Thursday, 21 November 2019

Success at killing vectors

With help from Desmos
I had more success at killing vectors in the second part of section 3.8. We were asked to prove that a linear combination of Killing vectors with constant coefficients is still a Killing vector and also show that the commutator of two Killing vector fields is a Killing vector field. The first was indeed trivial as Carroll promised. The second was not. The problem is to show that, when ## A,B## are Killing vectors (##\nabla_\nu A_\lambda+\nabla_\lambda A_\nu=0## and ##\nabla_\nu B_\lambda+\nabla_\lambda B_\nu=0##) that, $$
\nabla_\nu\left[A,B\right]_\lambda+\nabla_\lambda\left[A,B\right]_\nu=0
$$The first problem was that I only knew (from way back) that the commutator can be expressed as $$
\left[A,B\right]^\lambda=A^\rho\partial_\rho B^\lambda-B^\rho\partial_\rho A^\lambda
$$which is not much use in this case. But it was easy to get to these two$$
\left[A,B\right]^\lambda=A^\rho\nabla_\rho B^\lambda-B^\rho\nabla_\rho A^\lambda
$$$$
\left[A,B\right]_\sigma=A^\rho\nabla_\rho B_\sigma-B^\rho\nabla_\rho A_\sigma
$$which are much better. I also needed a covariant form of the Riemann tensor which is$$
R_{\ \ \sigma\nu\mu}^\tau X_\tau=\left[\nabla_\mu,\nabla_\nu\right]X_\sigma
$$They all earned a place in Important Equations for General Relativity.

It was another 13 equations to get to the desired conclusion with evil side-tracks and errors on the way. I have not included those in the answer.

The answer does include much index juggling, including use of antisymmetries to really throw the Riemann tensor indices about. That was the final breakthrough. It's at
Commentary 3.8 Symmetries and Killing vectors.pdf (pages 4-6).

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