Stokes and Spin Connection

The Stokes connection of the continuity equation in flow dynamics is shown to be a combination of spin connections of Cartan geometry, and the continuity equation shown to be the tetrad postulate of the density tetrad.

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The ECE Equations of Gravitation, the Density Tetrad

These are the four ECE equations of gravitation, introducing the density tetrad, the Newtonian limit and the static gravitomagnetic field (see papers 117 and 119). The density tetrad postulate is the continuity equation of flow dynamics, eq. (54).

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Derivation of the Navier Stokes Continuity Equation from Geometry

This is the derivation of the Navier Stokes continuity equation from the tetrad postulate. It is shown that the connection of the continuity equation is a spin connection of Cartan geometry.

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The Continuity Equation of the Navier Stokes System

It is shown that this equation is an equation of general relativity (eq. (12)) with a scalar valued connection gamma defined by eq. (11). The continuity equation is therefore a limiting form of the tetrad postulate, and the Stokes derivative is a covariant derivative of Riemann / Cartan geometry. The Navier Stokes system may therefore be derived straighforwardly as a limit of ECE theory.

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Looking Back at Twentieth Century Cosmology

In retrospect the era from 1915 to 2003 was a retrograde era in cosmology because its basic theory was incorrect. Not only that but it is irretrievably incorrect and unworkable, as argued in paper 139. Observational astronomy on the other hand has developed strongly and continues to provide a wealth of data. The basic problem is that the twentieth century cosmology was based from its very beginnings on a flawed knowledge of geometry. Its geometry is Riemann geometry, devised in the early nineteenth century and developed by Christoffel and others. Up to about the early nineteen twenties, the existence of torsion was not known, and the commutator method was not known. The commutator method shows that any symmetric connection is zero. So the whole of Einsteinian cosmology collapses because of its use of a symmetric connection. The latter was just guesswork introduced by mathematicians of around the turn of the twentieth century. Hardly any textbook of general relativity mentions torsion, whereas the latter is just as fundamental as the curvature. Torsion and curvature must always be non zero and the connection must always be antisymmetric in its lower two indices. Therefore all the lurid propaganda about black holes, big bang and dark matter is complete nonsense, it is an industry rather than physics. Intolerant editors who hardly ever read submissions automatically adhere to the hopelessly flawed cosmology. Prestigious chairs are made out of flawed marble cut statements, and the whole lot is hopelessly muddied by string theory. It is now known clearly that four dimensions as used in ECE theory are sufficient for a unified field theory. Some very strange pseudo theorems such as the Hawking Penrose singularity theorems came to dominate cosmology, but the data build up an overwhelming case against them. With the retirement of Stephen Hawking (and I wish him all the best), it is time to move on and build a simpler, error free, cosmology. The basics of this are already available in papers 122 onwards. I would say that string theory is mathematics, not physics at all, string theory is not applied anywhere except in string theory. It is not used in the rest of physics, not used in chemistry, and not used in engineering and medicine. Any good physics works itself through into other subjects, and string theory has failed to do this in half a century.

Paper 139 Posted

Paper 139 has been posted on the fundamental errors in the Einsteinian cosmology introduced by the use of an incorrect connection symmetry. It is simple to show that the connection must be antisymmetric in its lower two indices, as reviewed in paper 139. The latter contains an introduction designed to be understandable by the non specialist, followed by four technical sections giving examples of numerous errors in the Einstein field equation. The paper also points out that the Einstein field equation is hopelessly unable to describe galactic dynamics, and were it not for the fudge factor of “dark matter” it would have been abandoned forty years ago when astronomers discovered the velocity curve of a spiral galaxy. ECE theory describes spiral galaxies quite easily without the use of dark matter, and ECE contains none of the errors in geometry that Einstein made. These were errors that were made by mathematicians of Einstein’s era, because the commutator method of generating curvature and torsion was not known then. It appears that Cartan was the first to infer torsion in his differential geometry. Torsion and curvature are differential forms defined by the two Cartan structure equations. ECE theory is the first to realize fully that the connection is always antisymmetric. A symmetric connection as used in the old cosmology is always zero. The reason is that the connection in its lower two indices has the same well known antisymmetry as the commutator.

The Errors in Standard Geodesic Theory

Once it is realized that the connection must be antisymmetric, numerous sequential errors in the standard model of cosmology appear. The methods used by Einstein are quite easily shown to be incorrect and meaningless. The reasons are given in the attached note.

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Metrics of the Einstein Field Equation are Incorrect

Due to the fact that the Einstein field equation used the incorrect connection symmetry all metric solutions of the equation accumulated over the past ninety two years since its inception are also incorrect. This fact will be made clear in a forthcoming book: “Criticisms of the Einstein Field Equation” (Abramis in prep.). The first metrical solutions of the Einstein field equation are still those that are mainly in use to claim, quite wrongly, that there was a Big Bang, rejected by Hoyle and others as is well known to the general public. The first solutions to be discovered were by Schwarzschild in 1916. These solutions do not contain a singularity, a study of the original 1916 solutions will show that. Also, they are now known to be solutions to an incorrect equation, the Einstein field equation, so are meaningless. There is no basis therefore in the theoretical claim of Big Bang, and indeed books full of experimental evidence against it. Likewise the Friedmann Lemaitre Robertson Walker metric is incorrect, as shown in papers 93 and following on this site using computer algebra. Black hole metrics are shown to be incorrect in paper 120, again using computer algebra to work out the complicated formulae. A string theory metric was tested in paper 120 and shown to be incorrect. These incorrect ideas are kept going through the fact that major chairs of physics are held by the physicists of the old school. There has been a sudden and fundamental change of paradigm in physics, splitting the subject in two parts, ECE and standard model.

Hawking Penrose Singularity Theorems are Incorrect

These so called “singularity theorems” are incorrect because they use an incorrect connection symmetry. These are the so called “theorems” upon which Big Bang is based. They assert, quite wrongly, that there must be singularities which are physical, a contradiction in terms which should never even have been countenanced in physics. The Einstein field equation itself is based on the so called “second Bianchi identity”, which again assumed a symmetric connection. When the correct antisymmetric connection is used this identity no longer holds, it has torsion terms in it, and these will be written out in full in the forthcoming paper 139. These torsion terms are also mentioned by S. P. Carroll in chapter three of “Spacetime and Geometry – an Introduction to General Relativity” (Addison Wesley, New York, 2004). They are worked out in full detail for the first time in the books advertised on the ECE websites: notably the six volumes to date of M. W. Evans, “Generally Covariant Unified Field Theory” (Abramis Academic, 2005 onwards).

Antisymmetry of the Connection

This is probably the most important discovery in general relativity and cosmology in a hundred years of the subject. The structure of ECE theory is such that the antisymmetry works its way into dynamics and electrodynamics, as in papers 122 onwards of this site. The basic new law is that the connection takes the antisymmetry of the commutator. The latter is the round trip operator that generates the curvature and torsion. Both must be non-zero simultaneously. ECE theory is far ahead of the string theories that proliferate at present. It is almost certain that all types of string theory applied to gravitation use an incorrect connection symmetry. Added to this fatal error is the tremendous obscurity of string theory, based as it is on non-physical and unobservable dimensions.

FUNDAMENTAL_ERRORS_IN_THE_EINSTEIN_FIELD_EQUATION.doc