An analysis of the Sultana-Dyer cosmological black hole solution of the Einstein equations

TL;DR

This paper analyzes the Sultana-Dyer cosmological black hole solution, revealing an expanding singularity instead of a conformal Killing horizon, and studies its global structure and horizon properties.

Contribution

It provides a detailed analysis of the global structure and horizon dynamics of the Sultana-Dyer solution, correcting previous assumptions about its conformal Killing horizon.

Findings

Identifies an expanding spacetime singularity replacing the conformal Killing horizon.

Shows the singularity was naked early in the universe's history.

Describes the black hole's apparent horizon within a cosmological horizon.

Abstract

The Sultana-Dyer solution of general relativity representing a black hole embedded in a special cosmological background is analysed. We find an expanding (weak) spacetime singularity instead of the reported conformal Killing horizon, which is covered by an expanding black hole apparent horizon (internal to a cosmological apparent horizon) for most of the history of the universe. This singularity was naked early on. The global structure of the solution is studied as well.