Subject: 93(4) : Double Checking Carroll
Date: Wed, 22 Aug 2007 05:33:00 EDT
Attachment: a93rdpapernotes4.pdf
These are the eight correct Christoffel symbols by Carroll, double checked line by line. My task next is to correct the remaining non-zero ones, which Carroll gives incorrectly, either due to typo’s or erroneous calculation. All the Christoffel symbols mustbe rigorously correct of coruse, otherwise there will be confusion among students. This exercise is an important baseline check in order to adopt Crothers’ calculations for further development. As can be seen, these eight symbols so far all have the correct dimensions of inverse metres. This will teach anyone interested enough to read these notes a lot about Riemann geometry. So Carroll and I agree on these eight symbols. Carroll does not however give any details, which I do here. In GCUFT this pattern is repeated, I fill in the details left out by Carroll, and meticulously cross check everything, sometimes many times. All these calculations are based on the Einstein Hilbert theory:
R ^ q = 0 D ^ R = 0
which as paper 88 shows, is a very restricted geometry. The correct Cartan geometry is
D ^ T := R ^ q
not zero in general. This is the one and only one Bianchi identity.