**Solution to a problem with general relativity**

A Yang-Mills mechanism for quantum field theory exchange-radiation dynamics, with prediction of gravitational strength, space-time curvature, Standard Model parameters for all forces and particle masses, and cosmology, partly in advance of observations

This book is an updated and expanded version of a CERN Document Server deposited draft preprint paper, EXT-2004-007, which is now obsolete and can’t be updated there. Please see the additional new calculations and the duality between Yang-Mills exchange radiation and the dynamics of the Dirac sea spacetime fabric of general relativity (in chapter one).

**Abstract**

In the preprint EXT-2004-007, the observation was made that in space-time the Hubble recession of the mass of the universe around us can be represented either as (recession velocity, *v*)/(apparent distance in space-time, *s*) = Hubble parameter, *H = v/s,* or, equally well, as (recession velocity, *v*)/(apparent time past in space-time, *t*) = *v/t = v/(s/c) = cv/s = cH,* which is the *outward acceleration of the matter of the universe* as seen in our space-time reference frame (rather than how the universe might hypothetically appear if light and other fields travelled instantly, which cannot occur). Space-time was ignored by Hubble, which is why this fact was not recognised before. The immediate consequence is that we get an outward force for the big bang from this outward acceleration of matter, as given by Newton’s empirical second law of motion, *F = ma,* with *a = cH* and m the mass of the receding universe observable around us (because of various other considerations, such as an increase in density in space-time as we look to great distances and earlier eras of the universe, there are complexities which are analysed in chapter one). This outward force by Newton’s empirical third law of motion should be associated with an equal inward directed reaction force, which allows us to predict gravity as a local effect due to exchange radiation pressure due to the big bang. This prediction is substantiated because it is now proved that there are two distinct proofs which are dual of one another, one for material pressure (particles) and one for radiation pressure (waves). The result is a full prediction of empirically verifiable general relativity, not merely the inverse-square law, but the accurate prediction of the gravitational coupling constant *G* and the gravitational curvature produced by masses, as well as the elimination of all ‘dark matter’ and ‘dark energy’ problems from general relativity. The cosmological consequences of this mechanism go further, because the exchange radiation mechanism causes the big bang Hubble recession on large scales while causing gravitation and curvature on small scales. It unifies both electromagnetism and gravitation, in the process eliminating the unobserved Higgs mechanism for electroweak symmetry breaking. The 19 parameters of the Standard Model are all predicted by the simple replacement mechanism, providing a full and detailed prediction of strong, weak, electromagnetic and gravitational interactions. The author is aware now of a great deal of relevant independent work by other people, including, among others, Louise Riofrio, D. R. Lunsford (whose unification, see EXT-2003-090, of electromagnetism and gravitation by a space-time symmetry where there are three orthogonal space dimensions and a corresponding three time dimensions, leading him to prove the elimination of the cosmological constant, is a duality to the mechanism presented here), Thomas Love, Tony Smith, John Hunter, Hans de Vries, Alejandro Rivero and Carl Brannen.

[To be inserted here when book content is complete: Summary list of predictions and links to the places they occur in the body of the book]

**Acknowledgements**

Jacques Distler inspired the writing of this technical-level book by suggesting in a comment on Clifford V. Johnson’s discussion blog that I’d be taken more seriously if only I’d use tensor analysis in discussing the mathematical physics of general relativity. Walter Babin kindly hosted some papers on his *General Science Journal*, which is less prejudiced and thus more sceientific than a certain other glamorous internet archive, while editors at *Electronics World *printed them; Peter Woit, Sean M. Carroll, Lee Smolin and ‘Kea’ (Marni D. Sheppherd) discussed in various ways the facts on mainstream string theory propaganda. Edward Witten’s alternative idea, called stringy M-theory, turned out to be ‘not even wrong’. Thank you, Ed!

**Contents**

*Chapter 1:* The mathematics and physics of general relativity

*Chapter 2:* Quantum gravity approaches: string theory and loop quantum gravity

*Chapter 3:* Dirac’s equation, Spin and Magnetic Moments, Pair-Production, the Polarization of the Vacuum above the IR cutoff and It’s Role in the Renormalization of Charge and Mass

*Chapter 4:* The Path Integrals of Quantum electrodynamics, compared with Maxwell’s classical electrodynamics

*Chapter 5:* Nuclear and Particle Physics, Yang-Mills theory, the Standard Model, and Representation Theory

*Chapter 6:* Methodology of doing science: Edward Witten’s stringy definition of the word ‘prediction’; real predictions of this theory based purely on empirical facts (vacuum mechanism for mass and electroweak symmetry breaking at low energy, including Hans de Vries’ and Alejandro Rivero’s ‘coincidence’)

*Chapter 7:* Riofrio’s and Hunter’s equations, and Lunsford’s unification of electromagnetism and gravitation

*Chapter 8:* Standard Model mechanism: vacuum polarization and gauge boson field mediators for asymptotic freedom and force unification

*Chapter 9:* Evidence for the ‘stringy’ nature of fundamental particle cores (trapped, Poynting-Heaviside electromagnetic energy currents constitute static, spinning, radiating, charge in capacitor plates, the Yang-Mills exchange radiation being the zero point vacuum field)

*Chapter 10:* Summary of evidence, comparison of theories, limitations and further work required.

**Preface**

This errors in the unification of fundamental theories lie in the way general relativity is currently being used, particularly the continuum gravity source assumptions which are forced into the energy-stress tensor because you can’t mathematically use differential equations to model true discontinuities in practice. So the lumpiness of quantum field theory isn’t compatible with the continuum of general relativity for *purely mathematical reasons,* not physical reasons. It pays therefore to examine what is correct in general relativity, and to identify/isolate what is merely a statistical approximation. The errors are identified and corrected in chapter one, which leads to further ramifications for the rest of physics, that are analysed and solved in the rest of the book.

**Chapter One**

The mathematics and physics of general relativity

Until 1998, it was widely held that general relativity predicted a gravitational retardation in the recession of the most distant supernovas, which proved to be in contradiction to the observations of supernovae redshifts published that year by Perlmutter *et al*., and since corroborated by many others.^{1} However, in 1996 a mechanism of gravity had been advanced which offered an approach to predicting (uniquely) the universal gravitational ‘constant’, *G*, that resolves the problem and many others, including the flatness problem, the smoothness of the cosmic background radiation originating from 300,000 years after the big bang, Standard Model particle physics parameters, and the underlying mechanism for quantum field theory.^{2}

This chapter deals with the correct derivation and application of the Einstein-Hilbert field equation of general relativity, including quantum corrections that pertain to gravitational phenomena.

1.1 The mathematical physics of the Einstein-Hilbert field equation

The Einstein-Hilbert field equation, *R _{ab} – *½

*g*, of general relativity was obtained in November 1915 from solid mathematical and physical considerations. Einstein’s equivalence principle, that inertial accelerations and gravitational accelerations are indistinguishable, is one basis of the physical description of gravitation. Two other vital ingredients are non-Euclidean geometry, described by tensor calculus, and the conservation of mass-energy, which produces the complicated left hand side of the equation, specifically introducing the ‘- ½

_{ab}R = T_{ab}*g*’ term.

_{ab}REinstein’s first equated the curvature of space-time (describing acceleration fields and curved geodesics), to the source of the gravitational field (assumed to be some kind of continuum such as a classical field or a ‘perfect fluid’) by simply *R _{ab} = T_{ab}*. Here,

*R*is the Ricci tensor (a description of curvature based on Ricci’s developments to Riemann’s non-Euclidean spacetime ideas) and

_{ab}*T*is the stress-energy tensor.

_{ab}This simple equation, *R _{ab} = T_{ab}*, was wrong. It turned out that

*T*should have zero ‘divergence’ in order that mass-energy is conserved locally. The easiest way to describe this is by analogy to the Maxwell equation for the divergence of a magnetic field

_{ab}**B**, i.e., Ñ ×

**B**= 0. Because as many magnetic field lines radiate from the north pole of the magnet as from the south pole, and this means that the divergence of the field (which is simply the summation of the gradients of the field in the three orthogonal spatial dimensions), is always exactly zero. In the case of the stress-energy tensor,

*T*, the conservation of mass-energy density locally would be violated by, for example, the Lorentzian dependence of motion upon volume and hence upon the field density or the source of gravitation.

_{ab}

T_{ab} = r u_{a} u_{b}

Taking just the energy density component, *a = b* = 0,

T_{00} = r g ^{2} = r (1 – *v*^{2}/*c*^{2})^{-1}

Hence, *T*_{00} will increase towards infinity as *v* tends towards *c.* If, therefore, the curvature was equal to the stress-energy tensor, *R*_{ab} = *T*_{ab}, mass-energy is curvature is dependent upon the reference frame of the observer, increasing towards infinity as velocity increases toward *c.*

Einstein, in his 1916 paper ‘The Foundation of the General Theory of Relativity,’ recognised the need whereby ‘*The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. … The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant).*’ (Italics are Einstein’s own.)

In order to ensure that the source of the curvature describing gravitation is … [The ten chapters of the full book will be downloadable from a link at http://quantumfieldtheory.org/ when completed, shortly. It will replace the ramshackle, hit and miss compendium of ideas and calculations on pages like http://quantumfieldtheory.org/Proof.htm – which is where a hyperlinked index page for the new book will go – and the recent updates in numerous blog posts and comments, with a structured, *completely rewritten thesis,* eliminating repetitions and other annoying aspects of presentation.]

Illustration of physical mechanisms for exchange radiation in quantum field theory and the modification to the standard model implied therewith: SU(3) is OK, but SU(2)xU(1) and the Higgs mechanism are too complicated and SU(2) is rich enough (with a very simple mass-giving mechanism) to encompass the full electroweak phenomena, allowing the prediction of the strength of the electromagnetic force and weaker gravity correctly. So the standard model should be modified to SU(3)xSU(2) where the SU(2) has a mechanism for chiral symmetry and mass at certain energies, or perhaps SU(3)xSU(2)xSU(2), with one of the SU(2) groups describing massive weak force gauge bosons, and the other SU(2) is electromagnetism and gravity (mass-less versions of the W+ and W- mediate electric fields and the mass-less Z is just the photon, and it mediates gravity in the network of particles which give rise to mass). *It is simply untrue that electromagnetic gauge boson radiation must be uncharged: *this condition only holds for isolated photons, not for exchange radiation, where there is continual exchange of gauge bosons between charges (gauge bosons going in both directions between charges, an equilibrium). If the mass-less gauge bosons are uncharged, the magnetic field curls cancel in each individual gauge boson (seen from a large distance), preventing infinite self-inductance, so they will propagate. This is why normal electromagnetic radiation like light photons are uncharged (the varying electromagnetic field of the photon contains as much positive electric field as negative electric field).

If the gauge bosons are charged and massless, then you would not normally expect them to propagate, because their magnetic fields cause infinite self-inductance, which would prevent propagation. However, if you have two similar, charged massless radiations flowing in opposite directions, their interaction will be cancel out the magnetic fields, leaving only the electric field component as observed in electric fields.

This has been well investigated in the transmission line context of TEM (travsverse electromagnetic) waves (such as logic steps in high speed digital computers, where cross-talk, i.e., mutual inductance, is a limiting factor on the integrated circuit design) propagated by a pair of parallel conductors, with charges flowing in one direction on one conductor, and the opposite direction in the other. When a simple capacitor, composed of metal plates separated by a small distance of vacuum (the vacuum acts as a dielectric, i.e., the permittivity of free space is not zero), is charged up by light-velocity electromagnetic energy, that energy has no mechanism to slow down when it enters the capacitor, which behaves as a transmission line. Hence, you get the ‘energy current’ bouncing in all directions concurrently in a ‘steady, charged’ capacitor. The magnetic field components of the TEM waves cancel, leaving just electric field (electric charge) as observed. See the illustration in the previous post here.