The Penrose Property with a Cosmological Constant

TL;DR

This paper explores the Penrose property in various spacetimes with a focus on those with a cosmological constant, revealing differences from the asymptotically flat case and examining specific examples like wormholes and black holes.

Contribution

It generalizes the Penrose property to spacetimes with a cosmological constant and analyzes its validity in diverse geometries, including product spacetimes and non-singular black holes.

Findings

The Penrose property differs in spacetimes with a cosmological constant.

In asymptotically AdS spacetimes, the property requires redefinition.

Certain product spacetimes satisfy the Penrose property if their Lorentzian component does.

Abstract

A spacetime satisfies the non-timelike boundary version of the Penrose property if the timelike future of any point on $\mathcal{I}^-$ contains the whole of $\mathcal{I}^+$. This property was first discussed for asymptotically flat spacetimes by Penrose, along with an equivalent definition (the finite version). In this paper we consider the Penrose property in greater generality. In particular we consider spacetimes with a non-zero cosmological constant and we note that the two versions of the property are no longer equivalent. In asymptotically AdS spacetimes it is necessary to re-state the property in a way which is more suited to spacetimes with a timelike boundary. We arrive at a property previously considered by Gao and Wald. Curiously, this property was shown to fail in spacetimes which focus null geodesics. This is in contrast to our findings in asymptotically flat and asymptotically de Sitter spacetimes. We then move on to consider some further example spacetimes (with zero cosmological constant) which highlight features of the Penrose property not previously considered. We discuss spacetimes which are the product of a Lorentzian and a compact Riemannian manifold. Perhaps surprisingly, we find that both versions of the Penrose property are satisfied in this product spacetime if and only if they are satisfied in the Lorentzian spacetime only. We also discuss the Ellis-Bronnikov wormhole (an example of a spacetime with more than one asymptotically flat end) and the Hayward metric (an example of a non-singular black hole spacetime).