General Relativity-surface gravity in killing horizon

[nikhilb1997]

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

Homework 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.

 

[clamtrox]

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

Homework 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.

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 $$\nabla^{\mu} \chi^{\nu}$$

 

[nikhilb1997]

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 $$\nabla^{\mu} \chi^{\nu}$$

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