Understanding Killing Vectors & Schwarzschild Geodesics

[George Keeling]

TL;DR

I think I now realise stuff about Killing vectors. Am I correct?

I'm on to section 5.4 of Carroll's book on Schwarzschild geodesics and he says stuff in it which, I think, enlightens me on the use of Killing vectors. I had to go back to section 3.8 on Symmetries and Killing vectors. I now understand the following:

A Killing vector satisfies $$
\nabla_{(\mu}K_{\nu)}=0
$$and that implies that $K_\nu p^\nu$ is conserved along a geodesic. $p^\nu$ is a 4-momentum.

Carroll says "In fact, if a vector $K^\mu$ satisfies Killing's equation, it will always be possible to find a coordinate system in which $K=\partial_{\sigma\ast}$". $x^{\sigma\ast}$ was the coordinate of which the metric was independent. That is $\partial_{\sigma\ast}g_{\mu\nu}=0$. So in GR if the coordinate was $t$ then we would have a $K^\mu=\left(1,0,0,0\right)$. If the metric equation does not contain $t$ it must have such a Killing vector. It must then satisfy the equation $\nabla_{(\mu}K_{\nu)}=0$ and that implies that that $K_\nu p^\nu$ is conserved along a geodesic.

A few lines further on he says
"As we investigate in Appendix B, Killing vector fields on a manifold are in on-to-one correspondence with continuous symmetries of the metric [isometries] on that manifold".

From the second statement (slightly stretched) we infer that a time isometry must imply a Killing vector $K=\left(1,0,0,0\right)$ and with a Schwarzschild metric a $\phi$ isometry implies a Killing vector $R=\left(0,0,0,1\right)$.

Is that all correct? I want to check before I proceed, thanks.

 

[Orodruin]

Whether or not a timelike Killing field has components (1,0,0,0) or not depends on your coordinate system. However, it is always possible to construct a coordinate system in such a way that those would be the components. Note that this says nothing about your actually chosen coordinate system. For example, it is not the case for Kruskal-Szekeres coordinates on Schwarzschild spacetime.

There are also other rotations from those generated by $\partial_\phi$ that are symmetries of the Schwarzschild metric. They will not have that simple form (but you can construct a new coordinate system where they do).