How to evaluate components of covariant derivative

Question

I am stumbled how to define covariant derivative, so I can evaluate components of expresions such as:

-\nabla_{\alpha} p(r) = (\rho(r) + p(r))a_\alpha + u^\sigma \nabla_{\sigma} p(r) u_{\alpha}
where a_{\alpha} = u^\tau \nabla_{\tau} u_{\alpha}

presume I have define metric and expresion for Christofel symbols of 2nd kind in usual way. Also how to make difference if nabla is acting on vector or covector?

Answer 1

Yes you will need to determine the components of the metric and Christoffel symbols first, following e.g. the Schwarzschild tutorial notebook. You then need to expand your covariant derivatives into partial derivatives and Christoffels. You can do that with a simple substitution rule if you only have vectors and co-vectors, or you can use a more generic routine to do arbitrary tensors; see the example at the bottom of

https://cadabra.science/notebooks/ref_programming.html

for more on that. This is something that will become available in the form of a package in 2.2.2 or 2.2.4.