I want to prove the fundamental theorem of Killing forms, namely that

$$\frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) = 0 $$

If $P(\lambda)$ is a Geodesic curve, which implies that $\dot{P}^{\mu} \xi_{\mu}(P(\lambda))$ are constants of geodesic motion

This should be straightforward to prove, basically expanding the derivative expression

\begin{align*}
\frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) &= \\
&=\frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))+\frac{d P^{\mu}}{d \lambda}\partial_{;\nu} \xi_{\mu}(P(\lambda)) \frac{d P^{\nu}}{d \lambda} \\
\end{align*}

We now use the fact that $\xi_{\nu}$ is a Killing form, that is:

$$\partial_{;\nu} \xi_{\mu}(P(\lambda)) = - \partial_{;\mu} \xi_{\nu}(P(\lambda)) $$

And we expand the covariant derivative:

\begin{align*}
\frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) &= \\
&=\frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))-\frac{d P^{\mu}}{d \lambda}\partial_{;\mu} \xi_{\nu}(P(\lambda)) \frac{d P^{\nu}}{d \lambda} \\
&= \frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))-\frac{d P^{\mu}}{d \lambda} \Big[ \partial_{\mu} \xi_{\nu}(P(\lambda)) - \Gamma^{\theta}_{\mu \nu} \xi_{\theta}(P(\lambda)) \Big] \frac{d P^{\nu}}{d \lambda} \\
&= \frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))+ \Gamma^{\theta}_{\mu \nu} \xi_{\theta}(P(\lambda))\frac{d P^{\mu}}{d \lambda}\frac{d P^{\nu}}{d \lambda} - \partial_{\mu} \xi_{\nu}(P(\lambda)) \frac{d P^{\mu}}{d \lambda} \frac{d P^{\nu}}{d \lambda}
\end{align*}

But

$$\Gamma^{\theta}_{\mu \nu}\frac{d P^{\mu}}{d \lambda}\frac{d P^{\nu}}{d \lambda}= - \frac{d^2 P^{\theta}}{d \lambda^2} $$

Since the curve is a geodesic, which means that the expression simplifies:

\begin{align*}
\frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) &= \\
&= \frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))- \frac{d^2 P^{\theta}}{d \lambda^2} \xi_{\theta}(P(\lambda)) - \partial_{\mu} \xi_{\nu}(P(\lambda)) \frac{d P^{\mu}}{d \lambda} \frac{d P^{\nu}}{d \lambda} \\
&= - \partial_{\mu} \xi_{\nu}(P(\lambda)) \frac{d P^{\mu}}{d \lambda} \frac{d P^{\nu}}{d \lambda}
\end{align*}

So, I am able to get rid of those two terms, but there is still an uncancelled term with the coordinate derivative of the $\xi$ form. I don't know how to proceed next

Suggestions?

This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher