Past horizons in Robinson-Trautman spacetimes with a cosmological constant

TL;DR

This paper analyzes past horizons in Robinson-Trautman spacetimes with a cosmological constant, extending the Penrose-Tod equation and characterizing the horizons as spacelike outer trapping horizons.

Contribution

It generalizes the Penrose-Tod equation for nonzero cosmological constant and characterizes the nature and evolution of past horizons in these spacetimes.

Findings

Existence and uniqueness of past horizons are established.

Past horizons are shown to be spacelike outer trapping horizons.

The generalized Penrose-Tod equation is analyzed and visualized.

Abstract

We study past horizons in the class of type II Robinson-Trautman vacuum spacetimes with a cosmological constant. These exact radiative solutions of Einstein's equations exist in the future of any sufficiently smooth initial data, and they approach the corresponding spherically symmetric Schwarzschild-(anti-)de Sitter metric. By analytic methods we investigate the existence, uniqueness, location and character of the past horizons in these spacetimes. In particular, we generalize the Penrose-Tod equation for marginally trapped surfaces, which form such white-hole horizons, to the case of a nonvanishing cosmological constant, we analyze behavior of its solutions and visualize their evolutions. We also prove that these horizons are explicit examples of an outer trapping horizon and a dynamical horizon, so that they are spacelike past outer horizons.