Fwd: Checking the metric condition of paper 128



Subject: Checking the metric condition of paper 128
Date: Sun, 22 Mar 2009 03:50:53 EDT

Many thanks! This is an important computer result that checks that the mathematics of paper 128 are correct and that the computer code is of course correct. Paper 128 therefore derives a new result in Riemann geometry, a new metric compatibility condition. The various errors in the Einstein field equation show up only through the use of the dual identity as in papers 93, 95 and 120. I will now proceed to write up paper 128 co-authored with Horst Eckardt.

I implemented the new test in the Maxima program and investigated all metrics with non-diagonal elements. It should be noted that diagonal metrics automatically fulfil this condition since the contravariant metric elements then are the inverse of the covariant ones, giving a constant for the sum in the new condition

partial sub mu g sup nu lambda * g sub lambda rho

for each tripel (mu, nu, rho). I checked this with a number of diagonal metrics. For the non-diagonal metrics the condition was always met, there was no irregularity. In detail the following metrics were tested: M09 Spherically symmetric line element with off-diagonal elements M23* Kerr M27* Goedel M41 Eddington-Finkelstein M65* Alcubierre M73 Anti-Mach metric of plane waves hf homogeneous vacuum (vacuum metric) M74* Petrov M76* homogeneous non-null e-m Fields, type 2 M78* homogeneous perfect fluid, Cartesian M79* Petrov type N M91* Collision of plane waves

(The numbers are irrelevant, is my internal numbering). The details can be looked up in the Cambridge University book, this could be referenced in the paper.


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