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Fundamental Errors In The Einstein Field Equation




In this essay the fundamental errors in the Einstein field equation are pinpointed with precision for the impartial thinker. The academic subjects of general relativity and cosmology are so riddled with errors that they are meaningless, the academic system is a self perpetuating machine that funds lurid fantasies that are the antithesis of science. Science was defined by Bacon as being a subject that consists of the simplest possible hypotheses that can be tested against experimental data. In this book, several chapters have shown rigorously that the Einstein field equation is incorrect due to its arbitrary neglect of a fundamental property of spacetime called torsion. These chapters are not accessible to those without mathematical training, so in this essay an attempt is made to explain why the field equation is incorrect irretrievably. Matter of fact language is used, getting rid of the wild claims that saturate the academic literature on general relativity and cosmology, claims such as Big Bang, the existence of black holes, and the existence of dark matter. These are essentially anti Baconian, they are fantasies that can never be tested against data because the are just that - bad dreams that do not exist in nature.

The fundamental idea of general relativity is that physics, or natural philosophy, is based on geometry. This idea goes back to classical Celtic and Greek times, when geometry was considered the epitome of beauty. The idea was used for about a thousand years after the Greeks, right up to the time of Kepler around the turn of the seventeenth century. It was thought that the orbits of celestial objects were governed by the music of the spheres. Orbits were thought to be governed therefore by the beauty of geometry, in particular the circle. Real orbits were described by epicycles, circles added to circles. This idea was anti Baconian in the sense that nature was forced to conform to human ideal. This is precisely what is happening today in cosmology and general relativity, the general public is being told, quite cynically and wrongly, that there exist elements in nature that are figments of fantasy based on an incorrect equation.

There has therefore been a retrograde movement that has corrupted the scientific enlightenment brought about by figures such as Copernicus, Brahe, Galileo, Kepler, Bacon and Newton. These scientists and philosophers developed the fundamental methods of natural philosophy by developing a mathematical description of data. This was a long and slow process which culminated in the Newtonian synthesis. The equations of motion of all objects were described by three basic laws using mathematics that were correct within the context of its time. These mathematics were again based on geometry in Newton’s original development in a book called "The Mathematical Principles of Natural Philosophy". This book was based on experimental data meticulously gathered by astronomers such as Tycho Brahe and gradually synthesized with much effort into the three planetary laws of Johannes Kepler. Isaac Newton developed the mathematics to describe these laws as the title of his book suggests. It is not widely known even now that these mathematics were still based on geometry. However, the idea that nature and thus geometry were manifestations of subjective beauty was abandoned by Newton, geometry was used as a means of describing observation as demanded by the philosophy of Francis Bacon. This is known as "the idol of the cave" philosophy and is based on the ancient Greek philosophy of Plato. The word "idol" in this context is based on the classical Greek for "dream". The "cave" denotes the darkness of the human mind when unguided by experimental measurement, by data taken from nature. The human mind produces fantasies that become wilder and wilder, which is exactly what we see on our television screens today, Big Bang that never was, black holes that do not exist, dark

matter that is not there in reality. All these flow from a flawed geometry as shown in this book and in this essay.

Gradually the mathematical methods used by Newton were simplified and extended to include rotational motion. Many mathematical methods were developed in the seventeenth, eighteenth and nineteenth centuries by figures such Leibniz, Euler, Lagrange and Laplace. These extended the original Newtonian ideas into a subject known as "classical mechanics". In the context of cosmology "Celestial Mechanics" by Laplace was a high point of the enlightenment. Later, in the nineteenth century, Hamilton profoundly added to the subject. It was thought that the ideas of Newton, albeit extended, were adequate for the description of nature. However, all that changed dramatically in the late eighteen eighties following an experiment by Michelson and Morley which contradicted "common sense". An experiment which showed that the speed of light behaved in a way that was not compatible with Newtonian ideas. Around the same time Oliver Heaviside produced his vector equations of classical electrodynamics form the earlier quaternion equations of James Clerk Maxwell. The Maxwell Heaviside equations were not compatible again with Newtonian ideas, so classical dynamics and classical electrodynamics obeyed fundamentally different laws. The problem arose of reconciling what appeared to be two entirely different subjects of physics.

Following upon the results of the Michelson Morley experiment in about 1887, Oliver Heaviside began to correspond with George Francis Fitzgerald on how to produce a theory to explain the perplexing result of the experiment, that the speed of light appeared not to vary in different directions or frames of reference. The subject of relativity was brought into being by these discussions between Heaviside and Fitzgerald in the late eighteen eighties. The subject was later developed by many scientists, notably Henrik Anton Lorentz and Henri Poincare, who began to implement the then new tensor calculus. Around the turn of the twentieth century (1900) the Maxwell Heaviside equations were put into tensor format using the Lorentz transformation from one frame of reference to another. The Lorentz transformation is generally considered to be the key equation of special relativity, a subject that is defined by classical electrodynamics. This is because the Maxwell Heaviside equations obey the Lorentz transformation, not the Galilean transformation of the Newtonian laws. The Lorentz transform implements a vector in four dimensions, three space dimensions and one time dimension. Therefore time is no longer independent of space as in the Newtonian ideal. Spacetime is not an idea that was introduced by Einstein, as is commonly believed by the general public, it was introduced by several scientists, notably Heaviside and Lorentz.

So special relativity existed well before Albert Einstein arrived on the scene. The equations of special relativity may be thought to be the Heaviside equations of classical electrodynamics, written down around the time that Einstein was born in 1879. Einstein’s contribution to special relativity was to put the finishing touches to the work of those who had gone before him, as Einstein himself frequently mentioned. The big problem is that Einstein has been elevated into a kind of half-god in the public eye, and this process has led to a corruption of the scientific enlightenment. The tendency is to take an equation by Einstein and to try to prove endlessly and without purpose that that equation must be correct instead of finding the flaws in Einstein’s work and constructively correcting them as in this book. The contributions made by Einstein in 1905 were to propose that the speed of light is the same in a frame that moves at constant velocity with respect to another, and to propose, effectively, that the Lorentz transform applies to classical dynamics as well as to classical electrodynamics. This was therefore an early unified field theory, giving a unified view of part of classical dynamics and of electrodynamics. Later, Einstein also proposed that there exists a relativistic linear momentum. This is the idea that actually leads to his famous rest energy equation, that rest energy is mass multiplied by the square of the speed of light.

Around the same time, Minkowski spacetime was developed, a concept which simplifies the equations of special relativity. Horst Eckardt has recently uncovered several flaws in Einstein’s interpretation of his own equations of special relativity. The ideas by Eckardt are correct, but the less enlightened part of the contemporary academic world is so ossified that it refuses to countenance that Einstein could make mistakes. Einstein himself admitted frequently that he could make mistakes, and frequently corrected them in his papers.

Minkowski spacetime is frequently known as flat spacetime, and this idea suggests automatically that there can exist spacetimes that are not flat. These are the spacetimes of general relativity, in which physics is thought to be governed by geometry again. This time though, the equations based on this notion must be such that they can describe all of dynamics, now known as relativistic dynamics instead of classical dynamics. Specifically the notion of acceleration of one frame with respect to another has to be incorporated into relativistic dynamics because in special relativity a frame moves with a uniform velocity with respect to another, and does not accelerate. Einstein’s primary contribution, and the only one that has lasted the test of about ninety years of science history, is that general relativity can be based on geometry. This means that tensor equations retain their format in any frame of reference, i.e. in a frame that moves in any way with respect to another. This is known as the principle of covariance. It was applied originally by Einstein to dynamics, but not to electrodynamics, introducing a basic schism in physics. This schism has only recently been bridged by the emergence in 2003 of Einstein Cartan Evans (ECE) unified field theory, in which all the equations and laws of physics are generally covariant. This means that they retain their format in a frame of reference moving arbitrarily with respect to another.

The problem faced by Einstein and his contemporaries was what geometry to use. Minkowski geometry was known to them as being flat spacetime, so what represents non-flat spacetime? This question may seem bizarre to the uninitiated, but it is designed to find what represents acceleration in general relativity, a subject which is itself designed to ensure complete objectivity in physics. Complete objectivity is the Baconian ideal: the description of nature must be free from any anthropomorphic influence and that description is not thereby an idol of the cave. In the early nineteenth century Riemann had proposed the geometry that was ultimately used by Einstein having been introduced to it by mathematicians. In Riemann’s geometry two fundamental tensors are used to describe the way in which any geometry departs from the flat geometry of Minkowski. The latter is in itself a unification of the flat geometry of Euclid merged with time. These fundamental tensors are torsion and curvature. Einstein began to go wrong in several ways, the fatal mistake made by Einstein and all his contemporaries was to throw out the Riemannian torsion. This has led to a catastrophic corruption of the enlightenment, because the error has been repeated in academic physics, (the optimistically named "standard model" of physics) for more than a hundred years.

The torsion and curvature tensors are defined by the action of an object known as the commutator of covariant derivatives on any kind of tensor in any kind of space and any dimension. General relativity is restricted to four dimensional spacetime. The commutator is an operator, meaning that it must act on a tensor. It is sufficient to consider the commutator acting on a vector in four dimensions. The commutator is associated with two indices denoted as subscripts. When these indices are interchanged, the commutator changes sign. When the indices are the same the commutator is zero, it vanishes and in mathematical parlance is said to be a null operator. The latter acts on the four vector to produce zero torsion and zero curvature. When the two indices of the commutator are different the commutator acts on the vector to produce both non-zero torsion and non-zero curvature. The structure of the defining equation is such that the torsion and curvature must BOTH be non-zero. The key

point is that it is not possible to assert (or claim illogically) that there can be a non-zero curvature and a zero torsion. Unfortunately, this is the error committed by Einstein and his contemporaries, and repeated until the emergence of Einstein Cartan Evans (ECE) theory in 2003.

The error made by those contemporaries of Einstein now appears to be so glaring that one wonders why it was ever made. The error amounts to the incorrect assertion that there can be a non-zero commutator with equal indices. It is as simple as that. The only possible type of non-zero commutator must have indices that are different, and it must change sign when these indices are switched. It is said to be antisymmetric in its indices. This error generates many sequential errors, and a combination of all of them makes the Einstein theory completely unworkable and obsolete. The object that produces curvature and torsion is known as the connection. The most glaring sequential, or secondary, error in the twentieth century development of general relativity is that the connection was erroneously claimed to be symmetric in its lower two indices, whereas the correct mathematics shows that the connection has the same antisymmetry as the commutator. The whole of the Einsteinian method collapses in consequence of this error. This becomes clear as follows. After many false turns, Einstein finally decided to use an equation of Riemann geometry known to him as "the second Bianchi identity". Contemporary scholarship has shown that this equation holds if and only if the connection is symmetric. It is not a true identity because the torsion is omitted incorrectly. The arbitrary and incorrect choice of a symmetric connection means that the torsion is incorrectly zero while the curvature is incorrectly non-zero. The second Bianchi identity was made proportional through the Einstein constant to the covariant Noether Theorem - the conservation of energy / momentum theorem in non-Minkowski spacetime. A particular choice of integration of this assumed proportionality gives the fabled Einstein field equation.

The rest of twentieth century relativity rests on solving this incorrect equation and so the subject has caused great harm to scientific enlightenment. "Let Newton be and all is light", and now all is dark matter again. There are books full of solutions to an incorrect equation, each solution claiming, quite wrongly, to say something about physics. The use of an incorrect connection symmetry is the worst and most basic error, discovered by this author, and others have persistently criticized the field equation for over ninety years: for example Schroedinger, Dirac, Eddington and Levi-Civita. The latter frequently had to correct Einstein’s errors in Riemann geometry. With Ricci, Levi-Civita was one of the pioneers of tensors in about 1900. Historical scholarship is needed to explain why the developers of Riemann geometry in the early twentieth century made such a blunder as to use the wrong connection symmetry. Even more so, it must be explained why the error is repeated by academia and foisted on the unsuspecting general public in the form of TV shows and so on claiming the existence of lurid fantasies, the very idols against which Bacon warned. Part of the answer must surely be the incomprehensible and murky abstraction of academic physics and mathematics. So in the criticisms of this essay, the precise point of collapse of the academic cosmology has been pinpointed in the simplest possible way as described already. The error in the standard model is so blatant and so glaring that an intelligent thinker with no mathematical training at all can understand it.

For a year or so after the proposal of the field equation in 1915, no solution was found. Indeed Einstein thought it to be insoluble. In 1916, however, Schwarzschild published two papers which solved the equation analytically. In neither of these solutions did a singularity (or infinity) appear, as pointed out by Crothers in this book. Despite this, a solution was wrongly and cynically attributed to Schwarzschild, a solution with a singularity. This is the basis of the fantasy of "Big Bang", a derogatory term coined by Hoyle, as is well

known to the general public. Sir Fred Hoyle clearly did not take the idea seriously, the idea that the universe must have "started", and then started from a mathematically singularity. Shortly thereafter other closely related solutions were found by Friedmann, Lemaitre, Robertson and Walker (FLRW metric). As argued these are solutions to an incorrect equation, so are merely meaningless mathematics which should have been junked long ago. These solutions are expressed in terms of objects called metrics, which like the connection, measure the way in which spacetime departs from the Minkowski spacetime. General relativity was an obscure subject for some years after that until some bizarre solutions were proposed by Wheeler and coined "black holes". These are again solutions to an incorrect equation, so are meaningless to physics, in any context. The discovery of the way in which stars orbit in spiral galaxies finally showed the Einstein field equation to be unable hopelessly to describe the proliferation of experimental data in millions of galaxies. At that point in time, about forty years ago, cosmology disintegrated as an academic subject because of the introduction of "dark matter" to describe these galactic orbits and other data. It was cynically claimed that the universe consisted mainly of dark matter, snuffing out the Baconian enlightenment and introducing a concept that was and is as dark as the Baconian cave. In other words dark matter is just a fudge factor introduced at random, and not science at all. Dark matter introduces an appalling dichotomy into cosmology because the Einstein equation is still claimed, quite cynically, to be a precise descriptor of such things as solar system orbits, while at the same time it is abandoned as a descriptor of galactic orbits, abandoned in favour of a dark matter that is supposed to fill 95% or more of the universe. Into this absurd confusion stepped ECE theory in 2003, and ECE theory successfully describes all known orbits using the correct geometry.

One of the worst hikes pulled on the long suffering general public in the twentieth century was the claim that the Eddington experiment of the early twenties had "verified" the Einstein field equation through the observation of light bending. Eddington did not have the precision to make such a claim, later experiments proved that his type of instrument could not give reproducible results. Even within the Eddington experiment itself there were two (perhaps more) contradictory sets of data, only one seemed to verify the Einsteinian prediction that light should be bent by the sun by twice the Newtonian prediction. It seemed that J. J. Thomson arbitrarily and personally chose the data set that seemed to verify Einstein! The latter was catalysed into instant fame and ceased to be a fallible scientist. It is now known that no set of experimental data could ever have "verified" the Einstein field equation because of its glaring errors in geometry as argued already. In ECE theory therefore, new and simpler field equations have been produced based on a correct geometry, one that properly uses a non zero torsion and curvature. The bending of light by gravity (known technically as the relativistic Kepler problem) has been explained in a new way, using an orbital theorem based on spherical isotropy (uniformness) of spacetime.

The rigorously correct version of the flawed "first and second Bianchi identities" of the obsolete cosmology was given by Elie Cartan in the early twenties. This author has produced a new form of the Cartan identity using a mathematical method known as the Hodge transformation. This new identity is known as the Evans Identity, and proves itself without further ado. The Evans Identity plays a central role in this book, and shows that every single solution of the Einstein field equation fails because of the neglect of torsion. Both Cartan and Evans Identities are rigorously true and self checking - they prove themselves in that one side of the identity is precisely the same as the other when written out in a particular way. These rigorously correct identities are used as the basis for the field equations both of dynamics and electrodynamics, thus unifying physics for the first time in the shape of ECE theory. The latter has met with complete professional acceptance outside areas of vested

academic interest in failed cosmology. It is not in the interest of failed academic dogmatists to accept new reason, but these mindless zealots and anti-scientists are a tiny minority. Using site feedback data from computers it is now possible to measure precisely the impact of ECE theory, and to measure it in many ways. The impact is unprecedented and sustained, signaling the emergence of a major paradigm shift in physics.

The Evans Identity balances the covariant derivative of torsion on one side of the equation with a particular type of curvature on the other. The Identity may be used as in this book to test the many erroneous solutions that proliferate, solutions of the erroneous Einstein field equation. These solutions all assume that torsion vanishes, so the curvature tensor in the Evans Identity should vanish too. Of course it does not, the basic Einstein field equation itself is erroneous precisely because of its assumption of zero torsion, in other words its assumption of a symmetric connection. In order to demonstrate this to the impartial intellectual, many metrics were tested in preparation for this book. These metrics are all exact analytical solutions of the Einstein field equation. These metrics are the very basis for the lurid TV programmes on Big Bang, black holes and daleks. In this book all the main black hole and big bang metrics fail the test of the Evans Identity, so the Big Bang never existed, and there are books full of criticisms on it. The academic system that insists on a fundamentally erroneous physics has disintegrated into useless self glorification and has lost any authority it may imagine itself to have had over thought. There is never authority over individual thought. Time and time again in the course of history, dictators and bigots have learned this lesson.

The very simple method we have used to disprove the Einstein equation begs the question of why it has managed to survive for over ninety years in the teeth of so much criticism. Even stranger is the fact that millions are spent on spacecraft designed to "prove" an equation which is so hopelessly unable to describe the proliferation of galactic orbits throughout the universe. Indeed the latter has to be filled with dark fudge because of the failure of the field equation. Einstein himself was always uncertain about the basic validity of his equation, especially after discussions with Cartan in the twenties. These discussions showed Einstein that he had neglected torsion. At that point in time it should have been realized that the connection is antisymmetric and the Einstein field equation abandoned in favour of a torsion based cosmology. It is assumed that such a course of action was not taken because the commutator method of generating curvature and torsion simultaneously was either not known or not understood. The absurd notions of Big Bang and black holes were abandoned by Einstein in about 1939, in a little known paper that is not mentioned by contemporary TV zealots, our latter day corruptors of scientific enlightenment. It seems that self aggrandizement, thirst for fame and money, has taken over academia when it comes to general relativity and cosmology. It is so easy now to make those lurid animations and mathematical garbage can be beautified by computer.

In sinister and dark speculation it may be that the field equation is being kept alive merely to acquire money from the general public. It is tied up in string theory to make it look impressive, but all the strings in the world will not mend the ghastly error of the commutator symmetry. A string theory metric that pretends to be a solution of the failed field equation has been tested in this book, and fails the test of the Evans Identity. So hammered into the mind of students is the field equation, warts and all, that any notion, any inkling, that it may be wrong is dismissed as lunacy. This is the traditional way of all bigots. This mentality was first described by Plato, on which Bacon based his idols of the cave philosophy, the foundation of all science. The inhabitants of Plato’s cave are able only to see the dancing flicker of shadows on the walls, this is their only perception of reality. One of their number wanders into the world outside, which is filled with sunshine. He is amazed at the beauty of

nature, and describes this beauty to the cave dwellers. They do not believe him, their world is one of murky shadows, fantasies and dreams of their own making. The nightmare that besets us scientists now is that failed equation which is plastered into the minds of the unsuspecting public. The beauty of nature deserves better than this. As Walter Pater wrote: "Many attempts have been made by writers on art and poetry to define beauty in the abstract, to express it in the most general terms, to find some universal formula for it." That formula is not the Einstein field equation.

The wild fantasies of the twentieth century - idols in physics, idols in mathematics, do not describe nature, they are concepts conjured in darkness. The present author describes these idols as unobservables, things that are invented but do not exist in nature. There are many examples of these: string theory, superstring theory, indeterminacy, virtual particles, things going backwards in time, things that happen without a cause, things that move at any speed, renormalization, dimensional regularization, entirely abstract gauge spaces, structured vacua, spontaneous symmetry breaking, the god particle, asymptotic freedom, confined quarks that are designed never to be observed, Big Bang, black holes, dark matter, dark flow, physically meaningful singularities, all are parameters which are adjusted to fit experimental data. Given enough adjustable parameters, anything can be fitted, and there is no science, only epicycles, idols piled upon each other in the darkest recesses of the human mind. Where is the beauty in that nightmare? If all these big words cost nothing, no one would take notice of them, they would be the ravings of a lunatic. In our times however they have come to dominate the public psyche to such an extent that tens of billions of dollars are burnt at CERN in search of that god particle. Such is the futile arrogance of human nature at a time when the species faces extinction.

Such is the vested interest in a ghastly error, that commutator again, that strenuous efforts are made by the worst of the bigots to prove that a commutator may be symmetric. Elaborate and false proofs, deliberate and repeated and sometimes published fraud, hyper-abstract mathematics, geometries that have have nothing to do either with Riemann or Cartan, and the more murky but time honoured methods of the totalitarian regime. All have been turned against the unfortunate and simple minded commutator. There is dark matter, dark flow, dark jelly, dark custard and dark cheese, but that failed field equation is still the number one idol. Gravity Probe B for example, wasted millions on trying to prove it, and found nothing. In the meantime ECE theory has described everything that the incorrect field equation of Einstein fails to describe. Many questions beg to be answered. Why do purportedly intellectual journals publish articles on an incorrect commutator symmetry? Why are new thinkers described as lunatics by the editors of these journals, editors who do not read articles, and send them to referees who do not read articles? Why is the almost universal acceptance of ECE theory dismissed by bigots who are allowed to extract so much taxation from the long suffering public? Why is urgently needed research into new energy so neglected for a god particle that cannot exist in nature? Does animation of garbage have such power over the individual human mind?

These are questions about human nature itself, does it prefer extinction to thought?

Evans of Glyn Eithrym,
British Civil List Scientist


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