Will a mass $m$ rotating around a black hole with mass $M$ form a new black hole together with $M$, at a close enough distance?

Question

A mass $m$ rotates around a black hole with mass $M$ at a distance $r$ from the horizon. The combined system has an energy depending on the mass $M$ and the rest energy and kinetic energy of $m$.

When the mass $m$ is rotating close to the horizon, its speed will approach the speed of light. Time will go slower though near the horizon, so the mass $m$ seems to slow down and have less energy.

What will happen? Is there a distance at which the combined system will form a new black hole so the horizon of $M$ is enclosed by this new hole?

Answer 1

A massive object cannot be in a circular orbit arbitrarily close to a Schwarzschild black hole's event horizon. There is an innermost stable circular orbit at radius $3 R_s$ (where $R_s$ is the event horizon radius), which is the closest stable circular orbit for any massive object. Unstable orbits at closer radius are possible; but these are, well, unstable, and the slightest perturbation will lead to the massive object either falling into the black hole or flying off to infinity. What's more, such perturbations are literally unavoidable; see the next paragraph. So for any "stable" orbit, the mass $m$ is well-separated from the black hole; it can't orbit arbitrarily close to the event horizon.

Now, it's true that the mass $m$ will emit gravitational waves and thereby (slowly) lose energy. This will cause the radius of the orbit to (slowly) decrease, and eventually the mass $m$ will be orbiting so close that its orbit becomes unstable ($r < 3 R_s$). Then it will plunge into the black hole and merge with it. So in that sense, they will eventually merge. The final mass of the black hole will then just be $m + M$, minus whatever amount of gravitational wave energy is emitted over the many aeons of its inspiral and plunge into the black hole, and minus whatever binding energy the orbiting mass originally had.