# General Relativity-surface gravity in killing horizon

1. Feb 21, 2013

### nikhilb1997

1. The problem statement, all variables and given/known data
Prove the following-

$$\kappa^2=-1/2(\bigtriangledown_{\mu}V_{\nu})(\bigtriangledown^{\mu}V^{\nu})$$
Given, the following,
$$\chi^{\lambda}\bigtriangledown_{\lambda}\chi^{\nu}=-\kappa\chi^{\nu}$$
$$\bigtriangledown_{(\mu}\chi_{\nu)}=0$$
$$\chi_{[\mu}\bigtriangledown_{\nu}\chi_{\theta]}=0$$

2. Relevant equations

$$\kappa^2=-1/2(\bigtriangledown_{\mu}V_{\nu})(\bigtriangledown^{\mu}V^{\nu})$$
$$\chi^{\lambda}\bigtriangledown_{\lambda}\chi^{\nu}=-\kappa\chi^{\nu}$$
$$\bigtriangledown_{(\mu}\chi_{\nu)}=0$$
$$\chi_{[\mu}\bigtriangledown_{\nu}\chi_{\theta]}=0$$

3. The attempt at a solution
I do not know how to start as the equation to prove has a raised covariant derivative. I tried to use the metric to lower it but I got stuck at how the metric would affect the equation. So please help.

2. Feb 22, 2013

### clamtrox

Take $$\chi_{[\mu}\bigtriangledown_{\nu}\chi_{\theta]}=0$$, use $$\bigtriangledown_{(\mu}\chi_{\nu)}=0$$ to get rid of half of the terms, then contract with $$\nabla^{\mu} \chi^{\nu}$$

3. Feb 22, 2013

### nikhilb1997

Thanks a lot. I did the first two steps but I didn't know what to do next. Contracting gave the required result.