The area-angular momentum inequality for black holes in cosmological spacetimes

TL;DR

This paper establishes a new inequality relating area and angular momentum for stable marginally outer trapped surfaces in axially symmetric spacetimes with positive cosmological constant, sharpening previous bounds and identifying extremal configurations.

Contribution

It proves a refined area-angular momentum inequality for MOTS in cosmological spacetimes, including a universal upper bound on angular momentum and characterization of extremal solutions.

Findings

Derived a sharp inequality involving area, angular momentum, and cosmological constant.

Identified extremal Kerr-deSitter configurations saturating the inequality.

Established a universal upper bound on angular momentum for MOTS in these spacetimes.

Abstract

For a stable marginally outer trapped surface (MOTS) in an axially symmetric spacetime with cosmological constant $\Lambda > 0$ and with matter satisfying the dominant energy condition, we prove that the area $A$ and the angular momentum $J$ satisfy the inequality $8\pi |J| \le A\sqrt{(1-\Lambda A/4\pi)(1-\Lambda A/12\pi)}$ which is saturated precisely for the extreme Kerr-deSitter family of metrics. This result entails a universal upper bound $|J| \le J_{\max} \approx 0.17/\Lambda$ for such MOTS, which is saturated for one particular extreme configuration. Our result sharpens the inequality $8\pi |J| \le A$, [7,14] and we follow the overall strategy of its proof in the sense that we estimate the area from below in terms of the energy corresponding to a "mass functional", which is basically a suitably regularised harmonic map $\mathbb{S}^2 \rightarrow \mathbb{H}^2 $. However, in the cosmological case this mass functional acquires an additional potential term which itself depends on the area. To estimate the corresponding energy in terms of the angular momentum and the cosmological constant we use a subtle scaling argument, a generalised "Carter-identity", and various techniques from variational calculus, including the mountain pass theorem.