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Main Question or Discussion Point
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.
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.
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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