- #1
darida
- 37
- 1
Homework Statement
Metric signature: [itex] - + + + [/itex]
Schwarzschild metric:
[itex]
dS^{2}=-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}(sin\theta)^{2}d\phi^{2}
[/itex]
Second fundamental form:
[itex]
h_{ij}=g_{kl}\Gamma^{k}_{ij}n^{l}
[/itex]
where:
[itex]i=1,2[/itex]
[itex]j=1,2[/itex]
[itex]n^{l}=(0,1,0,0)= [/itex]normal vector
Mean curvature:
[itex]
H=g^{ij}h_{ij}
[/itex]
Hawking mass:
[itex]
m_{H}(\Sigma)=\sqrt{\frac{Area \Sigma}{16\pi}}(1-\frac{1}{16\pi}\int_{\Sigma}{H^2}d\sigma)
[/itex]
Homework Equations
1) Prove that in the Schwarzschild metric, the Hawking mass of any sphere [itex]S_{r}[/itex] about the central mass is equal to [itex]M[/itex].
2) How to find the normal vector [itex]n^{l}[/itex] (as shown above)?
The Attempt at a Solution
I have tried to find the Hawking mass but it's not equal to [itex]M[/itex]. Maybe it's because I used the wrong normal vector?