General Relativity-surface gravity in killing horizon

[nikhilb1997]

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

Homework Equations

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

Homework Equations

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


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

 

[nikhilb1997]

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

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