Abstract
For a stable, marginally outer trapped surface (MOTS) in an axially symmetric spacetime with cosmological constant and with matter satisfying the dominant energy condition, we prove that the area A and the angular momentum J satisfy the inequality , which is saturated precisely for the extreme Kerr–de Sitter family of metrics. This result entails a universal upper bound for such MOTS, which is saturated for one particular extreme configuration. Our result sharpens the inequality (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 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.
Highlights
Some remarkable area inequalities for stable marginally outer trapped surfaces (MOTS) have been proven recently [6,7,8,9,10, 14,15,16, 18,19,20]
In the cosmological case this mass functional acquires an additional potential term which itself depends on the area
8π which is saturated for extreme Kerr black holes
Summary
Some remarkable area inequalities for stable marginally outer trapped surfaces (MOTS) have been proven recently [6,7,8,9,10, 14,15,16, 18,19,20]. The situation bears some analogy to stable MOTS in (not necessarily axially symmetric) spacetimes with electromagnetic fields and electric and magnetic charges QE and QM In this case the inequalities A ⩾ 4πQ2 [9] with Q2 = QE2 + QM2 (saturated for extreme Reissner– Nordström horizons) and A ⩽ 4πΛ−1 imply the (unsaturated) bound Q2 ⩽ Λ−1. (1.5) is saturated precisely for the 1-parameter family of extreme Kerr–de Sitter (KdS) horizons while the universal bound (1.6) is saturated for one particular such configuration The proof of this theorem will be sketched, while details are postponed to section 5. Starting with the stability condition one obtains a lower bound for the area of the MOTS in terms of a ‘mass functional’ This is the key quantity in the proof, and depends only on the twist potential and the norm of the axial Killing vector.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
