Verifying the Komar Integral for the Kerr-Newman Solution

[latentcorpse]

Given the Komar integral

$J(V)=\frac{1}{16 \pi G} \oint_{\partial V} dS_{\mu \nu} D^\mu m^\nu$
where $V$ is the volume of the spacelike hypersurface $\Sigma$ with boundary $\partial V$ and $m=\frac{\partial}{\partial \phi}$ is the Killing vector field This particula Komar integral is associated with, I am asked to verify that $J=Ma$ for the Kerr-Newman solution with parameter $a$.

I have not really got any idea what to do here since the only definition in my notes of $a$ is $a=\frac{J}{M}$ and so I ended up going round in circles.
I was wondering if I am supposed to extract something from the formula for $J(V)$ that we can write as the mass or the ADM mass or something?

Thanks.

 

[fzero]

You'll want to express $$J(V)$$ and $$E(V)$$ in terms of Christoffel symbols. I haven't tried to compute the Christoffel symbols, but since many of them vanish, it's possible that only $$dS_{\phi t}$$ appears in both expressions.

 

[latentcorpse]

You'll want to express $$J(V)$$ and $$E(V)$$ in terms of Christoffel symbols. I haven't tried to compute the Christoffel symbols, but since many of them vanish, it's possible that only $$dS_{\phi t}$$ appears in both expressions.

Does this look as though it is correct:

$D^\mu m^\nu = g^{\mu \rho} D_\rho m^\nu = g^{\mu \rho} ( \partial_\rho m^\nu + \Gamma^\nu{}_{\rho \alpha} m^\alpha ) = g^{\mu \rho} \Gamma^\nu{}_{\rho \phi} m^\phi= g^{\mu \rho} \Gamma^\nu{}_{\rho \alpha}$ where I killed $\partial_\rho m^\nu$ since $m^\nu$ is always a constant (either 0 or 1 depending on whether $\nu=\phi$ or not) and then in the last line $m^\phi=1$

So we'd have

$J(V)=\frac{1}{16 \pi G} \oint_{\partial V} dS_{\mu \nu} g^{\mu \rho} \Gamma^\nu{}_{\rho \phi}$

Should I hit the $dS_{\mu \nu}$ with the $g^{\mu \rho}$ or is it better to just leave it in the form that it's already in?

The formula for $E(V)$ is

$E(V)=-\frac{1}{8 \pi G} \oint_{\partial V} dS_{\mu \nu} D^\mu k^\nu$

I'll not type that out yet until I check I got the first one right though!

 

[fzero]

$J(V)=\frac{1}{16 \pi G} \oint_{\partial V} dS_{\mu \nu} g^{\mu \rho} \Gamma^\nu{}_{\rho \phi}$

That's right.

Should I hit the $dS_{\mu \nu}$ with the $g^{\mu \rho}$ or is it better to just leave it in the form that it's already in?

I'd leave it the way it is until you get further along. Ideally, you'd be able to show relations between $$\Gamma^\nu{}_{\rho \phi}$$ and $$\Gamma^\nu{}_{\rho t}$$, but it might be that there are additional terms that vanish due to properties of the surface.

 

[latentcorpse]

That's right.



I'd leave it the way it is until you get further along. Ideally, you'd be able to show relations between $$\Gamma^\nu{}_{\rho \phi}$$ and $$\Gamma^\nu{}_{\rho t}$$, but it might be that there are additional terms that vanish due to properties of the surface.

By a similar procedure then, I find that

$E(V)=-\frac{1}{8 \pi G} \oint_{\partial V} dS_{\mu \nu} g^{\mu \rho} \Gamma^{\nu}{}_{\rho t}$

So I guess I need to use a relationship between E and V to go ahead? There isn't any such relationship mentioned in this section of my notes - should I just use the formula for the energy of a system with ang mom J i.e. $E=\frac{J^2}{2M}$?

 

[fzero]

I think that you want to show a relationship between the integrands. You'll need to compute the Christoffel symbols to some extent and understand the surface volume element to see what terms contribute.

 

[latentcorpse]

I think that you want to show a relationship between the integrands. You'll need to compute the Christoffel symbols to some extent and understand the surface volume element to see what terms contribute.

What is the relationship between E and J? Did I get that right?

Then, say for the Christoffel symbol in E we get

$\Gamma^\nu{}_{\rho t}=\frac{1}{2} g^{\nu \alpha} ( g_{\rho \alpha , t} + g_{\alpha t, \rho} - g_{\rho t,\alpha })$
I reckon I can get rid of the first term since no components of the Kerr-Newman metric have t dependence. So that would leave
$\Gamma^\nu{}_{\rho t}=\frac{1}{2} g^{\nu \alpha} ( g_{\alpha t, \rho}-g_\rho t, \alpha})$

 

[fzero]

What is the relationship between E and J? Did I get that right?

I doubt that $$E=J^2/(2M)$$ holds for a charged black hole. In any case, I don't think you need such a relationship to solve this problem.

Then, say for the Christoffel symbol in E we get

$\Gamma^\nu{}_{\rho t}=\frac{1}{2} g^{\nu \alpha} ( g_{\rho \alpha , t} + g_{\alpha t, \rho} - g_{\rho t,\alpha })$
I reckon I can get rid of the first term since no components of the Kerr-Newman metric have t dependence. So that would leave
$\Gamma^\nu{}_{\rho t}=\frac{1}{2} g^{\nu \alpha} ( g_{\alpha t, \rho}-g_\rho t, \alpha})$

As I've been saying, you'll also want to find out what conditions are put on $$dS_{\mu\nu}$$ from the spacelike hypersurface condition.

 

[latentcorpse]

I doubt that $$E=J^2/(2M)$$ holds for a charged black hole. In any case, I don't think you need such a relationship to solve this problem.




As I've been saying, you'll also want to find out what conditions are put on $$dS_{\mu\nu}$$ from the spacelike hypersurface condition.

If you check out section 5.3.1 in here:
http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf
it says just under eqn (5.43) that if V is on a t=const spacelike hypersurface then $dS_\mu m^\mu=0$ for the J integral and that for the E integral (see below eqn (5.41)) that we should pick V as a 2-sphere at spatial infinity.

Is this of any use?

 

[fzero]

If you check out section 5.3.1 in here:
http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf
it says just under eqn (5.43) that if V is on a t=const spacelike hypersurface then $dS_\mu m^\mu=0$ for the J integral and that for the E integral (see below eqn (5.41)) that we should pick V as a 2-sphere at spatial infinity.

Is this of any use?

Yes, but that leads to expressions involving the stress tensor. Maybe it works in the weak source approximation, but I'm not sure that it's valid.

 

[latentcorpse]

Yes, but that leads to expressions involving the stress tensor. Maybe it works in the weak source approximation, but I'm not sure that it's valid.

So what would you suggest for finding these conditions then?

 

[fzero]

So what would you suggest for finding these conditions then?

As I wrote:

You'll need to compute the Christoffel symbols to some extent and understand the surface volume element to see what terms contribute.

 

[latentcorpse]

As I wrote:

Surely I've already done the Christoffel symbols as far as they can be taken though?
I don't know what to say about $dS_{\mu \nu}$?

 

[fzero]

Surely I've already done the Christoffel symbols as far as they can be taken though?

It's possible that there's some simplifications if you actually compute them from knowledge of the metric.

I don't know what to say about $dS_{\mu \nu}$?

You'll want to use properties of the hypersurface, as in your post #9.

The sorts of simplifications you're looking for are analogous to the way that you'd get back to equ. (5.28) from (5.31) in Townsend's notes.