No-hair and almost-no-hair results for static axisymmetric black holes and ultracompact objects in astrophysical environments

TL;DR

This paper investigates the constraints on the geometry of static, axisymmetric black holes and ultracompact objects in realistic astrophysical environments, extending no-hair theorems to more practical scenarios.

Contribution

It demonstrates that only a one-parameter family of black-hole geometries matches a given external field and analyzes near-horizon horizonless objects, showing deviations diminish near the black hole limit.

Findings

Only a one-parameter family of geometries is compatible with a given external field.

Deviations from black-hole shape vanish as objects approach the black hole limit.

Results extend no-hair theorems to more realistic astrophysical conditions.

Abstract

No-hair theorems are uniqueness results constraining the form of the metric of black holes in general relativity. These theorems are typically formulated under idealized assumptions, involving a mixture of local (regularity of the horizon) and global aspects (everywhere vacuum spacetime and asymptotic flatness). This limits their applicability to astrophysical scenarios of interest such as binary black holes and accreting systems, as well as their extension to horizonless objects. A previous result due to G\"urlebeck constrains the asymptotic multipolar structure of static spacetimes containing black holes surrounded by matter although not revealing the possible structure of the metric itself. In this work, we disentangle some of these assumptions in the static and axisymmetric case. Specifically: i) we show that only a one-parameter family of black-hole geometries is compatible with a given external gravitational field, ii) we also analyze the case in which the central object is close to forming an event horizon but is still horizonless and show that the deviations from the natural black-hole shape have to die off as one approaches the black hole limit under the physical principle that curvatures are bounded.