Can the gravitational force of a black hole "lock" a particle in an unstable equilibrium?

Question

In classical mechanics, it is possible to have points of stable, unstable, and neutral equilibrium depending on the gradient of the potential field. Near a black hole, the gravitational potential becomes highly nonlinear and relativistic effects dominate.

1. Is it theoretically possible to find a location in the spacetime around a black hole where a test particle could be in an unstable equilibrium, balancing gravitational forces and other factors like radiation pressure or frame-dragging effects?

2. Would such a position depend solely on classical mechanics, or would general relativity fundamentally rule it out?

3. How could the dynamics of small perturbations evolve in such a scenario—would they exhibit chaotic or predictable behavior?

Answer 1

In the spherically symmetric spacetime around a massive, compact object, there are famously unstable circular orbits at $3r_s/2 < r < 3r_s$, where $r_s$ is the Schwarzschild radius. For these orbits to be possible, the orbiting body must have a specific angular momentum $> \sqrt{12}GM/c$.

Any inward perturbation or energy loss from an object in such an orbit will result in the orbiting body falling into the black hole; any outward perturbation or gain in energy will either result in it entering a bound, precessing orbit (if the specific angular momentum is originally $<4GM/c$) or the object escaping to infinity (if its specific angular momentum is $\geq 4GM/c$).

Plots of the effective potential (the sum of gravitational and centrifugal effects) are shown in solid lines for GR and dashed lines for Newtonian physics for different specific angular momenta. The unstable orbits are at the local maxima of the effective potential. There are no equivalent unstable orbits in Newtonian physics.

Any orbiting body cannot, by definition, be "locked" in an unstable orbit - it is unstable. The behaviour of the particle when it falls away from the unstable equilibrium is predictable.