Bound orbits of a slowly evolving black hole

TL;DR

This paper analyzes how bound orbits around a Kerr black hole evolve when the black hole's mass and spin change slowly, using adiabatic invariants to predict orbit adjustments in various astrophysical scenarios.

Contribution

It introduces a method to determine orbit evolution under slow black hole and mass changes by applying adiabatic invariance of orbital actions, extending prior static models.

Findings

Orbit actions remain constant during slow black hole evolution.

Black hole mass and spin changes cause small but significant orbit modifications.

Results improve understanding of inspiral dynamics and accretion effects.

Abstract

Bound orbits of black holes are very well understood. Given a Kerr black hole of mass $M$ and spin $S = aM^2$, it is simple to characterize its orbits as functions of the orbit's geometry. How do the orbits change if the black hole is itself evolving? How do the orbits change if the orbiting body evolves? In this paper, we consider a process that changes a black hole's mass and spin, acting such that the spacetime is described by the Kerr solution at any moment, or that changes the orbiting body's mass. Provided this change happens slowly, the orbit's actions ($J_r, J_\theta, J_\phi$) are {\it adiabatic invariants}, and thus are constant during this process. By enforcing adiabatic invariance of the actions, we deduce how an orbit evolves due to changes in the black hole's mass and spin and in the orbiting body's mass. We demonstrate the impact of these results with several examples: how an orbit responds if accretion changes a black hole's mass and spin; how it responds if the orbiting body's mass changes due to accretion; and how the inspiral of a small body into a black hole is affected by change to the hole's mass and spin due to the gravitational radiation absorbed by the event horizon. In all cases, the effect is very small, but can be an order of magnitude or more larger than what was found in previous work which did not take into account how the orbit responds due to these effects.