Subject: Some More on the Basic Flaw in EH Geometry
Date: Mon, 31 Mar 2008 07:48:17 EDT
This emerged in computer algebra calculations in paper 93 and extra plots for paper 93. All our code is given on _www.aias.us_ (http://www.aias.us) because this is a very important development, so we openly give all our code so that others can check the EH flaw for themselves, using any desktop computer. To check our code and computer algebra we first showed that the equation:
R ^ q = 0
is obeyed by all correct solutions of the Einstein Hilbert equation. We looked at a number of different solutions, vacuum and other types. This equation is referred to in the standard model as the first Bianchi identity. We also checked that the vacuum solutions of EH indeed give a zero Ricci tensor in our code. An example is the so called Schwarzschild metric of the standard model. Steve Crothers has pointed out repeatedly that this solution was not in fact given by Schwarzschild in 1916, it has been incorrectly attributed to him. All the exact solutions of EH use the Christoffel connection, which is symmetric in its lower two indices. This is also known as the Riemann, and Levi-Civita connection. This connection is built up using the assumption of the symmetric metric and on the basis of the metric compatibility condition. EH theory is all built on these assumptions. In other words the existence of balck holes, dark matter, gravitational radiation, Big Bang, all precision tests of GR and all similar claims, are based on these assumptions of Riemann geometry without torsion. We next computed the quantity:
R tilde ^ q
in its tensorial form, R sup kappa sub mu sup mu nu. This tensor was built up from the Christoffel connection and symmetric metric, the same ones as used for R ^ q. It was found by computer algebra that this tensor is NOT in general zero for a Chrstoffel connection and symmetric metric. However it is identically equal by the Bianchi identity as developed by Cartan to a well defined covariant derivative of the torsion tensor. The latter is by fundamental definition zero for the Christoffel connection, so we found that a Christoffel connection does not obey the Bianchi identity
D ^ T tilde := R tilde ^ q ————————- (1)
and so the EH equation is meaningless. We went on to develop a meaningful theory of gravitational relativity and have just succeeded (paper 108) in explaining the orbit of a binary pulsar without gravitational radiation. LIGO shows that there is no gravitational radiation and there are hugely expensive plasns to build even bigger LIGO’s.
These are major advances of ECE theory, so we give all our code in all detail, so that anyone can check our method. What has happened is that the Bianchi identity
D ^ T := R ^ q ——————————— (2)
is obeyed by a Christoffel symbol because T = 0 for a Christoffel symbol, and R ^ q = 0 for a Christoffel symbol. This is due to the way in which the Christoffel symbol is defined in terms of a symmetric metric from the metric compatibility condition. However, by taking Hodge duals, eq. (1) can be inferred from eq. (2), and eq. (1) is NOT obeyed by the Christoffel symbol, because T tilde = 0 for a Christoffel symbol, but T tilde ^ q is NOT equal to zero for the same Christoffel symbol as used in R ^ q = 0.
So the EH equation is self inconsistent at a fundamental level because it neglects torsion (T = 0, T tilde = 0) and is based directly on the second Bianchi identity of the standard model:
D ^ R = 0 ————- (3)
in which T = 0, T tilde = 0. The correct Bianchi identity is (1) or (2), and the correct second Bianchi identity (paper 88) includes non-zero torsion:
D ^ ( D ^ T) := D ^ (R ^ q) ————— (4)
The EH field equation is essentially:
D ^ R = k D ^ T1 = 0 ——————– (5)
where k is Einstein’s constant and where T1 is a well defined energy momentum tensor.
So the essential point is that a Christoffel connection and line elements which are solutions of EH give zero R ^ q but non-zero R tilde ^ q . Both were evaluated for many exact solutions of EH using computer algebra. They are far too complicated to evaluate by hand. So there is no logical way out for the standard model except to adopt ECE theory as developed in papers 93 to 108.
So resistance to this computer logic is anthropomorphic and unscientific. I tis wel lknown by now that there is illogical resistance of this kind becasue a lot of fudning depends on the incorrect EH geometry. That is an unfortunate legacy of the twentieth century, and must not be allowed to inhibit progress.