@ecterrab

Hi, I agree that the form I'm using is non-standard. This specific definition is used in Shankar's Principles of Quantum Mechanics and also in Dirac's original work (His textbook Fundamentals of Quantum mechanics use this convention). Shankar makes no specific reference to whether they are contravariant or covariant.  

Shankar:

Dirac's original: https://mathworld.wolfram.com/DiracMatrices.html (see eqns 20-23)

Thanks!

@ecterrab

Great thank you, we seem to be using a different convention in our lectures that still fulfil the anticommutation requirements so I'll just create a custom function.

Thanks! 

@ecterrab

Thank you for the in depth response and highlighting the error in the first equation. I will go away and get to work on what you have suggested.

Many thanks again! 

@ecterrab 

Hi there thanks for getting back. Ignore that line with Riemann[alpha,beta,mu,nu] etc., i was just experimenting and forgot to delte it out, oops.

I am effectively trying to reproduce the second equation shown in the below image by starting from the general expression from the Riemann tensor, lowering an index with the metric tensor and then use Riemann normal coordinates to work at a point on the manifold where I can treat it as being locally flat. This should make the connection terms zero but not their derivatives. All this should lead to the simpler second expression below in terms of the metric and its second derivatives. 

The tetrads package you mention looks like it might offer just what I need, I'll have a read.

Thanks again!