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

Abstract

For a stable, marginally outer trapped surface (MOTS) in an axially symmetric spacetime with cosmological constant λ > 0 and with matter satisfying the dominant energy condition, we prove that the area A and the angular momentum J satisfy the inequality 8π|J| ≤ A√(1 - λA/4π)(1 - λA/12π), which is saturated precisely for the extreme Kerr-de Sitter family of metrics. This result entails a universal upper bound |J| ≤ Jmax ≈ 0.17/λ for such MOTS, which is saturated for one particular extreme configuration. Our result sharpens the inequality 8π |J| ≤ A (Dain and Reiris 2011 Phys. Rev. Lett. 107 051101, Jaramillo, Reiris and Dain 2011 Phys. Rev. Lett. D 84 121503), and we follow the overall strategy of its proof in the sense that we first estimate the area from below in terms of the energy corresponding to a 'mass functional', which is basically a suitably regularized harmonic map S2 → H2. 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 generalized 'Carter-identity', and various techniques from variational calculus, including the mountain pass theorem.