Here's an interesting question inspired by a homework probem (not mine) - what is the orbital period (proper time) for someone in a circular orbit around a black hole as a function of the radius of their orbit (Schwarzschild r-coordinate).(adsbygoogle = window.adsbygoogle || []).push({});

By setting the derivative of the effective potential to zero I came up with the following expression.

[tex]\tau = \frac{2 \pi r}{c} \sqrt{\frac{2r}{r_s} -3}[/tex]

Here [itex]r_s[/itex] = 2GM/c^2 is the Schwarzschild radius of the black hole.

It seems reasonably sane, we get tau=0 at the photon sphere, and it varies as r^3/2 for large r.

The circular orbits will be unstable for r < 3 r_s

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# Black hole orbital period

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