On geometry of deformed black holes: II. Schwarzschild hole surrounded by a Bach-Weyl ring

TL;DR

This paper investigates how a Schwarzschild black hole's internal and external geometry is affected by a surrounding Bach-Weyl ring, revealing strong external influences and unusual curvature behaviors without rotation.

Contribution

It extends the analysis of black-hole geometries by examining the impact of a concentric Bach-Weyl ring on a Schwarzschild black hole, highlighting significant curvature effects and internal structure modifications.

Findings

External rings can cause the Kretschmann scalar to become negative inside the horizon.

The external source significantly influences the black hole's internal geometry.

Curvature landscapes outside the horizon are complex and shaped by the external ring.

Abstract

We continue to study the response of black-hole space-times on the presence of additional strong sources of gravity. Restricting ourselves to static and axially symmetric (electro-)vacuum exact solutions of Einstein's equations, we first considered the Majumdar--Papapetrou solution for a binary of extreme black holes in a previous paper, while here we deal with a Schwarzschild black hole surrounded by a concentric thin ring described by the Bach--Weyl solution. The geometry is again revealed on the simplest invariants determined by the metric (lapse function) and its gradient (gravitational acceleration), and by curvature (Kretschmann scalar). Extending the metric inside the black hole along null geodesics tangent to the horizon, we mainly focus on the black-hole interior (specifically, on its sections at constant Killing time) where the quantities behave in a way indicating a surprisingly strong influence of the external source. Being already distinct on the level of potential and acceleration, this is still more pronounced on the level of curvature: for a sufficiently massive and/or nearby (small) ring, the Kretschmann scalar even becomes negative in certain toroidal regions mostly touching the horizon from inside. Such regions have been interpreted as those where magnetic-type curvature dominates, but here we deal with space-times which do {\em not} involve rotation and the negative value is achieved due to the {\em electric}-type components of the Riemann/Weyl tensor. The Kretschmann scalar also shapes rather non-trivial landscapes {\em outside} the horizon.