

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assumed that the momentum is valid and zero when v = 0. That is to say, they think the standard momentum is always valid. It is not mathematically valid for v = 0 and again the Heisenberg uncertainty principle can say nothing about uncertainty in rest-mass, which at the quantum level is directly linked to the Planck mass particle that makes up all other masses. Moreover, the Planck mass particle is the very essence of gravity, it is the collision time that is missing in all non-gravitational areas of modern physics, and that is concealed in standard gravity theory through GM as explained above in this chapter.

25.

MODIFIED U NCERTAINTY P RINCIPLE ROOTED IN THE C OMPTON M OMENTUM

Our theory claims the Compton momentum is the real and observable momentum and that the standard momentum is just a derivative of this momentum. Let us start to work with the kg mass definition before we switch also to the collision-time mass, that is to say we have ∆pt ∆x ≥ ~ (126) where pt = mcγ rather than the standard momentum p = mvγ. Also here the uncertainty in the momentum comes from uncertainty in the velocity so we have mc q ∆x ≥ ~ 2 1 − (∆v) c2 ~1 ¯ c q λc ∆x ≥ ~ 2 1 − (∆v) 2 c

~ q ¯ 1− λ

1 q ¯ 1− λ

(∆v)2 c2

(∆v)2 c2

∆x ≥ ~ ∆x ≥ 1

(127)

This looks identical to the result we found from the Heisenberg uncertainty principle in the section above. However, there is a major difference. Here, we can have v = 0 and also ∆v = 0 as the Compton momentum and the mass rooted in the Compton wavelength that we have rooted the principle in, are also all valid for v = 0. However, the importance of this is first fully understood when we also switch to the collision-time mass definition. If we take the first line in the equation above and multiply by

l2p ~

on

Quantum Gravity Hidden in Newton Gravity ...

199

each side we get the collision-time mass lp2 lp2 mc q ∆x ≥ ~ 2 ~ ~ 1 − (∆v) c2 mc ¯ q ∆x ≥ lp2 (∆v)2 1 − c2 l l

p p ¯c q c λ ∆x ≥ lp2 (∆v)2 1 − c2

¯ we get And in the case ∆x = λ

1 q ¯ 1− λ

(∆v)2 c2

1 q

1−

(∆v)2 c2

∆x ≥ 1

(128)

≥1

(129)

This can only meet the condition of only being equal to 1, if ∆v = 0. And when working with collision-time mass we know from previous q analysis in this chapter that l2

the maximum velocity for a elementary particle is vmax = c 1 − λ¯p2 , and in the case of a Planck mass particle the reduced Compton wavelength is is the Planck length, so the maximum velocity of the Planck mass particle is zero. The velocity cannot be slower than zero, so if the maximum velocity is zero then the uncertainty in its velocity must also be zero. So, we have a fully consistent theory as both the Compton momentum and also the way to write the rest-mass as a function of the reduced Compton wavelength rather than the de Broglie wavelength is fully consistent with v = 0 and also ∆v = 0. ¯ we can then rewrite our uncertainty principle as Assume again ∆x = λ, ¯ ≥ l2 ∆p¯t λ p

mc ¯ q 1−

(∆v)2 c2

lp2 ≥ ¯ λ

l l

p p lp2 ¯c q c λ ≥ ¯ 2 λ 1 − (∆v) c2

lp lp q ≥ ¯ 2 λ (∆v) ¯ 1− 2 λ

(130)

c

That is the quantum probability for being in a collision state

¯ λ

l q p (∆v)2 1− c2

is simply

larger or equal to the quantum probability for that particle to be in collision state if

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it is at rest, over an observational time window of the Planck time. This means the uncertainty in the quantum probability is naturally linked q to the uncertainty in velocity. l2

However, the maximum velocity is given by vmax = c 1 − λ¯p2 . So the uncertainty in q l2 velocity must be 0 ≤ ∆v ≤ c 1 − λ¯p2 . Nonetheless, only for the Planck mass particle is the maximum velocity zero, as the reduced Compton wavelength of the Planck mass particle is the Planck length, so here we must have s lp2 0 ≤ ∆v ≤ c 1 − 2 lp 0 ≤

∆v ≤ 0

(131)

but that must mean we have ∆v = 0, which indeed is the case of the Planck mass particle, so in terms of the uncertainty principle, re formulated to collision state probabilities we get lp q lp 1 −

02 c2

1

≥

lp lp

≥

1

(132)

Naturally, we must have 1 = 1 as it gives no meaning to have 1 ≥ 1, that is simply the uncertainty principle is a certainty principle in the special case for a Planck mass particle. Still, this certainty only lasts for the moment of the Planck time as this is the lifetime of the Planck mass particle. Specifically, we can say the Heisenberg uncertainty principle when derived from a proper fundament, namely the Compton momentum that is also valid for v = 0, then there the uncertainty principle switches to a certainty principle, and this special case is for the Planck mass particle that is the building block of all matter. This could have a series of implications. For example, Einstein was very sceptical of the interpretation of entanglement, as he called it “spooky action at a distance”, and together with Podolsky and Rosen claimed it was likely due to hidden variables. However, testing according to Bell’s [118] theorem seemed to exclude hidden variable theories. However, Bell’s theorem is rooted in the idea that Heisenberg’s uncertainty principle always holds [119, 120]. As we have shown, the standard Heisenberg principle is built on a foundation that is not compatible with rest-mass, when built on a foundation that also holds for rest-masses the uncertainty principle becomes a certainty principle for the Planck mass particle. We would not be surprised if this again opens up for hidden variable theories in favour of Einstein’s scepticism towards standard quantum mechanical interpretations of entanglement. Table 3 shows the uncertainty interval for a series of “properties” for elementary particles. Be aware that under our theory the corrected Schwarzchild radius, the Compton momentum and the collision-length energy are exactly the same thing, just three

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Quantum Gravity Hidden in Newton Gravity ...

words for the same thing actually, which basically means we also have a strong simplification here.

Table 3. The table shows the possible uncertainty, all coming from uncertainty in velocity of elementary particle for different entities Schwarzchild radius Corrected Schwarzchild radius Collision length energy Compton momentum Collision time mass Collision state probability Non collision state probability Gravity acceleration

26.

¯ S ≥ 2lp l¯p 2lp ≥ ∆R λ ¯ S ≥ lp l¯p l p ≥ ∆R λ ¯ ≥ lp l¯p l p ≥ ∆E λ l lp ≥ ∆¯ pt ≥ lp λ¯p l tp ≥ ∆m ¯ ≥ tp λ¯p l 1 ≥ ∆Pc ≥ λ¯p l 0 ≤ ∆Pn ≤ 1 − λ¯p l

l2

p c2 Rp2 ≥ ∆g ≥ c2 λR ¯ 2

STANDARD D ERIVATION OF O UR N EW U NCERTAINTY P RINCIPLE AND F URTHER D ISCUSSION

Here, we will show a more formal way to derive the uncertainty principle that is often used in derivation of the Heisenberg uncertainty principle, but here with our new foundation rooted in the Compton momentum and collision-time mass. Based on our Compton momentum operator pˆ ¯t = ilp2 ∇, or in the special case when only dealing with ∂ i . We can the x axis dimension, we have the Compton momentum operator: pˆ ¯t = lp2 ∂x then check if our Compton momentum operator commutes with the space operator xˆ. We should have [pˆ ¯t , x ˆ]ψ

= = = =

[pˆ ¯t x ˆ − xˆpˆ ¯t ]ψ ˆt xˆψ − xˆpˆ p¯ ¯t ψ ∂ ∂ lp2 i i (ψx) − xlp2 i i ψ ∂x ∂x   ∂ ∂ ∂ ilp2 x i ψ + ψ i x − x i ψ = ilp2 ψ ∂x ∂x ∂x

(133)

That is to say, pˆ ¯t and x ˆ do not commute, just as in the case of the standard Heisen-

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berg uncertainty principle. We get ψ∗ [pˆ ¯t , x ˆ]ψdx|

∆p¯t ∆x ≥

|

∆p¯t ∆x ≥

|ilp2

∆p¯t ∆x ≥

|ilp2 |

∆p¯t ∆x ≥

ψ∗ ψdx|

lp2

(134)

This is no great surprise as our mass definition is equal to the kg mass definition multiplied by written as

l2p ~.

Interestingly, since G =

l2p c3 ~ ,

our uncertainty principle can also be

G~ (135) c3 but that would only conceal what it really is, due to the composite constant G not being 2 needed, G~ c3 is simply lp . In regard to our new uncertainty principle, we will think this is only valid for ∆p¯t ∆x > lp2 and not ∆p¯t ∆x ≥ lp2 as just derived in this section. According to the sections above, in the special case of a Planck mass particle we cannot have an uncertainty principle, but rather a certainty principle. The non-commuting relation we derived above is rooted in continuous space-time. Assume that observable space-time comes in discrete units linked to the Planck length and Planck time. Then, for example, when looking at an electron and looking at a small ∂ , this is probably still a very good approximation, as the Planck change in “space”; ∂x length is so incredibly short compared to the reduced Compton wavelength of the elec¯ tron. The reduced Compton wavelength of the electron corresponds to λlpe ≈ 2.38×1022 Planck length units. Thus, a Planck length change is very similar to an infinitesimally small change as compared to the reduced Compton wavelength of the electron. However, when it comes to the Planck mass particles then such approximation models using continuous space-time would probably no longer work as well. The Planck mass particle has a reduced Compton wavelength equal to the Planck length, and it only has a lifetime equal to the Planck time. By moving one Planck time in space the particle can no longer exist, so it is, in this special case, almost meaningless to look at changes in space (x) with respect to the Planck mass particle. How can we look at the change in a particle by moving a distance where it can no longer exist? It would not make much sense. The Planck mass particle is unchanged through its lifetime, which is only the Planck time. By looking at a Planck time change, or a Planck length change, the Planck mass particle would have dissolved, so we cannot describe it over time or over distance, except inside the Planck length or inside the Planck time where it is unchanged if it is observed. If we observe a Planck mass particle, we know all about it. We then know its position, and we know its total momentum, as it is always equal to p¯t = m ¯ p c = lp , and its kinetic ∆p¯t ∆x ≥

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Quantum Gravity Hidden in Newton Gravity ...

Compton momentum is always zero as only moving particles have a kinetic momentum above zero. Remember that Heisenberg [117] said that the uncertainty in momentum comes from the uncertainty in the velocity. The velocity of the Planck mass r particle is not uncertain, but always certain, as its maximum velocity is vmax = c

1−

l2p l2p

=0

as discussed in other sections. It is not moving because it can only be observed from itself, because it dissolves before any outside observer could observe it directly. It is the collision point between the building blocks of two photons, two indivisible particles colliding. And, during the collision duration of the Planck time, they stand still. They stand still because again we can only observe such a collision directly from the collision itself. Moreover, at the Planck time the Planck mass particle is the only particle that can be observed that is different from moving indivisible particles. Indivisible particles non-colliding are moving at the speed of light relative to colliding indivisible particles. If we try to observe a Planck mass particle from another frame then the Planck mass particle is dissolved before we have time to observe it. There are several other issues that could be worthy of discussion here. For example, what is meant by the position of a particle? For a point particle we can have an exact position along the x-axis (at least when ignoring the uncertainty principle). This is simply because a point can be fully described by x, y, z coordinates, at least when at rest. For a particle with spatial dimension (above zero), what is the position in a coordinate system, the centre of the particle? As the Planck mass particle is not a point particle, but has spatial extensions, we could claim it is an uncertainty in its exact position that is equal to its spatial extension, simply because its spatial dimension along the x-axis cannot be described by one number, but only by an interval. Then its uncertainty when coordinate along x − axis is tried to be described by one number would be the Planck length11, simply coming from it having spatial dimensions and not being a point particle. On the other hand, if we define its position as the centre of the particle then it has no uncertainty. However, when we move away from the Planck mass particle and to, for example, an electron, then we have a mass that changes over time during its lifetime and over distance, then we clearly have uncertainty as can be described by the uncertainty principle. For the Planck mass particle, we, on the other hand, do not think it makes much sense to describe it with an uncertainty principle, because we have in the previous sections shown the uncertainty in this case should be replaced with certainty; that is to say, if we know its position and we know its momentum at the same time. But again, one has to be careful with definitions here, such as what exactly is meant by the position of a particle, so much more can be discussed here than we get time for in this chapter. However, we are quite confident when we say the Heisenberg uncertainty principle potentially cannot be valid for ∆p∆x ≥ ~, but likely only for ∆p∆x > ~. 11

Or two Planck lengths, depending on whether we focus on the reduced Compton wavelength of the particle or the physical extension that, for a Planck mass particle, is 2lp in one direction, and lp in other directions, see Figure 9.

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This looks like a small change, but the = part is, as we have shown, likely to be directly linked to the Planck mass particle, that again is the duration of the internal collisions in matter, that can, in turn, be detected indirectly as gravity. The Planck scale is gravity, so we have detected the Planck scale, and we have presented a theory where energy, mass, momentum and gravity can all be described only with two constants, namely the Planck length and the speed of light (gravity) and naturally some variables needed to, for example, describe different mass sizes.

27.

T WO WAYS TO U NIFY: T HE B EAUTIFUL INTUITIVE WAY AND U GLY C ONCEALED WAY

In our view it is close to impossible to unify gravity and quantum mechanics without understanding that one in standard physics is indirectly using a different mass definition in gravity and in the rest of physics. In standard physics one is indirectly getting the more complete and correct mass definition by multiplying G with M , and the other ¯ So, if one understands what is lacking in the mass m cancels out. GM = c3 M. standard kg mass, then one can replace it with the collision-time mass everywhere, as we have essentially done in this paper. One could have unified indirectly, by simply everywhere that one has m in nongravity physics replacing it with cG3 m. For example this would lead that E = mc2 had ˘ to be replaced with is cG3 E = cG3 mc2 = G ¯ 2 , and c m, which is nothing more than E = mc G

M

G

m

M

G

m

a gravity formula of F = c3 c3 R2c3 = G Rc32 etc., this looks strange, many would think this to be wrong and definitely far from intuitive, but, as the mass m everywhere in physics is now replaced with cG3 m, in other areas of physics this will also lead to exactly the same results for observable phenomena when formulas for these are derived. For example, to find the escape velocity formula (when v  c) we now have M cG3 m 1G 2 mv − G =0 2 c3 R2

(136)

q which, solved with respect to v, gives the standard formula ve ≈ GM R , that is valid for ve  c. For example, the standard relativistic energy momentum relation would 2 2 2 now suddenly be Gc3 E 2 = Gc p2 + Gc2 m2 . In addition, one had to understand that the de Broglie wavelength is just a derivative of the real matter wavelength, the Compton wavelength. Then one could also have unified gravity and quantum mechanics this way. Still, it would mean all equations looked almost unrecognizable, with a G everywhere, that again if one does not understand it is a composite constant it gives little intuition. l l It is first when one understands that Gm represents; c3 cp λ¯p , and one also understands l l that cp λ¯p = m ¯ represents a more complete picture of a mass, that both have the par¯ and ticle aspect of the mass through lp and wavelength aspect of the mass through λ

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the dynamics of the particle aspect of the mass through c (the speed of the indivisible particle) that one can unify and also obtain a simple intuitive theory in terms of notation as we have attempted in this chapter, and we hope we have been quite successful in our attempt. This both explains why one has not been previously able to unify gravity and quantum mechanics, as it would be almost impossible to guess why one should replace m with cG3 m in all areas of physics without first understanding that G is a composite constant used to turn a incomplete kg mass into a collision-time mass. And yes, in addition one has to understand that the de Broglie wavelength is a derivative of the real matter wavelength, that we claim must be the Compton wavelength.

C ONCLUSION We have shown that standard physics theory uses two different mass definitions without being aware of it. We are using the kg mass definition in all non-gravity areas of physics. This is an incomplete mass definition. In gravity theory we think we are using the kg mass definition, but here the mass is corrected by multiplying it by G into what is actually a much more complete mass definition that we have called collision time. Furthermore, the de Broglie wavelength is just a derivative of the likely real matter wavelength which is the Compton wavelength. Much of standard physics is therefore unnecessarily complex, and it is in our view not possible to unify gravity with quantum mechanics before we have taken this into account, as we have tried to do in our work, where much is summarized and explored further in this chapter. We have not yet explored to what degree our theory is compatible or incompatible with general relativity theory. As we have seen, simple ad hoc adjustments of Newtonian theory seem to be in line with supernova data without the need of dark energy. Furthermore, our new insight on mass seems to be able to explain galaxy rotation without the need of dark matter. Also we have shown that the Heisenberg uncertainty principle in its standard form is built on a foundation not compatible with res-mass, and rest-mass is the very essence of the Planck mass particle and gravity.

A PPENDIX A In 1965, Adler and Bazin [121] showed that one could derive gravitational redshift completely independent of general relativity. From Einstein’s energy mass relation, we have E = mc2 (137) We also have that the equivalent mass of a photon must be m=

E hf = 2 2 c c

(138)

206

Espen Gaarder Haug Now, for the conservation of energy we must have hf

hf − G

M 2 Mm = hf − G c R R

(139)

Furthermore, when R → ∞ we must have f → f∞ . This means we have M hf c2 R Mf f −G 2 Rc f∞ − f f

hf − G

=

hf∞

=

f∞

=

−

GM Rc2

(140)

which is identical to the gravitational redshift predicted by general relativity in a weak field. See also [122] for discussions on this. We have just extended this to assume the large mass is moving relative to the observer, and claim we then must have hf − G

√

M m 2 /c2 1−vM

R

that will lead to f∞ − f =− f √

= hf − G

G√

M hf c2 R

M 2 /c2 1−vM

Rc2

(141)

(142)

The difference between the GR prediction and our prediction, which is γ = 1 is remarkably the small adjustment needed that leads to that our model 2 2

1−vM /c

seems to fit supernova data without relying on the hypothesis of dark energy.

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In: The Origin of Gravity from First Principles ISBN: 978-1-53619-566-8 Editor: Volodymyr Krasnoholovets © 2021 Nova Science Publishers, Inc.

Chapter 7

THE ORIGIN OF GRAVITY AND ITS EFFECTS: ACCORDING TO THE SUBQUANTUM KINETICS PARADIGM Paul A. LaViolette The Starburst Foundation, Schenectady, NY, US

ABSTRACT This overview explores the phenomenon of gravitation as represented in the novel physics methodology of subquantum kinetics (SQK). Model G of SQK is shown to spawn physically realistic particle-like structures having mass and charge which generate gravity potential fields capable of exerting forces on neighboring particles, but whose strength declines to zero beyond distances of 10 kpc, thus eliminating the need to assume the presence of dark matter. Gravity in this paradigm plays a far broader role than in standard physics. It creates subcritical conditions in intergalactic space that foster nonconservative photon energy damping, tired light redshifting, thus providing a static universe interpretation of the cosmological redshift. It also predicts the existence of supercritical conditions in gravity potential wells that allow spontaneous zero-point energy fluctuations to grow and continuously materialize subatomic particles. Furthermore, it predicts a mother-daughter particle nucleation process that allows matter creation to proceed at an exponential rate. Such supercritical regions induce nonconservative photon 

Corresponding Author’s Email: Paul A. LaViolette, PhD, [email protected].

218

Paul A. LaViolette energy amplification which accounts for the anomalous blueshifts observed in galaxy cluster spectra, and also resolves the puzzling Kaiser effect and Fingers-ofGod effect. Also gravity-potential-mediated photon energy amplification predicted within all celestial bodies is shown to account for the observed planetary-stellar mass-luminosity relation, to prevent the formation of black holes, to provide the energy powering supernova explosions, and to explain the tremendous outpouring of both matter and energy from galactic cores. All of these nonconservative energy effects are permissible since SQK conceives the material universe as an open system at the subquantum level. Other aspects of this physics covered here include the origin of the electrostatic, magnetic, and nuclear force, and the existence of a causal coupling between electric and gravitational potential fields. The gravity fields of SQK are also shown to account for the various general relativitistic effects.

OVERVIEW The concept of gravity that is explored here is different from anything one is likely to have encountered in the past. It emerges out of an approach to physics called subquantum kinetics. Before going specifically into an explanation for the origin of gravity and its effects, it is necessary to note a few things about subquantum kinetics.

•

Subquantum kinetics was first published in 1985 (LaViolette, 1985a, b, c), and has been in development for more than 40 years, being most completely presented in LaViolette (2012).

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To date it has had eleven papers published in refereed journals, numerous others published in books and conference proceedings, as well as additional unpublished white papers. Many of these may be accessed at the Starburst Foundation internet archive (https://starburstfound.org/paper-archive/).

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Although not taught in physics classes and university curricula, subquantum kinetics has nevertheless developed a strong and enthusiastic following. More recently, especially since 2010, researchers have joined to help in its development, more particularly in performing computer simulation of the partial differential equation system that lies at its heart which has had such great success in generating verified predictions.

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Thus far subquantum kinetics, a.k.a. SQK, has had 13 a priori predictions confirmed. This exceeds by far general relativity which historically has had three confirmations. A theory’s confirmations is generally recognized as a better indicator of its correctness than widespread acceptance of the theory by the scientific community, the latter being highly influenced by academic politics and by physicists’ difficulty to break away from old familiar concepts to explore new ones. The reason why subquantum kinetics has had so many more confirmations than general relativity has to do with its scope. General relativity focuses mainly on celestial gravitational phenomena and has had its predictions verified entirely by means of astronomical observation. SQK, however, covers phenomena on a wide scale, ranging all the way from the subatomic to the cosmological. As a result, it has accumulated a far greater number of prediction confirmations. But it is important to keep in mind that due to its greater scope, it has been far more vulnerable to disproof than, for example, general relativity.

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Subquantum kinetics grew out of the author’s basic belief that, at a very fundamental level, the physical Universe operates as an open system, rather than as a closed system as is the conventionally held view. The open system concept was a central idea that was being examined by a branch of science and philosophy called general system theory (GST), first developed in the 1930’s by biologist Ludwig von Bertalanffy (1968). GST investigates the commonalties of physical systems, be they physical, chemical, biological, social, or psychological. It seeks to study the fundamental characteristics that systems have in common. The open system concept, for example, is found to apply to a wide variety of physical phenomena ranging from cellular convection, to morphogenesis in open chemical reaction systems, to the formation of social order in social systems, and even to the emergence of emotionalcognitive thought structures in the human mind. While the open system paradigm had been found to be fundamental to most types of systems around us, until the advent of SQK, it had never been applied to microphysics.

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The formative ideas for SQK emerged at a time when the discovery of chemical wave phenomena in certain open chemical reaction systems was making news. Also at that time, in 1972, the work of the "Brussels school" on an interesting reaction-diffusion system called the Brusselator came to wide attention, e.g., see publications by Lefevre (1968), Prigogine, Nicolis, and Babloyantz (1972), and Nicolis and Prigogine (1977). The Brusselator is a nonlinear chemical reaction system described by four kinetic equations and two variables which, when simulated on the computer, is found to generate a variety of ordered structures from a simple initiating fluctuation. My initial insight at that time was that, the Brusselator reaction system could be properly modified into a reaction-diffusion model that would exhibit physically realistic behavior when simulated on the computer, the objective being to produce structural analogs of subatomic particles and fields. Inherent to this approach was the notion that all physical form emerges from an underlying stratum that functions as an open system.

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Subquantum kinetics is better thought of as an approach than a theory, one that differs radically from that used in standard physics. For example, it places modeling and theory antecedent to observation. It chooses a reasonable model to start with and then uses observations to check the model’s validity and to fine tune it by making proper adjustments to the model’s parameters. This approach differs from that of standard physics which puts observation first and theory second. In standard physics, theory serves as an after-the-fact attempt to explain existing observations. This approach is best depicted by the parable of the blind men and the elephant. Just as the blind men observe different parts of the elephant and form different theories of the whole, so too standard physics conducts numerous observations at various levels of Nature and this results in many and varied theories being devised. The theories that have resulted, though, fail to form a coherent and holistic explanation of physical reality, and in some cases contradict one another. In an attempt to “sew” them together into a coherent whole, the standard approach has relied on mathematical acrobatics and assumptions of higher dimensional spaces to create a so called “unified

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field theory.” These attempts have tended to be highly mathematical, very complex, and to produce results more reminiscent of a patch-work quilt. String theory is an extreme example of the abstractness and complexity that this approach leads to. The subquantum kinetics approach instead begins with a simple set of assumptions associated with a hypothetical Brusselator-like model that is expressed as three partial differential equations. These would then be computer simulated to produce particle-like structures, that are correlates of known subatomic particles, or energy waves. This core model, called “Model G,” is devised to describe reaction processes occurring at a subquantum level. Its operation, then, generates phenomena that should be observable on a more macroscopic level, subatomic, astronomical, or even cosmological. So, SQK in a manner of speaking, begins with an educated guess for the “elephant’s” genetic code, its genotype, and then simulates this to grow the elephant itself, the phenotype. When the code is properly engineered one gets a result that bears a good resemblance to what our observations tell us is out there. In this way, subquantum kinetics achieves simplicity and elegance right from the beginning. It has very few assumptions and, compared to quantum physics, it is mathematically far simpler and more commonsensical. It also avoids the many paradoxical results and pitfalls that standard physics suffers from.

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SQK postulates a “reaction-diffusion” ether pervading all of space. This differs from the nineteenth century mechanical ether theories, and instead resembles more the kind of multi-specie gaseous ether proposed by Dmitri Mendeleev (1904), or to the transmuting ether theory developed by the Chinese physicist T’an Ssu-t’ung (1865-1898), e.g., see Yu-Lan (1959). However, SQK has developed its approach in far greater detail and mathematical rigor than had these past theorists, partly because of the advent of the branch of physics and mathematics that deals with nonlinear chemical reaction systems and nonequilibrium thermodynamics. Significant also is our ability in modern times to carry out computer simulations of such nonlinear open reaction systems.

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Similar to these early theories, SQK postulates an ether consisting of different types of etheron constituents, components labeled as A, B, C,… X, Y, Z, Ω, etc., and which react with one another or transform one into the other. SQK specifies a particular preferred set of reaction pathways as well as specific rates at which these constituents diffuse through space. More specifically, it uses concepts and techniques which previously were developed in the field of chemical kinetics to describe the behavior of open chemical systems and for the first time applies them to physics at the subquantum etheric level. Thus the SQK approach should be more easily understood by nuclear and chemical physicists, who typically deal with reaction processes, as compared to physicists who have been immersed in the mechanistic concepts of classical physics.

Figure 1. Depiction of the hyperdimensional transmuting ether.

SQK postulates that the physical universe is inherently processual at the etheric level with etherons entering our universe from “upstream” states and leaving our universe to transform to “downstream” states; see Figure 1. Hence SQK requires more than three dimensions for its description. This is a significant departure from standard physics which is based on the assumption that the universe should function as a closed system, there being assumed to be no existence “beyond” the observable universe. As such, just as nonlinear open systems are known to have spontaneous amplification modes or spontaneous damping modes, SQK allows that under certain conditions a photon’s energy may either progressively increase over time or progressively decrease over time. Perfect energy conservation in SQK appears only as a special case with energy nonconservation being the more common. Due to this flexibility, SQK has an advantage over standard physics in that it permits, rather than forbids, over-unity energy production. Hence it provides a fruitful paradigm within

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which to assess the many over-unity energy devices that have been developed in past decades. This flexibility also allows it to predict a cosmology of continuous matter creation which is able to serve as a viable alternative to the trouble-ridden big bang/expanding-universe cosmology. This continuous creation scenario blatantly violates both the first and second laws of thermodynamics, but becomes permissible in SQK with its open system paradigm. Although, such energy conservation violations occurring in the cosmological setting take place on a scale so small as to be virtually impossible to detect in the laboratory.

Figure 2. The Brusselator reaction scheme.

Figure 3. The Model G reaction scheme.

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In SQK it is possible to postulate a variety of reaction-diffusion systems as ether models. But it mainly focuses on exploring the physically relevant properties of one reaction system called Model G, which strongly resembles the Brusselator reaction-diffusion system. Model G is obtained by modifying the two-variable Brusselator reaction system by adding a third reaction variable G to the Brusselator’s two variables X and Y; compare Figures 2 and 3. As a result, Model G is described by a total of five reaction steps, rather than four for the Brusselator. Figure 4 maps out the Model G reaction system specified in Figure 3. The Brusselator is unable to generate physically realistic structures since it spawns nonlocalized wave patterns, also termed dissipative space structures. That is, the wave pattern distributes itself at full amplitude throughout the entire space reaction-volume. The Model G alteration, on the other hand, is able to spawn localized wave patterns that are able to serve as analogs of subatomic particles. Model G not only spawns physically realistic soliton particles, but is both simple and, as will be seen, is able to account for a wide variety of physical phenomena, some not anticipated in the framework of standard physics.

Figure 4. A schematic representation of the reaction kinetics of Model G.

Almost two decades after the first papers on SQK were published, the modeling approach that it was proposing as a new approach to physics was lauded by physicist Stephen Wolfram, founder of Wolfram Research and creator of the Mathematica computing software. Wolfram (2002) published a book entitled A New Kind of Science in which he proposed the hypothesis that the universe may follow simple rules that can be described and even solved by

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a computer program. He suggested that underneath all of the richness and complexity we see in physics there may just be simple rules operating at some lower level, and that all of physics would be found to emerge from these rules. He pointed out that incredibly simple programs can do extremely rich and complex things. On page 4 of his ebook, entitled A Class of Models with the Potential to Represent Fundamental Physics (Wolfram, 2004), he gives examples of what he calls such “rules,” one being 1  2  3, and examines others that contain looping processes. Such rules bear some similarity to the idea of reaction kinetic processes. Although, they do not appear to incorporate concentration magnitudes or spatial diffusion coefficients. He believes he is getting closer to arriving at a workable set of rules for physics. If he eventually achieves that goal, the system of rules may turn out to look very much like Model G. He apparently was unaware of SQK, otherwise he would have mentioned this approach in his writings, and to be honest, I was not aware of his work until now while in the process of writing this present summary of SQK. Interestingly, in 2010, Wolfram’s Mathematica computing platform was used by our group to solve the set of three partial differential equations that constitutes Model G. As is discussed in the next section, the results were quite successful and confirmed the reality of the particle-like structures that had been inferred in previous publications of SQK.

THE CREATION OF GRAVITATIONAL AND ELECTRIC FIELDS The reactants specified in Model G are given values in terms of concentrations, all of which have positive magnitudes. However, in describing the evolution of physically realistic structures and fields, it is useful to define energy potentials which can take on either positive or negative values. If the analogy is made that etheron concentration is similar to ocean sea level, energy potentials would then refer to the waves on the ocean’s surface. The amplitude of such waves could take on either positive or negative values relative to sea level. Let us say that initially prior to the emergence of physical form, the etheron concentrations are uniformly distributed, a condition representing the vacuum state of space. Values G0, X0, and Y0 are assigned to Model G’s three variables to represent this initial vacuum state which is called the ether’s

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homogeneous steady state. Actually, these values are time averages since the etheron concentrations randomly fluctuate above and below these values due to the stochastic behavior of the etheron reaction processes. Gravity potential, then, could be referenced to the G0 “zero-point value,” and would be defined as g = G - G0. Gravity potential would be positive for G > G0 and negative for G < G0. In SQK, the electric field is represented by two etheron species X and Y that are involved in a self-closing reaction kinetic cycle, graphically portrayed in Figure 4. While use of two variables to represent the electric field (X and Y) may seem strange to engineers and physicists who have grown accustom to using a single variable notation in practice, it nonetheless accounts for the origin of charge polarity, something that is inadequately explained in standard physics. A high-Y/low-X concentration in a given locale would signify a positive charge whereas a low-Y/high-X concentration would signify a negative charge. In SQK, X and Y do not take values independent of one another; they co-depend in reciprocal fashion. As was done in defining gravity potential, electric potential would be defined as x = X - X0 and y = Y - Y0. Potentials x and y being understood to be co-dependent. If necessary, one could chart just one of these two variables, let us say y, to track electric field magnitude. The recursive X-Y loop transformation plays an important role in SQK. It is key to generating the emergent dissipative soliton wave pattern which can adopt two alternate polarities representing the matter and antimatter state of matter. This is examined in the next section. But, also significant, this bipolar character of the electric potential automatically leads to the creation of bipolar gravity potential. This is because the local value of the G-on concentration (gravitational mass) closely tracks the local value of the X-on concentration 𝑘−2

(electric charge) due to the presence of the reverse reaction 𝐺 ← X in step (b) of Model G; see Figure 3. This is the only significant reverse reaction, the reverse reactions in the other reaction steps being assumed to be essentially zero. As will be described later, this reverse reaction plays an important role in allowing the emergence of self-stabilizing solitons (subatomic particles). As a result of this electrogravitic dependence, the g potential at a given locale is able to adopt either a plus or minus polarity relative to the ambient ether

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concentration as it tracks the polarity of the x component of the electric field. Consequently, in SQK gravity potential is bipolar. This departs radically from the general relativistic notion that gravity should be exclusively monopolar. More specifically, a localized positive electric charge potential in SQK will generate a gravity potential well whose field gradient is able to gravitationally attract material bodies, whereas a localized negative electric charge potential will instead generate a gravity potential hill whose field gradient is able to gravitationally repel material bodies. This unique bipolar gravity prediction of SQK is consistent with the discovery by T. Townsend Brown (Brown, 1929) that there is an electrogravitic coupling between electric and gravitational fields, or in other words between charge and gravity. This coupling becomes most evident at electric potentials exceeding 50 kV. This electrogravitic coupling phenomenon is unexplained by Einstein’s general relativity theory. It was an issue of serious concern for him in his long attempt to find a way to somehow unify gravity with electromagnetism within his general relativistic framework. If it were not that the U.S. Navy had placed Brown under restrictions not to publish about his findings for national security reasons (see Brown, 1952), Brown, not Einstein, would today have been the revered icon in gravity theory. The history of the application of Brown’s findings to the aerospace field is described in the aviation intelligence report known as Electrogravitics Systems (Aviation Studies, 1956), a copy of which is archived at the Wright Patterson Air Force base library. This document, together with an extensive review of electrogravitics may be found in the book Secrets of Antigravity Propulsion (LaViolette, 2008). The point should be made here that in SQK electrically neutral matter produces a matter attracting gravity well, which is modest when compared to that of an isolated proton, because the proton’s gravity well is slightly deeper than the electron’s gravity hill. As for the demise of general relativity, physicists should not worry. For all the known general relativistic effects follow naturally from Model G. Relativistic effects are examined near the end of this chapter. As in quantum mechanics, SQK postulates that field potentials are the real existents whereas forces are effects produced on particles by gradients of these potentials. This differs from classical physics which instead regards force as the

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real existent and energy as a more abstract attribute. SQK regards energy potentials as real scalar quantities that correlate with the concentration of specific ether species at specific points in space, G-ons for the gravity potential and X-ons and Y-ons for the electric potential. There is no separate magnetic field component in SQK. Magnetic forces are theorized to be electrodynamic effects produced by translating electric potentials, or X-on and Y-on ether vortices. In a manner analogous to how electric potential fields generate electrodynamic magnetic effects, the relative motion of a gravity potential field is expected to generate gravitodynamic forces. The existence of such a gravitodynamic force has been reported by Henry Wallace (1971).

THE IMPORTANCE OF GRAVITATIONAL POTENTIAL IN THE CREATION OF MATTER In subquantum kinetics, G-on concentration, or alternatively gravity potential, serves as Model G’s bifurcation parameter determining whether or not a subatomic particle is permitted to nucleate. The system’s behavior depends on the value of the G-on concentration relative to the critical threshold concentration value Gc. Whether or not Model G allows a particle of matter to nucleate, depends on the prevailing steady state G-on concentration G0 relative to the critical threshold concentration value Gc. If G0 lies above the critical threshold, G0 > Gc, Model G remains subcritical (infertile) and is unable to spawn matter; see Figure 5. If G0 instead lies below this critical threshold, G0 < Gc, Model G becomes supercritical (fertile) and a subatomic particle, for example, a neutron, is allowed to emerge. Alternatively, it is useful to express this gravity bifurcation parameter in terms of a potential, referenced to some G concentration value. Earlier, the gravity potential was referenced relative to the steady state concentration value, G0, with g = 0 occurring when G = G0. Here it is more useful to set the critical threshold Gc as the zero reference. Hence with g(r) defined as G0(r) - Gc , a particle would be able to nucleate when g < 0 and would be prevented from nucleating when g > 0.

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But in addition to requiring a preexisting supercritical region, an energy potential fluctuation is also needed to initiate particle morphogenesis in this region. As mentioned earlier, the ether concentrations continuously fluctuate above and below their steady state concentration values. This subquantum noise is theorized to have fluctuation amplitudes distributed as a Poission function and would be the analog of the zero-point energy field in quantum physics. However, unlike the standard quantum fluctuation concept, these fluctuations would normally have energies far smaller than that of a fully formed subatomic particle, and they would not emerge as paired plus-minus polarities. Instead, they would emerge in an uncorrelated white-noise fashion. These fluctuations play an important role in SQK since these are what nucleate particles in empty space.

Figure 5. The ability for matter to nucleate in empty space depends on the ambient value of the gravity potential relative to the zero value which signifies critical threshold value, G c.

But this nucleating fluctuation, say x-y, must be of positive polarity and also have a sufficiently large magnitude. The role of polarity will become clear shortly. Under prevailing supercritical conditions, such a positive electric potential fluctuation would be able to grow in size and eventually develop into a dissipative soliton, e.g., a neutron. This process by which a fluctuation can initiate an ordered pattern in a nonlinear nonequilibrium reaction system is well known in studies of the Brusselator system, and has been termed by Ilya Prigogine as “order-through-fluctuation.” Details of this have been extensively worked out by Nicolis, Prigogine and their Brussels group in a field of study called “fluctuation theory.”

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In the case of SQK, we may view this particle materialization process by simulating Model G on a computer. Beginning from the Model G reaction system, Figure 3, we may write the following set of three partial differential equations as a representation of Model G:

(1)

Each partial differential equation here specifies the evolution of one of Model G’s three variables, G, X, and Y over space and time. This equation system is more easily solved if the variables are expressed in terms of potentials rather than concentrations. Matt Pulver has expressed equation system (1) in terms of potentials and through the use of dimensionless counterparts has computer simulated this equation system (Pulver and LaViolette, 2013). Figure 6, taken from this paper, presents the results of one such computer simulation which shows a positive electric potential fluctuation evolving into a “dissipative soliton.” This would represent the spontaneous nucleation of a neutron from “empty” space. The initiating fluctuation must be positive, because this generates a corresponding local gravity well, negative g potential, whose supercritical environment allows the emerging electric potential fluctuation to grow in size. A negative electric potential fluctuation, ultimately leading to the materialization of an antineutron, would fail to spontaneously self-amplify since it would generate a G-hill which would locally produce a subcritical environment, thereby snuffing its own growth. Because of this matterantimatter bias, the SQK matter creation process leads to a universe filled mainly with matter, rather than antimatter. This is an advantage since to date there has been no detection of antimatter galaxies. The apparent lack of an equal amount of antimatter in the universe has been a major setback for the big bang theory of standard cosmology. This is not to say that negative polarity particles

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such as the antineutron cannot form. These can nucleate in an existing supercritical region when initiated by a sufficiently large energy impulse.

Figure 6. Sequential frames from a three-dimensional computer simulation of Model G showing the emergence of an autonomous dissipative soliton particle: t = 0 the initial steady state; t = 15 growth of the positively charged core as the X seed fluctuation fades; t = 18 deployment of the periodic electric field Turing wave pattern; and t = 35 the mature dissipative soliton particle maintaining its own supercritical core G-well. Simulation by M. Pulver. May be viewed at: https://tinyurl.com/ybfphshf.

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Once the dissipative soliton has fully emerged, its growth ceases. Figure 7 depicts such a fully developed soliton representation of a neutron. The soliton maintains this final end state as an inhomogeneous steady-state condition, despite environmental perturbations that might attempt to destabilize it. It is important to note that the G-well present in the soliton’s core, provides a supercritical environment that ensures the particle’s continued survival. Thus in SQK such an emergent subatomic particle is both autonomous and autopoietic. That is, once formed it persists independently of environmental disturbances. This feature, whereby a particle becomes self-stabilizing by virtue of its own self-generated gravity well, is unique to Model G and gives this reactiondiffusion system the ability to produce physically realistic soliton structures.

Figure 7. Self-stabilizing soliton produced by a computer simulation of Model G which represents a neutron.

This representation of a neutron’s electric field differs substantially from previous representations that have come out of standard physics which show the neutron’s electric field potential rising asymptotically to a peak at the particle’s center, as in the model advanced by Schmieden (1999). However, particle scattering experiments subsequently conducted by Kelly (2002) show something very different. Compare Figure 6 to Figure 8-a which depicts the charge density profile of the neutron based on Kelly’s observations.

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Note that both show a rounded off central profile surrounded by a wave pattern that asymptotically declines in amplitude as it departs from the particle’s center. The surrounding periodicity is made more evident when Kelly’s data is plotted as a surface charge profile; see Figure 8-b. So Kelly’s findings unequivocally confirm the SQK a priori prediction of the electric field distribution in the neutron. This important finding is discussed in LaViolette (2006).

Figure 8. a) Charge density profile for the neutron predicted by Kelly’s preferred Laguerre-Gaussian expansion models and b) the corresponding surface charge profile (after Kelly, 2002, Figure 5 - 7, 18).

In SQK, the electric potential wave pattern in the core of a subatomic particle is termed its Turing wave, in honor of Alan Turing, the first to formulate a reaction-diffusion theory of morphogenesis. Similar wave patterns seen in chemical reaction-diffusion systems have been referred to as Turing patterns. SQK has predicted that the Turing wave should have a wavelength equal to a particle’s Compton wavelength; i.e., the wavelength denoting one complete

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wave cycle (LaViolette, 1985b). Kelly’s findings observationally confirm that the wavelength is indeed close to the Compton value. The SQK model of subatomic particles resolves the wave-particle dualism that has long plagued standard physics. As seen here, the SQK dissipative soliton is both a particle and wave at the same time. So there is no need to assume that a wave packet tags along with a particle as the particle moves, the idea advanced by deBroglie and adopted by quantum theory. In fact, LaViolette (1985b, 2012) has shown that the SQK soliton with its Turing wave field modulation accounts for the same phenomena that standard wave mechanics attempts to explain, such as particle diffraction and orbital quantization. Thus SQK offers an effective replacement for quantum mechanics while avoiding paradoxical pitfalls of the standard view, such as the particle wavelength being dependent on the relative velocity of its observer. Nikolic (2007) discusses a large number of the assumptions of standard quantum theory which are either questionable or still unproved. We will not go into this further here since the purpose of this chapter is to focus on the topic of gravity. SQK also theorizes that an ether vortex would develop in the particle’s core and that this produces what physicists refer to as particle spin magnetic moment. In the neutron, for example, this vortex arises because Y-ons continually diffuse outward from the core, where the Y concentration is high, into the adjacent inner shell where the Y concentration is low. Simultaneously, X-ons and G-ons continually diffuse inward from this inner shell, where the X and G concentrations are high, into the central core where the X and G concentrations are low. Since the core and shell concentrations are maintained indefinitely by the underlying reaction processes, these fluxes also continue indefinitely. Furthermore, these radial fluxes are theorized to develop into a vortex similar to water going down a sink’s drain. This vortex gives the particle a preferred spin axis. A particle’s spin vortex is what gives the particle its ability to bind tightly to another particle when both come in close proximity. Hence according to SQK, the nuclear force is simply the attractive force produced by the mutual alignment and entrainment of two adjacent spin vortex fluxes. These vortices must be so aligned that the ether fluxes travel in the same direction. Hence the vortices (spins) must be aligned antiparallel when the particle vortices are entrained equatorially, and aligned parallel (same direction) when the vortices

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are axially aligned. This leads to the spin alignment rules studied in quantum mechanics. But, since this chapter is devoted to the subject of gravitation this subject will also not be dealt with further here; for more information see LaViolette (2012), Ch. 5. Let us now examine the type of gravity field that a subatomic particle, e.g., a neutron, would generate as understood in the SQK approach. As discussed above, the g potential of the neutron will have a periodic aspect, as do all subatomic particle field patterns in SQK. Since the negative amplitudes of the neutron’s gravity field predominate over the positive amplitudes, the neutron produces a net consumption of G-ons in its core, and this in turn generates an extended gravity potential field which declines with distance from the particle’s center as 1/r. SQK expresses this as: (2) where 𝜑̅𝑔0 is the average value of the gravity potential 𝜑̅𝑔 (𝑟) at the particle core boundary 𝑟 = 𝑟0 . Indeed, in SQK, it is this net consumption of G-ons that determines the magnitude of the particle’s gravity field. This is because the conversion of Xons into Y-ons (reaction (c) in Figure 3) takes place more rapidly in the particle’s core than it does in the particle’s environment due to the presence of a higher concentration of Y-ons in the core. The resulting reduced X-on concentration, in turn, leads to X-ons being converted into G-ons at a slower rate (reaction (b) in Figure 3). The resulting deficit of G-ons in the core induces an inward G-on flux from the neutron’s surroundings. It is this inward G-on flux that determines the 1/r contour of the neutron’s gravity field. Hence the gravity field of the neutron is ultimately determined by this excess G-on consumption rate taking place in its core. The potential 𝜑̅𝑔0 at the core boundary given in (2) is expressed as: (3)

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where Dg is the diffusion coefficient of G specified in equation (1) and M g is the total G production rate balance accrued within the particle core, within a spherical boundary of radius r0. In other words, Mg represents the total rate of G-on consumption within the neutron’s core. This parameter is termed the particle’s active gravitational mass; i.e., the gravitational mass that determines the neutron’s extended gravity potential field. This gives an example of how known physically observed quantities may be expressed in terms of the SQK reaction-diffusion methodology. For a more detailed discussion of these mathematical expressions, one is referred to the main reference LaViolette (2012). This reaction-diffusion characterization of the particle’s gravitational mass and how it generates a surrounding gravity field may seem rather strange to the average physicist who is unfamiliar with SQK. But he should keep in mind that classical physics offers no explanation of the nature of gravitational mass or why it generates a gravity field; it merely quantifies these based on observation. Similarly, although general relativity claims that a mass creates its gravity field by warping space-time, it does not go into any specifics of how matter acts upon space-time to accomplish this. I tend to agree with the view of Nikola Tesla who said: “I hold that space cannot be curved, for the simple reason that it can have no properties. ... To say that in the presence of large bodies space becomes curved, is equivalent to stating that something can act upon nothing. I, for one, refuse to subscribe to such a view.”

At sufficient radial distances, the neutron should be found to produce a net G-on consumption. However, it should be recognized that close to the particle core the representation given in Eqns. (2) and (3) may be oversimplified. For, as is seen in Figure 7, the shell adjacent to the particle’s core has a gravity potential that is slightly positive, hence a G-on production rate surplus (negative mass), and the shell beyond that has a slight G-on production rate deficit (positive mass), and so on. The idealized representation of the neutron’s gravity field given in Eqn. (2) instead assumes that the G production rate balance outside of 𝑟0 is zero. So, based on the Figure 7 soliton simulation, we should

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expect that there will be departures from this idealized state in the immediate vicinity of the neutron’s core. Moving onto the topic of gravitational force, we see that the 1/r gravity potential field is able to induce an attractive pull on neighboring particles that declines as 1/r2 with distance. This effect is expressed in terms of the accelerating force intensity 𝐼𝑔̅ (𝑟) which depends on the gravity potential gradient as:

(4) where kg is a constant of proportionality. Hence SQK leads to gravitational field and force expressions that are consistent with classical theory. This equation holds at large distances from the center of the particle. As one instead approaches the center of the particle where its gravity field flattens out, one finds that the gravity gradient approaches zero. This leads to the conclusion that black holes should be unable to form. That is, if a star were to collapse and compressing itself under the action of its own gravity field, the inward pulling gravitational force would eventually approach zero. This conclusion is backed up by Kelly’s observations of the contour of the nucleon’s electric charge density which accordingly flattens out at the particle’s center. Kelly’s data, of course, does not directly make predictions about the particle’s gravity field. But, acknowledging the reality of a direct correspondence between the electric and gravitational fields, based on T. T. Brown’s findings, one is led to conclude that the particle’s gravity field should similarly plateau in the particle’s core, just as SQK predicts. There is also another reason why black holes cannot form in SQK, which we defer to the section on photon blueshifting. The manner in which a potential field causes a remote particle to move, due to the effect of its field gradient on the particle, differs substantially from the mechanistic explanations given in standard physics which rely on one’s personal “hands-on” experience of force. Note that classical physics offers no explanation other than to quantify a particle’s accelerating motion in the presence of a gravity field. Also general relativity, which relies on the concept of warped space-time, offers an understandable explanation of how a moving

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planet orbits its parent star. However, it falls short in offering an explanation of how an object initially at rest is caused to accelerate when subjected to a gravity field. By comparison, SQK offers the following explanation. When subjected to a gravity potential gradient (G-on concentration gradient), the G concentration well that forms the core of the particle is caused to migrate toward the gravity field source, that is, in the direction where the gradient’s G-on concentration is lowest. This is because the imposed gradient lowers the G-on concentration on the side of the particle’s core nearest the gravity field source and raises it on the opposite side furthest from the gravity field source. In fact, this external gravity gradient distorts the particle’s entire shell-like g space structure, altering its former spherically symmetrical shape in the field gradient direction and perturbing the homeostatic condition that maintains the particle’s entire space structure pattern. The particle consequently departs from its former stable attractor state and enters a condition of instability. It is through this stress, that the field gradient manifests its force on the particle. The magnitude of this force would vary in direct proportion to the magnitude of the imposed gravitational field gradient. The particle response to this stress follows in accordance with Le Chatlier’s Principle which holds that a system in dynamic equilibrium that is subjected to a stress will change so as to relieve that stress. Although originally developed to explain the equilibration of chemical systems, this principle applies equally well to describe how concentration gradients present in a reaction-diffusion ether are able to induce particle acceleration. The particle relieves the imposed stress by readjusting its form so that its shell-like wave pattern adopts a more symmetrical configuration in a new reference frame that is in motion relative to its old frame. In so doing, the particle moves down the gravity gradient and, as a result, in the old frame its space structure will appear distorted, slightly compressed on its bow side and slightly extended on its lee side in accordance with the requirements of the relativistic Lorentz contraction effect discussed toward the end of this chapter. As a demonstration that a gravity potential gradient in Model G is actually able to cause a soliton (neutron) to move, the reader is referred to the following

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YouTube postings by M. Pulver which show the results of a computer simulation in 1D of a particle moving down a constant 2% g slope. normal view: https://tinyurl.com/y7s9m9xl zoomed view: https://tinyurl.com/yc56t3oh

Figure 9. Computer simulation of a dissipative soliton particle moving in a 2% gravity potential gradient. Particle shown centered at position (x = +0.6). (Pulver and LaViolette, 2013).

Figure 9 shows a frame taken from this simulation. Analysis of these simulations shows that the magnitude of this velocity scales in proportion to the steepness of the imposed G gradient; see Pulver and LaViolette (2013). These simulations which were performed in only one dimension showed the soliton’s velocity converging to a constant velocity along the x-axis. To be a correct representation, the particle velocity should instead be found to continuously increase with distance traveled. More recent simulations of Model G, carried out in two dimensions by Brendan Darrer and the Model G/Vortical Motion Group, do show the soliton accelerating when placed in the presence of a G gradient; see video available at: https://youtu.be/1heLcC-xuiU. The two particles were nucleated on either side of the reaction volume each being subjected to a G concentration gradient (gravity potential gradient) that was inclined toward the volume’s center, as shown in Figure 10 (a). Figure 10 (b) shows the positions of the particles after t = 10 seconds in the simulation and

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also after t = 45 seconds, having each accelerated down its respective G gradient and approached closer to one another. In the video the particles are seen to ultimately collide and circle one another slightly before the simulation finishes. The position x of the left soliton was determined at 14 points in time during the simulation and plotted to form a time-distance graph; see Figure 11. A polynomial fit of the graph showed that it fit the form x = at2 bt + c, which corresponds to the formula exhibited by an accelerating body. This simulation was written in the TensorFlow/Python programming language and incorporated Navier Stokes equations in each of Model G’s three partial differential equations. So we find that when simulated in two dimensions, Model G solitons accelerate in a gravitational gradient just as would real masses. Other 2D simulations have demonstrated that these neutral charged solitons exhibit mutual attraction as would two gravitating bodies, as well as elastic particle collisions. Currently, we are working on also simulating soliton tunneling across a thin boundary as well as diffraction of a soliton as it passes through a single slit.

Figure 10. Illustration of movement of two Model G solitons down G gravity gradients toward one another. a) illustration of the G concentration contour imposed on the reaction volume, b) illustration of the relative positions of the two dissipative solitons at time t = 10 seconds and at t = 45 seconds into the simulation. Simulation by B. Darrer may be

viewed at: https://youtu.be/1heLcC-xuiU.

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Figure 11. Illustration of the acceleration curve determined for the left soliton in the simulation illustrated in Figure 10. The soliton is found to accelerate down the G gravity gradient in a realistic fashion.

The proton, like the neutron, would generate a G-well in its core. The electron, though, would instead produce a G-hill and production rate surplus in its core. In the case of an electron, G-ons would diffuse radially outward to produce a gravity potential field that declined as 1/r from the electron’s core. The electron then would generate a negative active gravitational mass and a field that was gravitationally repulsive rather than attractive. As mentioned earlier, this bipolarity of the gravity field, correlative with electric charge polarity, is supported by the experimental findings of electrogravitics. As mentioned earlier, G-ons are produced at a slightly lower rate in the electron’s core than they are consumed in the proton’s core. As a result, a neutral atom consisting of equal numbers of protons and electrons would be a net consumer of G-ons and would surround itself with a matter-attracting gravity well, consistent with observation. Charged subatomic particles, such as the proton and antiproton, would appear as sketched in Figure 12. Note that the y wave pattern of the proton is biased upward compared to what was shown for the neutron, and that the y pattern for the antiproton is biased downward; see hatched region. The sketch of the g field profile shown in Figure 12 is intended to represent the average

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value of the gravity potential Turing wave pattern. The x field pattern, which is not shown here, would mirror the y pattern. That is, x would be most positive where y would be most negative. For the proton, the x Turing wave pattern would be biased downward relative to the x zero reference value, and for the antiproton it would be biased upward.

Figure 12. Hypothetical electrostatic and gravity potential field profiles (in radial crosssection) of a 3-D localized steady-state dissipative soliton. Top: matter state (proton) and Bottom: antimatter state (antiproton).

Computer simulations producing these charged particle states have as yet not been performed, which is why SQK currently only offers a sketch of the

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expected appearance. Nevertheless, this a priori prediction (LaViolette, 1985b) was later confirmed by Kelly’s particle scattering results. Figure 13 (a) shows the charge density distribution and Figure 13 (b) the surface charge distribution as found by Kelly (2002) for the proton. The periodic aspect of the proton’s charge distribution is more evident in the lower surface charge plot. Notice its upward bias, especially as the center of the particle is approached. Compare to Figure 12 (upper profile).

Figure 13. a) Charge density profile for the proton (upper plot) predicted by Kelly’s preferred Gaussian models. b) Its corresponding surface charge profile (lower plot).

Such a biased ("charged") state is also observed for the dissipative structure pattern produced by the Brusselator. For example, Auchmuty and Nicolis (1975) have published a mathematical analysis of the Brusselator which examines circumstances in which the dissipative structure wave pattern is biased upward or downward relative to the homogeneous steady state concentration. Computer simulations performed on the Brusselator indicate that biasing of the dissipative space structure pattern occurs as a result of a

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secondary bifurcation of the first bifurcating branch and emerges abruptly at some finite distance from the primary bifurcation point at a point close to the next higher bifurcation (Herschkowitz-Kaufman, 1975). Since Model G is a modification of the Brusselator reaction system, the same phenomenon is expected for Model G. A bifurcation diagram similar to that published for the Brusselator, can be used to describe the nucleon’s transition to the charged state; see Figure 14. The neutron, which emerges spontaneously from a zero-point fluctuation, is shown to occupy the positive primary branch. The proton, which emerges as a secondary bifurcation of this primary bifurcation branch, occupies the new branch that splits off upward from the neutron’s branch. In Model G, the emergence of this charged state is identified with the beta decay nuclear reaction which occurs when a neutron changes abruptly into a proton.

Figure 14. A hypothetical bifurcation diagram for nuclear particles showing the neutron and antineutron and the charged proton and antiproton states emerging as secondary bifurcations at criticality point 𝛽′ .

The excess production of Y (and consumption of X) in the proton’s core would generate its extended positive polarity electric field. The antiproton, if it were formed through a particle scattering event, would accordingly generate a negative polarity electric field. In either case, these fields would disseminate as 1/r and their field gradients would exert electrostatic forces that declined as 1/r2, consistent with classical electrostatics. The generation of these fields and their

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exertion of force is similar to that described above for the gravity field. See LaViolette (2012) for more details on the mathematical description of this reaction-diffusion generation and deployment of the electric field. The electron’s x, y, space structure would look much like that of the antiproton, but would have a wavelength 2000 times larger. Feynman, Leighton, and Sands (1964) had proposed a reaction-diffusion concept of reality as a way of depicting the radial dependence of the electron’s field pattern. Making an analogy to the diffusion of neutrons out of the core of a nuclear reactor, they postulated a substrate of “little X-ons” created in the electron’s core and diffusing radially outward, to create a surrounding 1/r decrease in concentration. They note that the resulting mathematical relation is consistent with observation. They state that such a theory, where X-ons would be generated in distributed fashion within the electron’s core, would avoid the infinite energy absurdity of current electrodynamics, but note that such a workable theory had not at that time been developed. We feel that SQK is the “workable theory” that Feynman et al. were intuiting.

THE COSMOLOGICAL IMPLICATIONS OF GRAVITY As mentioned earlier, gravity potential serves as the bifurcation parameter controlling Model G’s modes of operation, determining whether the ether operates in a subcritical or supercritical state. The same would be true for the energy behavior of photons. Adopting the generalized wave equation proposed by Gmitro and Scriven (1966) for the description of reaction diffusion waves consisting of small amplitude excursions [𝜙] from the steady-state, we may write: 



(5)

where A0 = ||max , and R and i are the real and imaginary parts of the wave number  . The frequency and wavelength of the wave are given respectively as f =  /2 and  = 2/. Here [𝜙] represents a generic field potential, be it electric (𝜙𝑦 (r, t)) or gravitational (𝜙𝑔 (r, t)).

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The wave velocity relative to the ether rest frame is given as c0 = f  = / , where the wave number  and frequency must satisfy the determinental equation: (6) determined by the values of the kinetic constants [K] and diffusion coefficients [𝔇] chosen for the reaction system, where [I] is the identity matrix (LaViolette, 2012, Ch. 2). These reaction system parameters would be chosen so that the derived velocity would equal the speed of light. The oscillatory real term in Eqn. (5), first exponent on the right, is consistent with energy wave behavior in standard physics. The imaginary term in Eqn. (5), second exponent on the right, though, is a term that is new to physics. It dictates nonconservative wave damping when i > 0 (subcritical conditions) and nonconservative wave amplification when i < 0 (supercritical conditions). Equation (5) may be restated as Eqn (7) to portray the manner in which photon energy changes as a function of photon travel distance: (7) The term E(r) here signifies the wave’s maximum electric potential or energy over a wave cycle. Its initial energy E0 represents the wave amplitude at time t = 0 and is equivalent to the wave amplitude term A0 in equation (5). The exponent 𝑒 −(𝛼𝜑𝑔/𝑐)𝑟 would signify the second exponent in (5) where 𝜅𝑖 = 𝛼𝜑𝑔 /𝑐. Here  is a constant of proportionality given in units of s/cm2; g is the ambient gravity potential as noted in Figure 5; and c is the velocity of light. Negative values of g (r) would dictate supercritical conditions and photon energy amplification. Positive values of g (r) would dictate subcritical conditions and photon energy damping. Intergalactic regions of space which are largely devoid of matter would be subcritical (g > ). Hence photons traveling through such regions would experience energy damping, or redshifting; see Figure 15. Regions in the

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vicinity of galaxies and galaxy clusters, on the other hand, would be supercritical (g < ), and photons present in such regions would experience energy amplification, or blueshifting. Such supercritical regions would often span close clusters and even could include an entire galaxy supercluster. To be physically realistic, the degree of subcriticality or supercriticality affecting a photon’s energy must be sufficiently small that the photon’s departure from perfect energy conservation is unobservable in the laboratory. For energy nonconservation to be detected, an individual photon would need to be observed over a great length of time or over a great propagation distance. Hence such effects would become important only at the astronomical and cosmological scale. Photons traveling from distant galaxies will spend more time passing through subcritical void regions, than galaxy rich supercritical regions, and so would exhibit a net redshift. SQK then offers a way to account for the observed cosmological redshift of distant galaxies without adding in any extra assumptions about cosmic expansion. Equation (7) may be written as follows to express the average attenuation rate that a photon would experience over the course of its flight: (8) where the attenuation coefficient is equivalent to term 𝛼𝜑𝑔 /𝑐 in Eqn. (7). Expressed in terms of photon wavelength, , this would be rewritten as: (9) This is essentially the "tired-light" relation which was first proposed in 1929 by Fritz Zwicky to explain Hubble’s redshift-distance observations as an alternative to the expanding universe hypothesis. For his theory, Zwicky made the ad hoc hypothesis that photons have a non-zero rest mass and lose energy through an energy-conserving gravitational drag effect. In SQK, on the other hand, no such ad hoc photon interactions are needed since photons lose energy

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non-conservatively due to the fact that the open reaction-diffusion ether operates in a subcritical state.

Figure 15. Photon behavior depends on the ambient value of the gravity potential relative to the critical threshold value, g = 0. Photons progressively increase their energy within supercritical gravity wells surrounding galaxies and galaxy clusters (where g(r) < 0). Photons progressively decrease their energy in intergalactic space (where g(r) > ).

The value of attenuation coefficient  is given by the Hubble constant, i.e.,  = H0 /c . Sandage, et al. (2006) have estimated the Hubble constant using Cepheid stars in nearby galaxies as redshift-distance-indicators and find that it has the value H0 = 62.3 ± 1.3 km/s/Mpc. Higher values are also quoted in the literature such as 70 km/s/Mpc and 74 km/s/Mpc of Freedman, et al. (2019 and Riess, et al. (2019). But these are based on redshift calibrators found in the Large Magellenic Cloud which lies about 400 fold closer to us than the galaxies Sandage used and which along with the Milky Way resides on the edge of a void, hence in a region of space that is likely more subcritical than average (LaViolette, 2021). Also the value quoted by Sandage, et al. is substantiated by Marosi (2014) who finds a similar Hubble constant value of 62.6 km/s/Mpc by fitting magnitude-redshift data of supernovae out to a redshift of z = 8.1. Adopting the H0 value of Sandage, et al. yields a tired-light photon energy decline rate of  = 6.4 ± 0.1% per billion light years (bly). This may instead be expressed in units of time by multiplying  by the factor c = 3.17  10-17 bly/s , which yields an energy loss rate µ = -c = -2.03  10-18/s. This is about 10 orders of magnitude smaller than the smallest change observable in the laboratory.

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To test this SQK prediction, the stationary universe, tired-light prediction of SQK and the prevailing expanding universe big bang hypothesis were compared against observational data on four different cosmology tests (LaViolette, 1986). There it was shown that the tired-light model fit the data better than the standard expanding universe model on all four tests. This result presented a coup against standard cosmology since the tired-light model made its good fit without requiring the addition of any evolutionary corrections. In the past, big bang cosmologists would typically add evolutionary corrections to make their expanding universe model fit the observational data trend. The four-test approach published in 1986 (LaViolette, 1986), however, blocked this cunning practice. For it was shown that any attempt to add evolutionary corrections to make the big bang prediction fit the data better on one test, would make its prediction depart from the data trend on one or more of the other tests. This multi-test approach traps the big bang theory much like a Chinese handcuff. So the judgment against standard cosmology is final. One cosmology test taken from this multi-test study is a version of the angular-size-redshift test that plots the harmonic mean angular separations of bright galaxies in a galaxy cluster against the cluster’s redshift, where the mean galaxy angular separations are used as a measure of distance; see Figure 16. Three cosmologies are plotted against the z data set of Hickson and Adams (1979), which include the simple linear Hubble relation, the no-evolution tired light model (static universe), and the no-evolution, q0 = 0 Friedmann model (expanding universe). There is no question that the static universe, tired light cosmology falls closer to the data trend. The fit of these three models was assessed by comparing the variances between the  data points and the prediction that each model makes. The variances for the three models (linear   1/z, tired light, and q0 = 0 expanding universe) were respectively in the ratio 1 : 1.2 : 5.0 . Repeating the calculation for the 31 most distant clusters (z > 0.1) gives relative variance ratios of 1 : 1.4 : 10. Thus the static, Euclidean tired light cosmology is seen to be significantly favored over the q0 = 0 expanding universe model. The q0 = 0.5 Friedmann model, which many cosmologists have favored which incorporates the assumption of hidden mass plots above the q0 = 0 prediction and makes an even worse fit.

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Actually, the initial purpose in conducting the 1986 study was to check whether the SQK energy attenuation relation could present a valid explanation of the cosmological redshift. In fact, it was only in the course of doing background research for this test that I had found that the tired-light model had been known for some time, and had been originally conceived as an alternate explanation for the cosmological redshift. The outcome of the 1986 study was that cosmological test data confirmed the SQK prediction and showed that it offered a new theoretical explanation for tired-light behavior. Over the intervening years, many other studies have been published also showing that the tired-light model makes the better fit. A number of these more recent studies have been included in an updated multi-test evaluation of the expanding and static universe cosmologies; see LaViolette (2021).

Figure 16. Log normalized galaxy angular separation vs. log galaxy cluster redshift.

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MATTER CREATION With the big bang theory and its underlying expanding universe hypothesis disproved, a new theory of cosmic creation must be sought, one that allows matter to emerge spontaneously in a cosmologically static universe. Hence a cosmology must be sought that proposes repeated minor materialization events taking place in distributed fashion throughout all visible space rather than relying on the occurrence of a singular materialization event of unbelievable magnitude, as the big bang theory had suggested. One such hypothesis is the theory of continuous creation as advanced by astronomers such as Sir James Jeans (1928) or William McCrea (1964), both of whom were led to the idea on the basis of astronomical observation. SQK, though, is the first theory to explain in detail how such materialization might occur. By its nature, a theory of continuous creation would involve a violation of energy conservation on a cosmological scale, the appearance of matter being a negentropic event. But the spontaneous formation of ordered structures is a phenomenon frequently observed in open systems. So, the demise of the big bang theory automatically leads to the idea that our universe likely functions as an open system. Indeed, if the big bang theory is invalid and continuous creation is the only remaining alternative, there appears to be no choice but to arduously embrace the open system view of physical reality. So there is hope that the physics community as a whole is now ready to seriously consider the paradigm of subquantum kinetics. As was discussed above, Model G allows the spontaneous creation of matter wherever the initially subcritical ether has its G-on concentration sufficiently close to the critical threshold as to allow a zero-point energy fluctuation of critical size to nucleate a neutron. Such conditions would likely present themselves over large stretches of space throughout the cosmos. In SQK, this matter creation process is called parthenogenesis, which signifies the spontaneous self-formation or birth of matter. However, there is another matter creation feature of SQK, namely, the ability of existing particles of matter to create fertile environments in their immediate vicinity in which “progeny” particles may spawn themselves. That is, once a particle has formed, the G-well present in the particle’s core forms a

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supercritical environment that nurtures any emerging seed fluctuation, allowing it to rapidly grow into a new particle. Thus in the SQK cosmology, matter spawns more matter (LaViolette, 1985c, 1994, 2012). This process of exponential growth much resembles biological reproduction. Projected to a very late stage of matter evolution, after star filled galaxies have developed, SQK predicts that the supermassive core positioned at the galaxy’s center, with its very deep gravity well, should spawn matter at a prodigious rate. Computer simulations of Model G in 1D have verified this maternal birth process by showing a nascent neutron coming into being within the supercritical gravity well of an already existing nucleon. A randomly emerging zero-point energy fluctuation is found to spawn a progeny neutron most effectively if it emerges in one of the two inner most gravity well shells that surround the core of an existing nucleon the supercritical conditions in these shells providing fertile environments where these fluctuations are able to self-amplify (Pulver and LaViolette, 2013). Also simulations have shown that a neutron is able to self-materialize in the supercritical region located between two pre-existing nucleons; see Figure 17 taken from LaViolette (2012), also discussed in Pulver and LaViolette (2013). Actually, maternal nucleation would take place more commonly from “mother protons” rather than from “mother neutrons” since once it had emerged a neutron would subsequently decay into a proton and electron through the beta decay process. So hydrogen nuclei would more commonly be present in interstellar space than lone neutrons. Also due to their positive charge, isolated protons would have a much deeper G-well than isolated neutrons, this being because in the positive charge state, the nucleon’s core x well is biased downward, which in turn downwardly biases its g well, making it more supercritical. This is essentially the electrogravitic field coupling feature discussed earlier. So materialization would proceed in a mother-daughter fashion in which a daughter proton acts as a nucleation center which spawns a daughter neutron which then decays into a daughter proton which itself acts as a nucleation center to spawn a daughter neutron. Consequently, one nucleon becomes two, which become four, which become eight, and so on in an exponential growth process.

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Figure 17. Sequential frames from a one-dimensional computer simulation of Model G showing the spontaneous emergence of a third soliton from between two pre-existing Turing wave parents. Simulation by M. Pulver may be viewed at:

https://tinyurl.com/y6vunl7a.

The continuous materialization of neutrons and their subsequent beta decay produces a diffuse interstellar hydrogen gas throughout the universe which becomes heated into an ionized state by the 0.78 Mev beta particles produced as a result of neutron decay. Through collisional energy transfer, this gas consequently becomes heated into an X-ray emitting plasma, thereby explaining

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the observed diffuse X-ray emission that is observed coming from all directions of the sky. This X-ray emitting intergalactic gas has been referred to as the Warm Hot Intergalactic Medium, or WHIM. Its presence is also indicated by the so-called Lyman alpha forest, diffuse Lyman alpha emission radiated by the ionized portion of this gas. These observations have led astronomers to conclude that there may be as much matter in the universe residing in the “voids” between galaxies as resides in the galaxies themselves. This has posed a problem for the big bang theory because it indicates that there is far more ordinary matter in the universe than big bang models predict. Also the big bang theory cannot account for its temperature because it predicts that the gas of the initially hot fireball should have long ago cooled down. The source of ionizing radiation for the WHIM has puzzled astronomers since no stars are visible in these clouds and radiation from active galactic cores falls short of the energy requirements. LaViolette (2012), however, has shown that beta decay of parthenogenic neutrons would supply more than enough energy to power this emission. Crawford (1987, 2011) has shown that the electrons in such a heated X-ray emitting plasma have a temperature and density sufficient to generate the observed 2.73° K cosmic microwave background radiation (CMBR). Moreover Arp, et al. (1990) have argued that iron whiskers present in intergalactic space could thermalize the 3°K radiation field while allowing transparency at other wavelengths. So with the theory that the WHIM is the source of the CMBR, we still retain the idea that the microwave background is of cosmic origin. The only difference is that instead of the emission arising all at once from a single big bang event, it is now understood as being continuously generated by beta decay electrons associated with the ongoing creation of matter. Once the WHIM had formed, some regions could cool sufficiently to allow neutral hydrogen gas to form and to condense into droplets of liquid hydrogen. These would eventually coalesce to form comet sized bodies which would grow both through accretion and through internal matter creation. This preliminary phase of creation could have transpired over as much as a few trillion years. Note that the SQK continuous matter creation cosmology has no 13.8 billion year restriction in which to generate the cosmos as does the big bang theory. This early phase and subsequent phases of cosmic evolution are depicted in

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Figure 18. Continuing the sequence, a comet sized body would eventually grow into a brown dwarf, then into a primordial star, whose stellar wind would spawn gaseous planets. The daughter planets of this solar system would themselves evolve into stars and eventually a star cluster would form. As a given stellar body grew in mass, so would its surface gravity potential and degree of supercriticality. As a result, its rate of matter creation and rate of energy output due to spontaneously generated energy would both progressively increase. The phenomenon of spontaneous energy production due to photon blueshifting that would take place in these bodies is discussed in the next section.

Figure 18. Cosmogenic evolution: Sequential development from primordial self-nucleating particles to mature galaxies.

The primordial "Mother star,’ the first star to have formed, would be the most massive star in the star cluster, having had a head start over its daughter stars. It would reside at the cluster’s center with the other less massive stars orbiting around it. By this point, it would have grown to a mass of hundreds to thousands of solar masses. As the star cluster continued to proliferate and grow in size, it eventually would turn into a dwarf elliptical galaxy. As it grew in mass and became increasingly supercritical, the Mother star would begin to produce energetic outbursts. Upon reaching a mass of a hundred thousand solar masses or more, its outbursts would become increasingly violent similar to the

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outbursts seen to come from the cores of Seyfert galaxies. These outbursts would propel stars and gas outward to gravitate toward the evolving galaxy’s orbital plane causing the dwarf elliptical to evolve into a compact spiral galaxy and then into a mature spiral galaxy. The Mother star core would at this time produce a gravity well so deep and so supercritical that its matter creation rate would far outstrip the combined matter creation rate of all the stars in its galaxy. In fact, this cosmology predicts that 99% of the matter in our own galaxy has originated from our core Mother star, Sagittarius A-star (Sgr A*). Some of the more violent outbursts would cause the Mother star to fission and spew out a part of itself as a star cluster or even a dwarf daughter galaxy. Astronomer Halton Arp has catalogued many cases of such core ejections. These daughter bodies would orbit the spiral galaxy, forming a star cluster halo around the galaxy as well as spawning nearby galaxy progeny. Such is seen happening in our own Milky Way. Eventually, as a result of continued core ejections, the spiral galaxy would evolve into a giant elliptical galaxy. A number of prominent astronomers, such as Jeans, Ambartsumian, Sérsic, Arp, Hoyle, Narlikar and McCrea, have voiced the opinion that galaxy evolution involves a kind of continuous matter/energy creation that is most evident in galactic cores and that likely involves some sort of new physics. We might also add Edwin Hubble to this list of sympathizers. Hubble (1926) proposed an evolutionary galaxy classification sequence that had the shape of a tuning fork, which indicated that as an elliptical galaxy evolved from compact to lenticular morphology it would subsequently evolve into either of two spiral galaxy morphologies. In studying galaxies of a fixed total magnitude (mT = 10), he found that as one proceeded through the morphology sequence from compact E0 elliptical on the left to mature Sc spiral on the right, the major axis diameter of a galaxy progressively increased. Also a number of recent observations of how galaxies evolve as redshift decreases are compatible with the SQK matter creation prediction. For example, the data of Buitrago, et al. (2013) shows that massive galaxies exhibit an average 5.2 billion year doubling time over the past 18 billion years (here using the tired-light distance metric for converting redshift). Attempts to save the standard big bang cosmology paradigm by hypothesizing that galaxies merge with one another, however, fail to explain

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these galaxy growth findings (LaViolette, 2012, Ch. 8). Lopez-Corredoira (2010) has also concluded that galaxy merger theories fail to explain observation. More about how SQK and its continuous matter creation prediction is able to account for astronomical and cosmological observation can be found in chapters 7 and 8 of LaViolette (2012). Another benefit of SQK is that it has no need of adding dark matter assumptions as is done in standard astrophysics. Earlier we had noted that the gravity field behaved as a Newtonian field, having a 1/r potential decrease. However, a different situation presents itself at astronomical scales. The gravity field ultimately must taper off to a zero gradient outside a galaxy as the G-on concentration approaches the ambient steady-state concentration value present in intergalactic space. Thus gravity does not have an infinite reach as it does in standard physics. The departure of a star’s gravity field from Newtonian expectations would begin to become noticeable beyond a distance of 3 kpc, where its field gradient would begin a more rapid decline (LaViolette, 2012, Ch. 8). For this reason, there would be no long-range force influencing distant galaxies that might cause an initially static universe to collapse. So SQK resolves the gravitational potential summation problem that has been an area of difficulty for Euclidean static universe cosmologies. The gravity field tapering prediction of SQK parallels ideas presented in Milgrom’s modified Newtonian dynamics theory (MOND). Whereas the MOND theory emerged from the standpoint of observing stellar orbits in galaxies, the SQK gravity field tapering outcome emerges as a theoretical prediction of the Model G model. Various studies have shown that MOND is able to explain the rotation velocity profiles of spiral galaxies without the need of introducing assumptions about the presence of dark matter, one such study being that of McGaugh (2011). The gravity prediction of Model G should yield similar results.

THE PHOTON BLUESHIFTING EFFECT When the ether is supercritical, that is in regions where 𝜑𝑔 (𝑟) < 𝜑𝑔𝑐 , photon energy will progressively increase where Eqn (7) now becomes expressed as:

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case where g < 0. In effect,  now acts as a negative Hubble constant, dictating photon energy amplification, rather than photon energy damping.

Figure 19. Redshift-distance trend lines for galaxies in the vicinity of the Virgo cluster. The circle, triangle and square data points indicate the redshifts and distances of various galaxies in the Virgo cluster that have been transformed from the heliocentric frame into the cosmic microwave background frame. The vertical scatter of the sample is due to the peculiar velocities of the galaxies in the cluster. Tired-light photon redshifting temporarily changes to progressive photon blueshifting when photons traverse the Virgo cluster (diagram after S. Mei, et al.).

Such blueshifting would occur when a photon passes through a galaxy cluster where the gravity potential would adopt a negative value (LaViolette, 2012, 2021). One example of this can be seen in studying the Virgo cluster. Being 16.5 Mpc away, it is the closest massive galaxy cluster to our own Local Group (LG) galaxy cluster. This blueshift effect is apparent in Figure 19, which plots the data of Mei et al. (2007). Following the redshift distance relation (heavy black line) outward from a distance of 7 Mpc, the nearest graphed

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distance value, we see that initially this relation on their graph proceeds outward with a slope of 78 km/s/Mpc. Then around 4 Mpc from the cluster’s center, this redshifting trend reverses and turns into a blueshifting trend, with a slope of approximately -115 km/s/Mpc. Then, going further out, around 4 Mpc from the cluster’s center on its far side, this blueshifting trend tapers off and reverses to pursue the same redshifting trend of 78 km/s/Mpc. The vertical displacement of the two redshift-distance trend lines relative to one another, due to this intervening blueshifting, amounts to 850 km/s. Consequently, the total energy gain for the blueshifted photons in their passage through the cluster amounts to 0.28%. The fact that the extra cluster Hubble flow has an H0 value considerably greater than the 62 km/s/Mpc adopted over cosmological distances does not affect the present discussion. This cluster photon blueshifting can also alter the shape of galaxy clusters when they are plotted in redshift-distance space. For example, due to moderate blueshifting experienced by photons originating from peripheral galaxies, the cluster at its periphery will appear to be flattened or pancaked along the line-ofsight when plotted in redshift space; see Figure 20. This has been called the Kaiser effect. In addition, the significant blueshifting experienced by photons coming from galaxies located on the far side of the cluster and passing through the cluster center as they travel toward the Earth observer, causes the cluster at its center to appear elongated toward the observer when its redshift-distance values are plotted in redshift space. That is, galaxies in this region will be highly blueshifted which will reduce their overall redshift value relative to the cluster average, causing them to appear to be positioned much closer to the observer. This has been called the Fingers-of-God effect because it gives the impression that the central portions of these galactic clusters are pointing towards the Earth (in redshift-distance space). Cluster redshift space diagrams appear to point toward us regardless of which cluster in the sky they are located in. This phenomenon appears surreal to astronomers who have no good explanation in standard physics. In both the Kaiser effect and the Fingers-of-God effect, when the galaxy positions are mapped out in redshift space, this blueshift distortion will cause galaxies on the far side of the cluster to appear to be positioned on the cluster’s near side and vice versa. Thus the positions of the galaxies in the cluster will

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have the false appearance of being flipped front-to-back and back-to-front along the line of sight. While standard astronomy regards both of these effects as an unexplained mystery, they have a natural explanation in the paradigm of SQK. More about how SQK accounts for these phenomena with its photon blueshifting prediction can be found in LaViolette (2012) Ch. 7 and LaViolette (2021).

Figure 20. The Kaiser effect and Fingers-of-God effect created when photons coming from the far side of a galaxy cluster blueshift during their passage through the cluster’s supercritical region.

GENIC ENERGY SQK predicts that photons should blueshift their wavelengths in supercritical regions such as that surrounding the Milky Way and those surrounding all other galaxies as well. This blueshifting prediction then should dramatically impact our understanding of all stellar astronomical phenomena. In this respect, it is useful to calculate the excess energy, E = E(t) - E0 , produced as a result of photon energy amplification. This may be regarded as a new source of energy, previously unrecognized by physics, but just as important as the energy released from chemical reactions or nuclear fusion. SQK calls this spontaneously generated energy: genic energy. To calculate it, the spontaneous progressive increase of photon energy is best expressed as a function of time rather than distance. Hence Eqn. 10 would be changed to the equivalent form given as:

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Here 𝜇 = −𝛼𝜑𝑔 signifies the amplification coefficient where  is the proportionality constant, and where the c denominator has been absorbed in converting r to t. Differentiating Eqn. (11) yields the genic energy luminosity Lg: (12) That is, as a result of its continuous increase in energy, a photon would produce genic energy at rate Lg. By assigning a value of 7 X 10-33 s / cm2 to , Eqn. (12) is able to account for the intrinsic luminosities of jovian planets and lower main sequence stars. At the surface of the Earth, where g ~ -2.1 X 1014 cm2 / s2, this value for  yields a photon amplification rate of  = 1.48 X 1018 / s, which is equivalent to a 4.7% increase in energy every billion years. This is about 73% as large as the observed cosmological redshift attenuation rate, but of opposite sign. Also, it is less than half of the -115 km/s/Mpc blueshifting rate described earlier for the Virgo cluster which calculates to be  = -3.7 X 10-18/s. This created genic energy ultimately arises as a result of the continuous operation of the underlying subquantum reactions specified by the Model G equation system (Figure 3). Whereas the nineteenth century mechanical ether was an inert and inactive substance, the subquantum kinetics ether continuously transforms. Its reactions operate in a continuous state of flux, continuously building up the G, X, and Y concentrations that compose the physical universe. Although these unobservable ether reactions are assumed to behave in a conservative manner, the physically observable field amplitudes which they produce can behave nonconservatively as was shown above. One consequence of relation (12) is that the energy stored in a celestial body will evolve genic energy at the rate: (13)

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where H represents the body’s total heat capacity, given by the product of the ̅. body’s mass M, its average specific heat C̅, and average internal temperature T As before, the parameter φ ̅ g represents the body’s average internal gravity potential, the variable determining the prevailing degree of supercriticality. Based on Eqn. (13) one may conclude that all celestial bodies must produce genic energy and that the amount of their genic energy luminosity should vary in direct proportion to their mass. This in turn leads to the prediction that planets and red dwarf stars alike should generate genic energy and hence that their mass and luminosity values should lie along a common mass-luminosity relation. A test of this prediction confirmed it (LaViolette, 1992, 2012, Ch. 9). Mass and luminosity values for a sample of red dwarf stars belonging to the lower main sequence stars were plotted on a log-log plot along with the least squares fit performed by Harris et al. (1963); see Figure 21. That fit indicated an M-L dependence of L  M 2.76 ± 0.15. In addition, as a test of the SQK prediction, the mass and luminosity values for the jovian planets, Jupiter, Saturn, Neptune, and Uranus obtained from satellite IR data were plotted on the same graph. As is apparent, these were found to lie along this same M-L relation. This new result indicated that a common energy process must operate in both planets and stars. It could not be nuclear since jovian planets have insufficient mass to support nuclear reactions. It could not be stored heat acquired during primordial mass accretion because such heat should long ago have been radiated away due to the high luminosity of red dwarf stars. Genic energy, however, would be a viable candidate since the mass exponent for genic luminosity has an estimated value of 2.7 ± 0.9, conforming to the M-L exponent observed by Harris, et al. (LaViolette, 2012, Ch. 9). As a result, the planetary-stellar mass-luminosity relation presents an important confirmation of SQK. Sometime later, massluminosity values for a number of brown dwarfs were published, and these also were found to fall close to this same relation, further strengthening this SQK prediction; see again Figure 21. The presence of genic energy leads one to conclude that nuclear burning begins to occur in red dwarf stars at a higher mass than previously supposed, that is above about 0.45 M rather than above 0.08 M (LaViolette, 1992). Note that at this same mass of 0.45 M, the M-L relation takes an upward bend to form the steeper upper main sequence relation where it follows a power law of

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L  M4. Standard theoretical models place the convective-to-radiative core transition for stars at around 0.35 M . Now with the assumption of two energy sources being present, genic energy plus nuclear, this transition is expected to be pushed to a higher mass range (0.4 - 0.5 M). Hence this leads to the conclusion that the onset of the transition to a radiative core and steeper M-L slope is associated with the onset of nuclear burning. The luminosity difference between the upper and lower M-L relations indicates the added contribution of nuclear energy. Projecting this curve upward predicts that 16 ± 6 percent of the Sun’s energy should be of genic energy origin, with the other 84% being due to nuclear fusion.

Figure 21. The position of the jovian planets and several brown dwarfs shown in relation to an extension of the lower main sequence stellar mass-luminosity relation.

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It is also noteworthy that this higher mass nuclear onset coincides with an inflection point in the stellar luminosity function. Thus the luminosity function is actually bimodal. Its lower mass main lobe would be populated by stars powered only by genic energy, while its higher mass, more minor lobe would be populated by stars powered by both genic and nuclear energy (LaViolette, 1992). This indicates that most of the stars populating the Galaxy (the main lobe) are powered by genic energy and that nuclear burning is present only in the rarer more massive mature stars occupying the minor lobe. The SQK continuous creation cosmology requires that lower mass red dwarf stars eventually grow to become sun-like stars, then giant stars, and finally supergiant stars. This turns standard astronomy’s assumption about the ages of stars on its head. According to SQK, blue supergiant stars would be among the oldest and red dwarf stars among the youngest, just the opposite of what is conventionally assumed (LaViolette, 2012, Ch. 10). The assumption that a blue giant star would burn itself out after just a few million years becomes an absurdity since it continually materializes new hydrogen in its interior at a rate faster than it can be burned through nuclear fusion. As a result, in spite of its very high luminosity, a blue supergiant would keep growing and increasing its luminosity until it ultimately ejected its atmosphere, leaving behind a hot core, commonly called a white dwarf. White dwarfs would not be dead stars cooling off as astronomy conventionally believes. Instead, they would be stars that were almost entirely powered by genic energy. Their energy output would be so great that they would be unable to maintain an atmosphere. Nevertheless, they would continue to grow in mass and energy, eventually turning into X-ray stars, and finally into supermassive X-ray and gamma-ray-emitting stellar cores. These stellar cores need not be electron degenerate, as standard physics teaches, since the star’s genic energy output would be sufficient to support the weight of the star’s entire mass. As for neutron stars, there is a big question as to whether they even exist. Astrophysicist Sorin Cosofret (2019) has raised serious questions about the standard pulsar model and the existence of neutron stars. He has pointed out that if pulsars are neutron stars, their extreme mass and density should gravitationally bend starlight and gravitationally lens stellar images as they travel through space. However, no such stellar aberration phenomena have been

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reported around pulsars. Also it is theorized that there should be a vast pool of dead neutron stars, perhaps on the order of a billion, wandering through space invisible to telescopes which also should be deflecting starlight. But not a single instance of this has been reported. The neutron star concept, first proposed in 1934 by Baade and Zwicky, was not taken seriously by the astrophysical community until 1968 when Thomas Gold proposed the neutron star light house model to explain the pulses produced by the newly discovered Crab pulsar. Gold proposed that the Crab pulsar’s signals, which pulsed at 30 Hz, were produced by a radiation beam coming from the pole of a spinning neutron star. Only a star as compact as a neutron star (20 km) would make such a model sensible. For example, if an electron degenerate stellar core such as a white dwarf, were spinning at 30 Hz, its surface rotational velocity would exceed the speed of light, an unacceptable outcome for most physicists. Cosofret (2019) has effectively disproved the validity of the lighthouse model. Also in earlier publications, I have presented multiple reasons why the pulsar lighthouse model is invalid (LaViolette, 2000, 2005, 2016). There I show that the signals are better explained if the radio pulsar synchrotron radiation is directed towards us as a stationary beam and artificially modulated to produce the observed complex signal patterns. With stellar rotation no longer required, there is no need to assume their beams originate from cosmic-rayemitting neutron stars. In their stead, X-ray emitting stellar cores serve as adequate cosmic ray sources powering the beams. In the case of supernova explosions, SQK again offers a ready explanation for their source of energy. Namely, the enormous amount of energy that is released in a supernova explosion would be primarily genic energy. With the genic energy mechanism present, a stellar collapse phase may not be necessary. It may just be an issue of how fast a star is able to discharge energy from its core through radiation and convection. If the energy is unable to discharge fast enough, the amount of heat in the core would build up exponentially producing a rapid rise of temperature. As noted in Eqn (13), the rate of genic energy production is directly related to temperature. Thus if heat were not able to be convected or radiated outward sufficiently rapidly, a star would at some point enter an unstable feedback mode in which increased core temperature leads to increased genic energy output, which leads to increased core temperature, and

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so on in an ever increasing spiral. This nonlinear energy increase becomes ultimately relieved in the supernova outburst. Genic energy also offers a reasonable explanation for hypernova, extremely energetic supernova releasing energies greater than 1052 ergs. At such high energies, standard theory is hard pressed to offer a reasonable explanation. The type-II supernova SN 1987A, which was observed in the Magellenic Cloud in 1987, has demonstrated the inadequacy of the conventional explanation. According to conventional astrophysics, a type-II supernova occurs when a star is in its red supergiant phase and is assumed to have totally exhausted its supply of nuclear fuel. However, the progenitor star for SN 1987A was not found to be a red supergiant as expected, but an unusually luminous type B 3 blue supergiant known as Sandulek – 69° 202. Instead of being a star that was about to exhaust its energy supply and flicker out, it was seen to be a star emitting energy at a prodigious rate. Since this is the only supernova in which the nature of the precursor star was known to astronomers, we must assume, until proven otherwise, that this supernova progenitor was typical. As mentioned above, the conventional idea of a stellar collapse may not be required to explain Sandulek’s explosion. But in the possibility that its supernova was initiated by an inward collapse, radiation pressure produced by the skyrocketing temperature and genic energy luminosity would likely bring the collapse to a halt prior to the supernova. Black hole singularities, then, should be unable to form in SQK. Genic energy radiation pressure should increase according to 1/R4 during a collapse, whereas the inward pull of gravity causing collapse should increase only according to 1/R2 (LaViolette, 2012, Ch. 9). Hence because outward genic energy radiation pressure rises faster than the inward pull of gravity, a point is eventually reached as radius decreases where genic energy dominates and halts the collapse. Moreover even if one were to suppose that a collapsing stellar core were somehow able to become compressed to the point that subatomic particles became closely pressed to within half a Compton wavelength of one another, at such high densities the inward pull of gravity would vanish. This is because the gravity potential field contour at the center of a subatomic particle would plateau to a finite value (recall Figure 7), this “haystack” shape being backed

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by particle scattering data. So the classical physics notion that the energy potential gradient and force of attraction should approach infinity at the particle’s center is no longer tenable, neither in subquantum kinetics nor in recent determinations of nuclear structure. So a collapsing stellar core would theoretically be unable to attain the singularity state. Discoveries have recently been reported of stellar mass black holes having masses in the range of 3 M to 10 M which have no detectable X-ray emission. However, SQK interprets these as being low temperature stripped stellar cores which are kept from collapse by internal genic energy production. With improved astronomical measurements in the future, these objects should be found to produce low luminosity emission in the far UV to soft X-ray bands. Probably one of the best disproofs of the black hole theory comes from recent observations of the core of our galaxy, Sgr A*. Observations of Sgr A* at a wavelength of 1.3 mm made by Doeleman, et al. (2008) using long baseline interferometry show Sgr A* to be a luminous region having a diameter of 37 (+16, -10) microarcseconds. Adopting a distance to Sgr A* of 7.86 kpc, this luminous region would measure 0.29 AU in size. If Sgr A* were a black hole, it should have a gravitationally lensed Schwarzschild diameter of 0.42 AU given that it has a mass of 4.0 X 106 M . However, the observation of Doeleman, et al. shows a luminous region, not a black hole, and this luminous region turns out to be smaller than the event horizon diameter, about 70% of its calculated size. Standard black hole theory would hold this to be impossible since observation suggests that radiation is coming out from a region smaller than the size of Sgr A*’s assumed event horizon. The authors of the study attempt to circumvent the problem by claiming that this luminous region is a jet of material aimed towards us that lies outside of the event horizon in our direct line of site. The problem with this is that material in this central region is rotating very rapidly around the Galactic core, and no motion has been detected for this emission. Hence the radiation we are seeing coming from Sgr A* must be coming from the surface of a star-like body and not from a black hole’s event horizon. Based on the measurement of Doelman et al., an upper limit measurement of the radius of the Sgr A* celestial body would be 0.145 AU, or ~32 R . As a lower limit radius, Sgr A* must have a radius greater than 4.3 R , otherwise

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its radio emission would be so concentrated as to be self-absorbed. So the true radial size of Sgr A* must lie in the range 4.3 R < R < 32 R . Let us suppose, then, that Sgr A* has a radius of about 21 R . Given that it has a mass of 4.0 X 106 M , its density would calculate to be 600 g/cm3, much less than that of a low density white dwarf. More details about Sgr A* according to the SQK paradigm are given in LaViolette (2012), Ch. 8. Based on our observations of our own nearby core, Sgr A*, we may conclude that the supermassive objects observed at the centers of other galaxies in the heavens would not be black holes, but non-electron-degenerate stellar cores of extremely high mass and density. As mentioned earlier, SQK refers to such celestial bodies as mother stars. This name highlights the trait that a mother star would serve as the primary matter and energy birthing site in a galaxy, being the galaxy’s most supercritical region. That is, rather than being matter consumers, as astronomers typically characterize black holes, supermassive galactic cores would be matter and energy producers. They would tend to both grow in mass and to continually expel a portion of their continuously created matter to their surroundings. Here, the significant suggestion of Sir James Jeans (1928) comes to mind: ‘The type of conjecture which presents itself, somewhat insistently, is that the centres of the extragalactic nebulae [galaxies] are of the nature of ‘singular points,’ at which matter is poured into our universe from some other, and entirely extraneous, spatial dimension, so that, to a denizen of our universe, they appear as points at which matter is being continually created.”

Jeans comes very close to the idea that our universe functions as an open system, something that becomes most obvious when one considers the type of phenomena that take place in galactic cores. The tendency for mother stars to increase in mass as a result of their ongoing matter creation process would drive them to states of increasing supercriticality. As a result, the genic energy process would at times become very unstable and there would be instances in which the mother star would “explode,” that is, go into an active state where its energy output might abruptly jump 6 orders of magnitude or more. As in its quiescent state, in the active state it would be creating and discharging primarily genic energy. It is apparent that such objects

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would display some of the most egregious violations of physicist’s cherished First Law. Thus the genic energy hypothesis offers a ready means to explain not only extreme supernova, but also extremely energetic outbursts coming from galactic cores, which standard physics is at a loss to explain. Only, in this case, the celestial body would be much more massive and the outburst would take place on a much more energetic scale. Some physicists may feel that it is cheating to introduce such dramatic violations of the First Law to explain astronomical phenomena. But keep in mind this blueshifting phenomenon is not an ad hoc assumption. Rather it follows naturally from Model G which has had its predictions verified in many areas of physics. When so much that is otherwise puzzling to standard theory becomes explained in such a unified manner, there should be no reason to be concerned. Things become much simpler to understand when one realizes that we live in a universe that functions as an open system.

SPECIAL RELATIVISTIC EFFECTS Subquantum kinetics adopts the Lorentz transformations of standard physics, however, it interprets the effects of moving frames differently. Its interpretation conforms with that offered by the rod-contraction-clockretardation ether theory. For example, subquantum kinetics predicts that when a reference frame is moving relative to the ether rest frame, clocks in the moving frame will slow down, an interpretation that accords with the view that Lorentz expressed in 1909. It does not accept the special relativistic idea that the time dimension itself dilates in the moving frame. According to SQK, clocks and all physical processes proceeding in the moving frame should slow down because of the effect of the ether wind on reactions taking place in the moving frame. That is, such motion would increase the average relative velocity between etherons in the moving frame which would have an effect similar to reducing the values of the ether reaction rate constants. Such reduced reaction rates would cause clocks and oscillations to slow down, as if time had slowed down. Subquantum kinetics also predicts that the soliton wave pattern that makes up a subatomic particle should become compressed on the upwind side (on the side facing the direction of travel), and should become expanded on the

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oppositely facing downwind side. Although, this has not yet been demonstrated through computer simulation of Model G, it is expected that this will soon be shown. While the space structure of each individual subatomic particle would become skewed in this fashion, the distances separating adjacent particles in a molecular structure should not be expected to change much. In fact, experiments conducted at CERN which accelerate ions to relativistic velocities appear to substantiate this. It has been reported that the pancaking of the ions, expected on the basis of special relativity, has not been observed (Rak, 2019). While this result has not yet formally been announced, when eventually published, it should deliver a serious blow to special relativity theory. But special relativity has already been disproved based on findings which indicate that the speed of light can be surpassed. For example, Podkletnov and Modenese (2001) report sending a collimated scalar longitudinal gravity impulse across their laboratory at a speed of 64c. The electrogravitic impulse was generated by discharging a 2 Mv pulse from a Marx bank through a 10 cm diameter superconducting electrode that was magnetized such that magnetic field lines were oriented along the direction of beam propagation. In a later experiment, using an improved apparatus that discharged 10 Mv pulses Podkletnov (2007) reported that aided by atomic clocks he had determined the speed of these gravity impulses to be thousands of times the speed of light. Experiments that I had conducted with Guy Obolensky at his laboratory in 2005 and 2006 used 200 kv discharges to produce scalar longitudinal coulomb waves (electric field shocks) that propagated from the cathode at speeds as high as 6.5c (LaViolette, 2008, Ch. 6). Another example of superluminal propagation may be found in the work of Gasser (2016). He found that a 10 kv spark discharge between two spheres would produce a Coulomb shock that traveled at speeds ranging from 1.4 c to 5 c. LaViolette (2008) has reasoned that, in such pulse discharge experiments, the electric and gravity potential pulse, attains its superluminal velocity because it is riding an ether wind that propagates with the pulse in its direction of travel. Hence the observed pulse velocity may be expressed as v’ = c + v, where c is the velocity of light and v is the velocity of the forward propagating ether wind that it surfs upon. According to SQK, such pulses are scalar reaction-diffusion waves that move forward because their concentration gradients produce a forward etheric

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diffusion flux vector that raises the concentration of X, Y, or G in the forward direction, thereby causing the wave to shift forward. Such waves, termed scalar longitudinal waves, differ from Hertzian waves in that they have no transverse magnetic field component, i.e., no curl terms. A Hertzian wave would be created when a reaction-diffusion wave is modulated or polarized to have an oscillating transverse reaction-diffusion component (LaViolette, 2012, Ch. 6). NonHertzian scalar longitudinal waves are not predicted by classical Maxwellian electrodynamics as modified by Heaviside. But they are predicted when classical electrodynamics is modified into a new version called Extended Electrodynamics (EED); see Hively and Loebl (2019) and Reed (2019). EED is an improvement in that it resolves many of the flaws inherent in the classical version. Moreover, it is completely compatible with the gradient-driven scalar and Hertzian waves that SQK predicts. For a discussion of some of the flaws of Maxwell’s equations, see LaViolette (2012), Ch. 6 and references therein.

GENERAL RELATIVISTIC EFFECTS: EFFECTS DUE TO GRAVITY Gravitational Length Contraction, Mass Dilation, and Orbital Precession Dicke (1961) and Clube (1977, 1980) have shown that the standard tests of general relativity may be reproduced using Lorentz invariant laws of mechanics in flat space-time provided that inertial mass is allowed to increase and the speed of light is allowed to decrease with increasing gravitational potential according to certain specified relations. They propose the following variation:

(14)

where g = MG/r is the gravity potential expressed as a positive quantity and where m0 and c0 indicate the values of mass and the speed of light at a point

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remote from visible matter. In terms of subquantum kinetics, we might define m0 and c0 as representing m and c in intergalactic space. The above gravity potential expression effectively replaces that of general relativity which attributes orbital precession and the bending of light to a gravitational "warping" of space-time. Hence any (e.g., pulsar) test claiming to confirm the predictions of general relativity will also confirm this variable m and c theory. Such alterations of mass , length and temporal period are predicted by Model G. As the value of the bifurcation parameter G is decreased (i.e., as gravitational potential is made more negative), the Turing wave wavelength of a subatomic particle should become shorter and hence its inertial and gravitational mass should become greater. Accordingly, if a body were to move deeper into a gravitational potential well, its inertial and gravitational mass would correspondingly increase. If a planet such as Mercury, which has a highly elliptical orbit about the Sun, were to vary in mass, becoming most massive at perihelion and least massive at aphelion, its orbit should be found to precess as is observed. Future computer simulation of Model G hopefully will show quantitatively how soliton wavelength and velocity vary with gravitational potential.

Gravitational Clock Retardation According to SQK, a light clock in a gravity potential well would slow down, thus reproducing the effect known in physics as gravitational time dilation. The lower G concentration prevailing in the potential well would reduce the reaction rate in the X-Y ether reaction loop which, in turn, would cause a photon’s field potential to oscillate at a slower rate. By the same token, a photon traveling inside a gravity potential well would be expected to travel slower than normal. General relativity secures these related effects by maintaining that gravity in some unexplained way dilates the time dimension. In SQK the time dimension remains unchanged, the effect instead arising because the altered ether reaction rates reduce the photon’s frequency and velocity.

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Gravitational Redshifting The gravitational redshift phenomenon, which is most evident in the spectra of white dwarfs, is related to the gravitational clock retardation effect. Since light clocks situated deep in the central part of a gravity potential well would run slower than clocks residing at more peripheral locations, spectral line photons emitted at the surface of a star would have lower frequencies, be redshifted, relative to the frequency that they would have if those same spectral line photons had been emitted at a point distant from this gravity well. As these redshifted photons leave the surface of the star, their frequency and velocity would both increase, preserving their original emitted wavelength to be seen as a redshift.

Gravitational Bending of Light As mentioned above, as G is decreased, g becoming more negative, the frequency and wavelength of a propagating photon (reaction-diffusion wave) will decrease and its propagation velocity will be slowed. From this it follows that if a photon were to encounter a gravitational potential well, these effects would cause its trajectory to bend inward as if the photon were being pulled inward toward the center of the well. Hayden (1990) has shown that such light refraction by a gravitational potential field is able to completely account for the gravitational bending of light. General relativity, on the other hand, explains this same effect by assuming that the space-time fabric in the vicinity of a star is warped by the star’s gravity field, and thereby causes the photon’s space-time trajectory to bend. But how a gravity field causes space-time to warp is not explained. It is important to note that the Model G cosmology resolves the gravitational potential summation problem that has been an area of difficulty for Euclidean cosmologies such as those proposed by Dicke and Clube. This problem may be stated as follows. Namely, if gravitational potential fields extend to infinity, as is assumed in classical field theory, then it can be shown that the gravitational potential at any given point in space would approach infinity as the potential contributions were summed up over an infinitely large

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region of the universe. This in turn would lead to the absurdity that in all regions of space particle masses should be infinitely large and photon velocities infinitely small. This problem does not emerge in SQK. The finite extension of gravitational potential fields emerges as a natural corollary of the methodology. As mentioned earlier, the gravitational field of a celestial body would be expected to terminate at a substantial distance from the body as the prevailing G-on concentration plateaus to the steady-state local intergalactic value, G0 , determined by the homeostatic equilibrium established by input and output reaction kinetic processes.

TECHNOLOGICAL APPLICATIONS OF SUBQUANTUM KINETICS Subquantum kinetics has been usefully applied to explain several propulsion and over-unity energy production technologies which standard physics has been at a loss to explain (LaViolette, 2007, 2008, 2012). Propulsion technologies that have a direct relevance to the subquantum kinetics theory of electro-gravity include Townsend Brown’s electrogravitic thruster, the Podkletnov gravity impulse beam, and the Searl disc. The operation of other devices such as T. T. Brown’s asymmetrical capacitor thruster and the Nassikas superconducting thruster (Nassikas, 2012) may be understood in the SQK framework when it is realized that the electric field (potential gradient) and magnetic field (vortical ether currents) are not attached to their field sources but instead are seated in the ether. Hence their fields are able to exert unbalanced forces on the devices that produce them, allowing these devices to achieve propulsion without the action of outside forces. Both of these technologies, of course, violate Newton’s third law. One version of Brown’s asymmetrical electrokinetic thruster has been reported to have a thrust-to-power ratio of 70,000 Nwt/kw, over 4000 times that of the jet engine. Since a jet engine is known to have an efficiency of 15%, this projects an efficiency of over 26,000 percent for Brown’s thruster. So his technology in addition violates the First Law of Thermodynamics, something that is permissible in the SQK paradigm. SQK is also able to explain the operation of the Sun cell invented by Randal Mills of Brilliant Light Corp., which produces 1 MW of power by inducing the

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electron in the hydrogen atom to jump to orbits below the recognized ground state. Such transitions, thought to be impossible in standard physics, become highly probable when one substitutes the SQK Turing wave model of the electron in place of the trouble-ridden wave packet model of standard quantum mechanics. More broadly speaking, SQK provides a new paradigm for viewing the world, one in which over-unity energy generation and violations of Newton’s third law becomes the new norm rather than the enemy to be suppressed. Energy technologies such as Kim Zorzi’s over-unity Shauberger vortex air turbine, Walt Jenkins’ water gas combustion engine, and many others that are ready to change our way of life are even now waiting to be implemented. Field propulsion technologies such as those developed by Townsend Brown have been already implemented by the military but have been locked in secrecy since the late 1950’s. Meanwhile standard physics has frozen physicist’s thinking in a paradigm that has yet to produce any major breakthroughs in the last century other than nuclear power. It is time to realize that physics should not be left to university professors who sit in their armchairs spinning abstract string theories. Our daily lives are directly impacted by the unworkable theories sanctioned by physics that have failed to solve society’s mounting problems. It is time that we awaken to the fact that the physical world around us does not behave in the way physicists and astronomers have long taught us and to open ourselves to new possibilities such as the perspective offered by subquantum kinetics.

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In: The Origin of Gravity from First Principles ISBN: 978-1-53619-566-8 Editor: Volodymyr Krasnoholovets © 2021 Nova Science Publishers, Inc.

Chapter 8

DERIVATION OF GRAVITY FROM FIRST SUBMICROSCOPIC PRINCIPLES Volodymyr Krasnoholovets* Department of Theoretical Physics, Institute of Physics, Kyiv, Ukraine

ABSTRACT A submicroscopic theory of real physical space is outlined, showing that space is constituted as a mathematical lattice of primary topological balls, known as the tessellattice. The main parameter – mass – is introduced as a local fractal volumetric deformation of a cell of the tessellattice. Due to the interaction with ongoing cells of the tessellattice, a moving particle is desintagrated to a cloud of spatial excitations named inertons. So, the moving particle together with its inerton cloud is mapped to the quantum mechanical formalism as a particle’s wave function. Inertons oscillating around a massive object behave like standing spherical waves and establish a peculiar landscape in the tessellatice around the object. In the mean-field approximation the landscape looks like Newton’s gravitational potential -Gm / r . The tangential movement of a test object creates an additional term to Newton’s gravity proportional to u tang / c . Such a 2

2

submicroscopic gravitational potential allows one to solve all the problems predicted by the phenomenological theory of general relativity and also shows how *

Corresponding Author’s Email: [email protected].

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Volodymyr Krasnoholovets inerton physics can work out a number of new problems, in particular the emission of new unknown radiation from the Sun. Arguments are given in favor of introducing a new branch of astrophysics – inerton astronomy.

Keywords: space, tessellattice, fractals, mass, inertons, gravity

INTRODUCTION Textbooks of physics inform us that gravity is a force that makes things move towards each other and that gravitational theories deal with attraction of massive bodies. According to Newton, objects are attracted through the gravitational force. In Einstein-Hilbert theory, gravitation arises from the warping of space and time, such that a curvature instead of a force appears around a massive body. This implies that the curvature of space-time and the absence of the force of attraction radically violate Newton’s theory. Nevertheless, the first term in general relativity is Newton’s potential – Gm/r (see e.g., Bergmann, 1976), despite the fact that the origin of Newton’s force is unclear. General relativity suggests that matter bends space-time and in turn the bending creates a peculiar force that acts on the object under consideration. That is, the force appears because the object follows its geodesic path through spacetime. Hence a landscape described by the object’s geodesic path plays the role of the source of attraction. Two difficult problems remain unsolved. The first problem relates to the behavior of gravity at scales close to the particle’s de Broglie wavelength, where quantum mechanical laws start to become relevant. This includes smaller length scales which lay behind the quantum mechanical formalism. In other words, how does gravity arise for microscopic systems? The second problem concerns the definition of mass. In fact, the current framework of physics does not offer a rigorous definition of mass. This further inhibits our first-principle based understanding of gravity. We must begin by defining the concept of mass, because it is a mass m that bends the ambient space (or space-time). An exact formulation can be given only in terms of a physical theory that is beyond the quantum mechanical formalism, because the modern theory of quantum physics does not offer a selfconsistent definition of mass. Attempting to do so from a quantum-mechanical

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perspective, we find ourselves in an unknowable world that is inaccessible even to the intricate formal methods of physical mathematics, which actively promotes an abstract discipline known as quantum gravity developed mainly on the bifurcated foundations of strings and loops.

THE NOTION OF MASS It is evident that the physics of deep space must have its own nuances. In the 1860s and 1870s, Bernhard Riemann and Hermann von Helmholtz, when considering the movement of an object through space, noted that although the space around the object is curved, the object itself remains ‘rigid.’ In other words, they defined the first essential feature that distinguishes an object from space: the moving object has to be ‘rigid’ and the space has to be ‘soft.’ Clifford (1882) considered the creation of matter and its movement as the appearance of curvature of local pieces of space, which occurred in four stages: 1) small curved portions of space could be treated as matter because the ordinary laws of geometry are not valid in them; 2) a curved/distorted property could travel from one portion of space to another, like a wave; 3) this variation of curvature happens in the motion of matter; 4) in the physical world only this variation occurs. Michel Bounias (1990; Bounias and Krasnoholovets, 2003) suggested the constitution of physical space as something that initially consisted of objects and intervals. That is, it appears that from the mathematical point of view a solid itself is an element of space, and because of that a moving solid has to be considered as part of that space as well. An analogy can be made with water and ice. Although ice cubes float in water, they nevertheless remain water, but are in a different physical state. Set theory, fractal geometry and topology allowed us (Bounias and Krasnoholovets, 2003) to construct the real physical space as a mathematical lattice of primary topological balls. Such a lattice was named the tessellattice by M. Bounias; topological balls play the role of cells in the tessellattice. The tessellattice is empty if no deformations are available. But what types of deformations can exist in the tessellattice? In general they are:

284

Volodymyr Krasnoholovets   

a contraction of the cell; an inflation of the cell; a shift of the cell from its equilibrium position.

Furthermore, the tessellattice as a whole should possess elastic properties, since it is a substrate. We have been talking about primary topological balls whose dense lattice forms the tessellattice, thereby making up the inner structure of ordinary physical space. But what could be the size of such a ball, or cell, in the tessellattice? Logically, the size of a ball in a degenerate state can be related to the Planck length

m.

Our studies have shown that contractions and inflations of a cell could not be stable if they occurred without changes of homeomorphism (Figure 1a). A morphism with a dimensional change, which occurs upon infinitely repeated iteration, forms a stable local deformation in the cell under consideration (Figure 1b). A stable local deformation can be presented as a convoluted product of 3D space coordinates, which results in the appearance of an embedding D4 part as below   D4 =    d x d y d z  * d  ( w)  dS 

(1)

where dS is an element of space-time and dY(w) is a function accounting for the extension of 3D coordinates to the fourth dimension through convolution () with the volume of space. In such a way, the local deformation with dimensionality 3D + D  4D formed by means of volumetric fractal iterations becomes a rigid object in the unmanifested space. Hence a cell with such deformation is different from any other degenerate cell of the tessellattice (i.e., the unmanifested real physical space) and the objective characteristic of such difference is a ratio (3D + D)/3D > 1.

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Figure 1. Homeomorphic variation (a) and dimensional change (b) of a cell in the tessellattice.

In Figure 1b, at the ith iteration corresponding subvolumes (V i ) depend on the subvolumes of the previous iteration, which in the simplest case are related as Vi  Vi 1  (1/ r0 )3 where r0 is the characteristic linear size of a degenerate cell. The total volume occupied by the subvolumes formed by fractal iteration to infinity is the sum of the series V

frac



  (r

i 1k

0



  r )i  Vi 1  (1 / r0 )3 ,

(2)

which means that a fractal decomposition consists in the distribution of the members of the set of fractal subfigures. These subfigures are constructed on the primary cell and are similar to it.

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Volodymyr Krasnoholovets

The deformed ball as considered above provides a formalism describing the elementary particles. It follows that mass is represented by a fractal reduction of the volume of a ball. If V deg is the volume of a cell in the tessellattice (the degenerate state of a ball), then the reduction of volume resulting from a fractal concavity is V

particle

V

=V

particle

deg

-V

V

deg

frac

= DV > 1, or in line with expression (2)

   1   ( r0   r )i  Vi 1  (1 / r0 )3  . i  





(3)

Now we can introduce the notion of mass mB of a particled ball B as a function of the fractal-mediated decrease of the volume of the ball, i.e.,   mB  C  V /V  1    (r0   r )i  Vi 1  (1 / r0 )3   i   deg particle  C V / V  (eV  1)ev  1 deg

1

particle

(4)

where (e) is the Bouligand exponent and (eV - 1) depicts the gain in dimensionality given by the fractal iteration. Here, V

deg

>V

particle

and (eV - 1)

is positive and even more, (eV - 1) > 1. This means that the right hand side of expression (4) is positive and greater than unity; C is a dimensional constant. Thus, expression (4) defines the physical concept of mass from first principles.

DECOMPOSITION OF THE PARTICLE MASS TO INERTONS Since a particle is created in the environment of other similar cells, they have to react to its structure. If the particle is contracted following the law of fractal iterations, the surrounding cells must be stretched, that is, they must have a tension that keeps them in a tense state. Gradually, the tension of surrounding cells must decrease, and the entire tense region has to be characterized by a

Derivation of Gravity from First Submicroscopic Principles

287

certain radius l Com / 2 . Here l Com represents the Compton radius of the particle in question. The behavior of cells in this region, also known as the deformation coat (Krasnoholovets, 2017), obeys the Klein-Gordon equation (Christianto et al., 2019). The space beyond this region is the usual degenerate tessellattice. When the particle starts to move it must interact with neighbouring cells of the surrounding space. This movement implies an ongoing exchange of properties between mass and tension: step by step, the particle loses its fractals (i.e., fragments of mass) and passes them on to the neighbouring cells, and on their turn the cells pass their tension to the moving particle. Hence a cloud of spatial excitations appears around the moving particle. These excitations carry fragments of mass; they were termed inertons (Krasnoholovets, 2017) because they represent the force of inertia and appear owing to the resistance of the physical space (the tessellattice) to the movement of an object. As excitations of physical space, inertons migrate by a relay mechanism hopping from cell to cell, while the particle itself moves by squeezing itself between cells of the tessellattice. Since the moving particle loses its mass and momentum (and its velocity) due to scattering with ongoing cells, it must eventually stop. The section in which the particle emits all its mass through inertons corresponds to its de Broglie wavelength

. This region demonstrates the submicroscopic

dynamics of the particle (Krasnoholovets, 2017). After passing the section the particle loses its mass and becomes massless; however, it acquires a tension that coincides with the initial velocity vector of the particle. The motion of the particle’s inertons shows that they migrate through the tessellattice in the direction transversal to the particle path, up to the distance L = l dB c / u . This distance Λ can be termed the amplitude of the particle’s inerton cloud. Here, c represents the sound velocity of the tessellattice, also known as the speed of light, and u is the initial velocity of the particle. It should be noted that in the tessellattice, the longitudinal speed of sound 𝑐̂ (the speed of free inertons) may exceed the transverse speed of sound c (the speed of light); a preliminary estimate is (Krasnoholovets and Tane, 2004).

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Volodymyr Krasnoholovets

What happens to an inerton when it reaches the distance L ? The cell to which the inerton has arrived becomes shifted from its equilibrium position in the direction away from the particle. At this point the inerton stops – its velocity becomes zero but the tension in the cell reaches its maximum value. Due to the elasticity of the tessellattice, the cell is pushed back to its equilibrium position. This causes the tessellattice to launch the inerton back to the moving particle, which caused the excitation. While arriving back to the particle following a speed-dependent potential, the inerton loses the tension

and gains its mass

m. Upon arrival, the inerton passes the particle mass fragment m back to the particle, thereby restoring its original mass. The oscillating inerton motion is described by the equation (5) where

is the coordinate of the inerton and g is the elasticity constant of the

tessellattice. The equation shows that the initial kinetic energy of the inerton, when it is emitted by the particle energy g w 2 L 2 / 2 at the point

, passes to the potential , before being returned again to

m c2 / 2 and undergoing further mass-tension cycles. Here, the cyclic frequency is given by the elasticity of the tessellattice, which guides the inerton, and the inerton mass m (i.e., the degree of volumetric fractal contraction of the cell):

w = g / m. To show the migrational transformation of the inerton from mass to tension state, we have to construct the appropriate Lagrangian which includes both mass and tension terms, as well as a term accounting for the interaction of both states: . Here, the variables

and

(6)

characterize the mass and tension

of the inerton, respectively; l is the characteristic inerton wavelength (the de

Derivation of Gravity from First Submicroscopic Principles

289

Broglie wavelength in the case of a particle); m0 is the initial mass of the inerton and the initial velocity of the particle. Since the Lagrangian (6) contains the term , the Euler-Lagrange equations for variables m and are (ter Haar, 1974) ,

(where

), which leads us to the following differential equations in

explicit form (7) From these equations we obtain the typical wave equations of motion for each of the two variables (8) Therefore, the equations for the variation of the mass m and tension

,

which move in antiphase, show that an inerton moves as a typical wave. Stated otherwise, the fractal contraction of a cell (the mass m ) is periodically replaced by the stretching of the cell (the tension

).

The wave equations (8) allow two different solutions, corresponding to a standing and a travelling wave. For inertons which are emitted by a moving particle and then reabsorbed by the particle after reflection from a distant point Λ of the tessellattice, their trajectories look like strings attached to the point-particle. That is why formally, the appropriate solutions for the inerton mass and tension could look as follows

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Volodymyr Krasnoholovets ,

(9)

(10) where the wave number k = 2p / (2L) and the cycle frequency w = 2p / (2T ). Here, L is half of the spatial period (amplitude) of the oscillating inertons and T is half of the period of the standing inerton wave. At the point , which is associated with the position of the particle, the inerton mass is maximum distant point

the inerton mass becomes 0. The tension

oscillates in antiphase to the mass: at the point the point

; at the

the tension becomes

the tension is 0 and at

.

The solutions (9) and (10) assume the validity of the dispersion relation w = ck . The traveling wave solutions of equations (8) can then be written as ,

(11)

.

(12)

The solutions (11) and (12) describe the migration of a free inerton released from the particle’s inerton cloud. The small local contraction (the mass m ) and the corresponding small local stretching (the tension

) are two opposite

features of the inerton. The wave number k = 2p / l of the free inerton is defined by its wavelength l that has been imparted upon it at the moment of release from the particle’s inerton cloud. The same holds for the cyclic frequency w = 2p / T in which T is the period defined by the relationship c= L/T .

Derivation of Gravity from First Submicroscopic Principles

291

To summarize, we have considered how particle mass is decomposed to inertons and illustrated their characteristic properties, concluding that inertons exhibit wave attributes typical for waves in elastic condensed matter systems.

THE STANDING SPHERICAL INERTON WAVE In the previous section we have examined the behavior of inertons from a particle’s inerton cloud and showed that their motion can formally be described by a standing wave with solutions (9) and (10). Let us now consider the emission of such inertons in more detail. We start by fixing the frame of reference to the particle, so that the particle is stationary in the corresponding ). In this reference frame, the particle emits coordinate system (its velocity inertons in a sphere, and the inertons reaching the boundary located at a distance r = L are reflected back to the particle (Figure 2). The emission of inertons in the sphere occurs sequentially, starting with the 1st shell, followed by the 2nd shell, and so forth, up until the Nth shell. In addition to this, an order of emission is postulated – the inerton cloud unfolds like a spiral, which is caused by surface fractals that characterize the electric and magnetic properties of the particle cell (Krasnoholovets, 2017; 2019). In Figure 2, the spiral develops counterclockwise. Thus, inertons oscillate from the central point to the boundary of the sphere distant by L from the center. This means that the problem of oscillation of inertons can be reduced to the problem of radial vibrations of a gas in a sphere (Koshlyakov et al., 1970). As it follows from Euler’s hydrodynamics, sound waves in ideal liquids and gases are longitudinal, which makes it possible to describe them using a single scalar potential φ called the velocity potential. The velocity Then the vibrational speed is expressed in the form potential satisfies the wave equation (see expressions (8)). The oscillating behavior of the radial velocity potential u and the variables m and x coincides because of the same initial and boundary conditions, which for our case of inertons are:

292

Volodymyr Krasnoholovets m t=0 = m (r), ¶ m / ¶r r=L = 0,

¶ m / ¶t t=0 = F(r),

x t=0 = x (r),

¶x / ¶t t=0 = F(r),

¶x / ¶r r=L = 0.

(13)

Using the expression for the Laplace operator in spherical coordinates we can rewrite the wave equations (8) in the form

¶2 u 2 ¶u 1 ¶2 u . + = ¶r 2 r ¶r c 2 ¶t 2

(14)

Figure 2. Spherical oscillations of inertons around a particle.

Partial solutions are sought in the form u(r, t) = w(r)T(t) and the equations become

2 w¢¢(r) + w¢(r) + k 2 w(r) = 0 , r

(15)

T ¢¢(t) + k 2c2 T(t) = 0 .

(16)

The general solutions to Eqs. (15) and (16) are, respectively

Derivation of Gravity from First Submicroscopic Principles

w(r) = C1

| sin kr | | cos kr | + C2 . r r

293

(17)

T(t) = C11 | sin(w t) | + C22 | cos(w t) | ,

(18)

where w = kc . The modulus sign is needed because the values of mass and tension cannot be negative. Based on the initial and boundary conditions (13) and the conditions that at the point r = 0 the mass of the ith inerton is m |r=0 = m0 and its tension is x |r=0 = 0 , and at the point r = L the mass of an inerton is

m |r=L = 0 and its i

tension is x |r=L = xmax, i , we finally obtain the solutions for the inerton i

amplitudes of mass m i and tension x i :

m i (r, t) = m0

æ 2p r ö r01 cos ç , r è L i ÷ø

(19)

æ 2p r ö L sin ç r è L i ÷ø

(20)

x i (r, t) = xmax, i

where the minimum value of r is even less than the size of a cell, i.e., Planck’s length; r is limited by the constant r01 (Krasnoholovets, 2017; p. 350): the minimal length of the quantum motion of the particle is its Compton wavelength

l Com ; the number of inertons emitted by the particle while passing its Compton wavelength is l Com / lP  10 20 , and the constant becomes

r01  lP / ( Com / lP )  10 55

m.

Here ki = 2p / L i , Li = l dB c / ui and i = 1, 2, ..., N / 2 ~ 1025 . Recall that the particle velocity reduces gradually from

to zero in the section l dB due to

collisions with ongoing cells of the tessellattice, resulting in emission of approximately 1025 inertons.

294

Volodymyr Krasnoholovets The solutions for the entire inerton cloud look as follows:

m(r, t) = m0

r01

æ 2p r ö æ 2p t ö cos ç cos ç , ÷ r è L ø è T ÷ø

(21)

æ 2p r ö æ 2p t ö L . sin ç sin ç ÷ r è L ø è T ÷ø

(22)

X(r, t) = Xmax

These solutions show that the mass of any massive particle is distributed around it as a standing spherical wave. Such a distribution occurs in the interior of a sphere with radius L . The radius L describes the limit to which the quantum mechanical formalism is applicable; beyond L the macroscopic world starts, and quantum-mechanical rules transform to the rules of classical physics. In other words, the particle with its inerton cloud is mapping to the quantum mechanical formalism as the particle’s wave -function. In this context, a force in space can be described as arising during a length equal to the particle’s de Broglie length l dB , and reaching up to the amplitude of particle’s inerton cloud

L = l dB c / u . The de Broglie wavelength l dB of an electron in an atom is 10-10 m, hence L ~ 100 l dB ~ 10

-8

m. Thus L determines the distance to which the

gravitational interaction of the particle under consideration is able to propagate.

NEWTON’S LAW OF GRAVITAION Newton’s physics is the physics of the macroscopic world. Hence we have to move from considering a single massive particle to considering an ensemble of massive particles. A typical case is represented by a solid in which atoms are packed in a crystal lattice. In the lattice, atoms oscillate near their equilibrium positions. Any motion of a material object is accompanied by the appropriate motion of its inerton cloud (Krasnoholovets, 2017), and in a solid object the inerton clouds of neighbouring atoms overlap.

295

Derivation of Gravity from First Submicroscopic Principles

The spectrum of acoustic waves can be presented in the form of l n = 2 g n where g is the lattice constant; n = 1, 2, ... N / 2 and N is the number of entities (i.e., atoms) in the solid under consideration. Then the fundamental wavelength is l N = g N . In 1 cm3 of a solid there are about 1022 atoms. Then the fundamental wavelength for this piece of matter is l fund ~ 10 m. The 12

corresponding de Broglie wavelength for this fundamental acoustic excitation becomes Lfund = l fund c / u ~ 1018 m. In 1 m3 of a solid there are about 1028 atoms, then lfund ~ 1018 m and Lfund = l fund c / u ~ 1024 m, which is already approaching the visible radius of the Universe. In other words, L fund is the amplitude of the object’s inerton cloud and its extent is enormous. The object’s inerton cloud oscillates in such a way that the mass and tension change in the radial direction following the laws (21) and (22). Let us consider the behavior of the standing massive variable m(r, t) :

m(r, t) = m0

æ 2p r ö æ 2p t ö cos ç cos ç ÷ ÷, r è L fund ø è Tfund ø

r01

(23)

where m0 is the mass of the macroscopic object in question, and r is the distance from the object to the front of the object’s inerton cloud. We assume the dispersion law c = Lfund / Tfund holds in the cloud. Since any real distance

r