Surface gravity calculation, where am I wrong?

[hjq1990]

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.

 

[grav-universe]

QUOTE BY hjq1990

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:
$$\chi^ \mu \nabla_ \mu \chi^ \nu=- \kappa \chi^ \nu \hspace{10 mm} (1)$$
$$\hspace{10 mm} \nabla(_ \mu \chi_ \nu)=0 \hspace{10 mm} (2)$$
$$\hspace{10 mm} \chi_{[ \mu \nabla_ \nu \chi_ \sigma]}=0 \hspace{10 mm} (3)$$
Prove: $$\kappa^2=-1/2( \nabla_ \mu \chi_ \nu)( \nabla^ \mu \chi^ \nu)$$
I calculated it as follows:
from (2) and (3),
$$\hspace{10 mm} \chi_ \mu \nabla_ \nu \chi_ \sigma+ \chi_ \nu \nabla_ \sigma \chi_ \mu+ \chi_ \sigma \nabla_ \mu \chi_ \nu=0 \hspace{10 mm} (4)$$
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,

$$\kappa^2=1/2( \nabla_ \mu \chi_ \nu)( \nabla^ \mu \chi^ \nu) \hspace{100 mm}\square$$


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.

END QUOTE

Hello. Does this look a little closer to what you were trying to post? I basically just replaced the $'s with [/tex] and [tex] and added spaces before the symbols so they show properly. You can use "quote" on this post to see what I did.