Einstein's Vacuum Exploring the Metric & Killing Vectors

[Pouramat]

Homework Statement

Let $K$ be a Killing vector field. Show that an electromagnetic field with potential $A_\mu = K_\mu$ solves Maxwell's eqs if the metric is a vacuum solution to Einstein's Eqs.

Relevant Equations

N/A

Einstein's vacuum solution metric:
$$
ds^2 = -(1-\frac{2GM}{r})dt^2 +(1-\frac{2GM}{r})^{-1}dr^2+r^2 d\Omega^2
$$
which $g_{\mu \nu}$ can be read off easily.
metric Killing vectors are:
$$
K = \partial_t
$$$$
R = \partial_\phi
$$
How can I relate these to Maxwell equation?

 

[George Jones]

Homework Statement:: Let $K$ be a Killing vector field. Show that an electromagnetic field with potential $A_\mu = K_\mu$ solves Maxwell's eqs if the metric is a vacuum solution to Einstein's Eqs.
Relevant Equations:: N/A

Einstein's vacuum solution metric:
$$
ds^2 = -(1-\frac{2GM}{r})dt^2 +(1-\frac{2GM}{r})^{-1}dr^2+r^2 d\Omega^2
$$
which $g_{\mu \nu}$ can be read off easily.
metric Killing vectors are:
$$
K = \partial_t
$$

You are not supposed to assume this specific vacuum solution, you are supposed to assume a generic vacuum solution. Note that this exercise comes before the chapter on the Schwarzschild solution.