Trapping Horizons in the Sultana-Dyer Space-Time

TL;DR

This paper analyzes the trapping horizons in the Sultana-Dyer space-time, revealing the nature and properties of horizons in a black hole embedded in an expanding universe.

Contribution

It provides a direct calculation of the expansions of null vectors to identify the trapping horizons and their characteristics in the Sultana-Dyer space-time.

Findings

The trapping horizon outside the event horizon is a past trapping horizon.

The past trapping horizon can be outer, degenerate, or inner depending on the radial coordinate.

The analysis clarifies the horizon structure in this cosmological black hole model.

Abstract

The Sultana-Dyer space-time is suggested as a model describing a black hole embedded in an expanding universe. Recently, in \cite{0705.4012}, its global structure is analyzed and the trapping horizons are shown. In the paper, by directly calculating the expansions of the radial null vector fields normal to the space-like two-spheres foliating the trapping horizons, we find that the trapping horizon outside the event horizon in the Sultana-Dyer space-time is a past trapping horizon. Further, we find that the past trapping horizon is an outer, instantaneously degenerate or inner trapping horizon accordingly when the radial coordinate is less than, equal to or greater than some value.