Monday 30 September 2019

The Riemann tensor

I'm now reading section 3.6 about the Riemann tensor which expresses the curvature of a manifold. In component form it is$$
R_{\ \ \ \sigma\mu\nu}^\rho=\partial_\mu\Gamma_{\nu\sigma}^\rho-\partial_\nu\Gamma_{\mu\sigma}^\rho+\Gamma_{\mu\lambda}^\rho\Gamma_{\nu\sigma}^\lambda-\Gamma_{\nu\lambda}^\rho\Gamma_{\mu\sigma}^\lambda
$$I want to check some of Carroll's assertions and challenges.

  • Equation 3.111 which he said was very straightforward
  • Prove that the Riemann tensor really is a tensor
  • Showing that the Riemann tensor as a map is the same as its component form

I succeeded on the first and the last. The second sounds odd but what is needed is to use the transformation law for ## \Gamma##, which is not a tensor, and see if the transformed equation gives the proper transformation for the Riemann tensor. Basically tons of stuff needs to cancel. The first step gives this

and the second this (the line numbers on the left refer to the parts from above)

That was 22 terms in total. The green ones were the ones that were wanted. The rest had to vanish. I only succeeded in removing four.

On the way I relearnt a few techniques and so created the tensor tricks section of Important equations. I also learnt how to take a 'second order covariant derivative'. I had failed to spot the example Carroll gave in equation 3.111. It's quite interesting. The problem is to work out ##\nabla_\lambda\left(\nabla_\eta Z^\rho\right)##. If one was doing an ordinary second order derivative one would differentiate the inner part then differentiate the outer part. The reverse happens with a covariant derivative: One must first take the covariant derivative of the inner part and then calculate the covariant derivatives that are left over.
Full details at  Commentary 3.6 Riemann tensor.pdf 

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