I'm looking to derive the surface area of the event horizon of a Schwarzschild black hole. I was just wondering if it were possible for someone to explain to me this:

$$ \sqrt{g_{\theta\theta}g_{\phi\phi}} $$

I don't know what it is, or how to compute it. I don't even know what it is called? Could anyone shed some light on this.

I give the potential answerers the full right and privilege to patronise me.

I do, however, have one hunch that it is some sort of killing vector, or closely related to the killing vector...?


In general, the square root of the determinant of the metric will give the volume element. That is, $dV = \sqrt{|\det g|} dx_1 \cdots dx_n$. Here you have the metric restricted to surfaces of constant $t$ and $r$, so you will get the area element on these. Since the metric is diagonal, the determinant is just $g_{\theta\theta}g_{\phi\phi}$.

  • $\begingroup$ I seem to feel that the expression in my question evaluates to $r^2\sin\theta$ in spherical polar coordinates... $\endgroup$ – DarthPlagueis Oct 22 '15 at 22:28
  • $\begingroup$ Do you know of any freely available resources which could describe how to compute these metrics in more detail? $\endgroup$ – DarthPlagueis Oct 22 '15 at 22:38

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