AI Insight
This study analytically calculates how a spherically symmetric massive scalar field "cloud" surrounding a Schwarzschild black hole affects the black hole's spacetime geometry. Working in the regime where the scalar field mass times the black hole radius is much smaller than unity, the researchers derive explicit expressions for metric perturbations and determine how the black hole's mass increases as it absorbs the decaying scalar cloud. The analysis includes determining the quasi-resonant frequency of the fundamental scalar wave mode and tracking the evolution of the event horizon as the system evolves.
Why it matters
This work provides precise analytical tools for understanding how quantum fields behave around black holes and how they modify black hole properties. Such calculations are relevant for modeling hypothetical scalar field configurations around astrophysical black holes and for testing general relativity in strong-field regimes.
Understand the Science
Abstract: We analytically determine the leading backreaction of a spherically symmetric massive complex scalar quasi-bound state (with mass $mu$) on a Schwarzschild black hole with (initial) gravitational radius $r_0$. Working in the small-coupling regime, $r_0 mu ll 1$, we evaluate the stress-energy tensor of the fundamental scalar $s$-wave and solve the Einstein equations through quadratic order in its amplitude in ingoing Eddington-Finkelstein coordinates. We also determine the small-mass quasi-resonant frequency of the fundamental $s$-wave analytically by matched asymptotic expansions and validate it numerically using Leaver’s method. Unlike steady-state treatments, the calculation retains the exponential decay of the quasi-bound state. We obtain explicit expressions for the metric perturbations and Misner-Sharp mass and derive the evolution of the future outer trapping horizon. The black-hole mass grows monotonically with the decaying horizon flux and saturates when the finite scalar cloud has been absorbed, with the decrease of the cloud mass exactly balancing the horizon growth at the perturbative order considered. We also determine the domain in which the scalar small-coupling approximation and the gravitational perturbative expansion are simultaneously valid.
Source: Analytic backreaction of a scalar wig on a Schwarzschild black hole