I've been playing around with Maxima and it's ctensor library for tensor manipulation. I decided to have a crack at deriving Schwarzschild's solution for the interior of a constant-density sphere.(adsbygoogle = window.adsbygoogle || []).push({});

I've managed to derive a static, spherically symmetric solution, but am struggling a bit with the boundary conditions. I've used a coordinate system based on the Schwarzschild exterior solution - ##\theta## and ##\phi## have the usual definitions and ##r## is defined in the same sense as the exterior (##\sqrt{A/4\pi}##). I ended up with three constants of integration, and a need to somehow associate the density used in the interior solution with the total mass that the exterior solution uses - four constraints needed. I've got three.

In the circumstances described above (static spherically symmetric solution), am I allowed to require ##g^{\mathrm{exterior}}_{tt}=g^{\mathrm{interior}}_{tt}## and ##g^{\mathrm{exterior}}_{rr}=g^{\mathrm{interior}}_{rr}##? I think, given that I'm using the same coordinate basis vectors (give or take a scale factor), I'd simply be insisting that that scale factor is 1. I think that's OK, but am a bit worried that I'm basing something physical on coordinates.

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# I Schwarzschild interior boundary conditions

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