On Page 245 of book <<space time and geometry: an introduction of general relativity>> by Sean M. Carroll, the author gave the definition surface gravity and gave its property. I tried to do the calculation for several times, only to find the result different. So could you please take a look and tell me where I have make a mistake. Thanks a lot.(adsbygoogle = window.adsbygoogle || []).push({});

Provided:

$\chi^\mu\nabla_\mu\chi^\nu=-\kappa\chi^\nu$ (1)\\

$\nabla(_\mu\chi_\nu)=0$ (2)\\

$\chi_[\mu\nabla_\nu\chi_\sigma]=0$ (3)\\

Prove: $\kappa^2=-1/2(\nabla_\mu \chi_\nu)(\nabla^\mu\chi^\nu)$\\

I calculated it as follows:\\

from (2) and (3),\\

$\chi_\mu\nabla_\nu\chi_\sigma+\chi_\nu\nabla_\sigma\chi_\mu+\chi_\sigma\nabla_\mu\chi_\nu=0$ (4)\\

thus,\\

$\kappa^2\chi^\mu\chi_\mu=(-\chi^\mu\nabla_\mu\chi^\nu)(-\chi^\sigma\nabla_\sigma\chi^\nu)$\\

$=(\chi^\sigma\nabla^\mu\chi^\nu)(\chi_\mu\nabla_\sigma\chi_\nu)$\\

$=(\chi^\sigma\nabla^\mu\chi^\nu)(-\chi_\sigma\nabla_\nu\chi_\mu-\chi_\nu\nabla_\mu\chi_\sigma)$\\

$=(\chi^\sigma\nabla^\mu\chi^\nu)(\chi_\sigma\nabla_\mu\chi_\nu)+(-\chi^\sigma\nabla^\nu\chi^\mu)(\chi_\nu\nabla_\sigma\chi_\mu)$\\

$=\chi^\sigma\chi_\sigma\nabla^\mu\chi^\nu\nabla_\mu\chi_\nu-(\chi^\sigma\nabla_\sigma\chi^\mu)(\chi_\nu\nabla^\nu\chi^\mu)$\\

$=\chi^\sigma\chi_\sigma\nabla^\mu\chi^\nu\nabla_\mu\chi_\nu-\kappa^2\chi^\mu\chi_\mu$\\

thus,\\

$\kappa^2=1/2(\nabla_\mu\chi_\nu)(\nabla^\mu\chi^\nu) \hfil \square$

Sorry for being lazy, but I really hate it of typing the codes to this site again. I wondering why it is not compatible with latex. I hope you could view the picture I uploaded, if you do not want to copy the code into your editor and compile it.

PS: I am a freshman as for Latex, thus if I put any code not proper, please point them out and give me some advice, if possible.

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# Surface gravity calculation, where am I wrong?

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