Find Flux Through Black Hole: Metric g & Gauge Field A

[praharmitra]

Given a black hole metric $g_{\mu\nu}$ and gauge field $A_\mu$, how do I find the flux through a sphere of a black hole.

In simpler terms how do you find $\star F$ (the hodge star, I believe) using the metric ??

 

[praharmitra]

Well, I found out that
$$(\star F)_{\alpha\beta\gamma} = F^{\mu\nu} \sqrt{-g}\epsilon_{\mu\nu\alpha\beta\gamma}$$
(I am in five dimensional AdS space)

How do I integrate this now to find the total charge? Please help

 

[Mentz114]

In tensor terms, use the covariant derivative of the field tensor to get the current and integrate the time component over a volume.

In this case J = d*F ( 4-form ?) which I presume is integrable.

 

[Ben Niehoff]

It seems like you're just asking how to integrate forms.

To integrate a n-form K over an n-surface $\Sigma$, first you must find the pullback of K to $\Sigma$; then you just do an ordinary n-dimensional integral. Suppose $x^a$ are the coordinates on your manifold. Then K can be written

$$K = \frac{1}{n!} K_{a_1 \ldots a_n} \; dx^{a_1} \wedge \ldots \wedge dx^{a_n}$$

Now, if $y^b$ are the coordinates of the n-submanifold $\Sigma$, then

$$\int_\Sigma K = \underbrace{\idotsint}_n K_{a_1 \ldots a_n} \; \frac{\partial x^{a_1}}{\partial y^1} \ldots \frac{\partial x^{a_n}}{\partial y^n} \; dy^{1} \ldots dy^{n}$$

To find the electric charge that sources some Maxwell field F, you integrate $*F$ over a closed (d-2)-surface that contains the charge. In your case, you are in 5 dimensions, and you probably have spherical symmetry. So use spherical coordinates, and integrate over a 3-sphere centered on the black hole (set R and T to constants).