On Schwarzschild causality in higher dimensions

Abstract

We show that the causal properties of asymptotically flat spacetimes depend on their dimensionality: while the time-like future of any point in the past conformal infinity I − contains the whole of the future conformal infinity I + in (2 + 1) and (3 + 1) dimensional Schwarzschild spacetimes, this property (which we call the Penrose property) does not hold for (d + 1) dimensional Schwarzschild if d > 3. We also show that the Penrose property holds for the Kerr solution in (3 + 1) dimensions, and discuss the connection with scattering theory in the presence of positive mass.

Conformally isometric embeddings and Hawking temperature

“…The null geodesic γ will reach a point on I + therefore the integral in (6.53) converges as r → ∞ (which can also be verified directly). In [19] Penrose applies some estimates to the integral (which are valid if r 0 > 5m) and shows that 5 ∞ r 0 χ(ρ)dρ < r 0 + const 4 This property also holds in 2 + 1-dimensions, where the metric is locally flat but admits a conical singularity, but it does not appear to hold in higher dimensions [4]. 5 The proof goes as follows:…”

“…Penrose then argues that P1 holds for the Schwarzschild space-time in 3+1 dimensions 4 . Consider a geodesic Lagrangian for the Schwarzschild metric…”

The Penrose Property with a Cosmological Constant

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A spacetime possesses the Penrose property if the timelike future of any point on I − contains the whole of I + . This property was first discussed by Penrose in [15], along with two other equivalent definitions, and was considered further in [6]. In this paper we consider the Penrose property in greater generality.

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“…We find that two of these generalise to a sensible notion of the Penrose property which remain equivalent, while the third (the "finite version") does not. We then move on to consider some further example spacetimes (with zero cosmological constant) which highlight some features of the Penrose property which were not discussed in [6]. We discuss the Ellis-Bronnikov wormhole (an example of a spacetime with more than one asymptotically flat end), the Hayward metric (an example of a non-singular black hole spacetime) and the black string spacetime (which is topologically R d+1 × S 1 so is not asymptotically flat).…”

“…We discuss the Ellis-Bronnikov wormhole (an example of a spacetime with more than one asymptotically flat end), the Hayward metric (an example of a non-singular black hole spacetime) and the black string spacetime (which is topologically R d+1 × S 1 so is not asymptotically flat). The Penrose property in each of these spacetimes is discussed using similar techniques to those established in [6].…”