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## Homework Statement

Calculate the volume of a sphere of radius ##r## in the Schwarzschild metric.

## Homework Equations

I know that

\begin{align}

dV&=\sqrt{g_\text{11}g_\text{22}g_\text{33}}dx^1dx^2dx^3 \nonumber \\

&= \sqrt{(1-r_s/r)^{-1}(r^2)(r^2\sin^2\theta)} \nonumber

\end{align}

in the Schwarzschild metric.

## The Attempt at a Solution

Well the integral I get for the sphere's volume,

\begin{equation}

V = \int dV \nonumber

\end{equation}

gives an imaginary volume! What's going on? Of course the volume will be imaginary because ##dV## is imaginary when ##r<r_s## (plus, there's a singularity at ##r=r_s##, which complicates things if we want to integrate up to the Schwarzschild radius). There's obviously something I'm missing here, but I have no idea what it is.

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