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General Relativity-surface gravity in killing horizon

  1. Feb 21, 2013 #1
    1. The problem statement, all variables and given/known data
    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]



    2. Relevant 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.
     
  2. jcsd
  3. Feb 22, 2013 #2
    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]
     
  4. Feb 22, 2013 #3
    Thanks a lot. I did the first two steps but I didn't know what to do next. Contracting gave the required result.
     
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