I can show that Birkhoff's Theorem is wrong, and I think I might be able to prove it in a second independent way, which would require me to use that particular form of the Riemann Tensor.
Thanks.
I know that a person can get the Riemann Tensor in the form I want by substituting the metric and its first derivatives into the Christofell gammas in the standard Riemann Tensor formula.. As you noted, it is easy to make errors and that is why I would not trust me doing it, and would...
Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula.
Thanks.