Inner Cauchy Horizon of Axisymmetric and Stationary Black Holes with Surrounding Matter in Einstein-Maxwell Theory

Abstract

We study the interior electrovacuum region of axisymmetric and stationary black holes with surrounding matter and find that there exists always a regular inner Cauchy horizon inside the black hole, provided the angular momentum J and charge Q of the black hole do not vanish simultaneously. In particular, we derive an explicit relation for the metric on the Cauchy horizon in terms of that on the event horizon. Moreover, our analysis reveals the remarkable universal relation ð8 JÞ 2 þ ð4 Q 2 Þ 2 ¼ A þ A À , where A þ and A À denote the areas of event and Cauchy horizon, respectively.

New entropy formula for Kerr black holes

“…In four and five spacetime dimensions Cvetič and collaborators have found that the product of black hole and "inner horizon" entropies is some integer multiple of 4π 2 , see [44] and references therein as well as papers by Ansorg and collaborators, e.g. [51]. Our results (31) and (32) actually suggest that already the individual entropies are some integer multiples of 2π, provided the zero mode charges J ± 0 and K ± 0 are quantized in the integers.…”

Universal properties and the first law of black hole inner mechanics

“…This suggestive general relation for black holes prompted the purely gravitational investigation of the product of areas of black holes [9], where the topology of the horizon is a sphere. The product of areas had also been investigated for Kerr-Newman black holes in [10,11]. Higher dimensional gravity allows for more exotic species of regular solutions.…”

CFT and Logarithmic Corrections to the Black Hole Entropy Product Formula

“…Therefore, the theorem of Ansorg-Hennig [1], "the area product formula of H ± being independent of mass," is no longer true when we have taken into consideration the leading order logarithmic correction.…”

“…There has been considerable ongoing excitement in physics of BH thermodynamic product formula [particularly area (or entropy) product formula] of inner horizon (H − ) and outer horizon (H + ) for a wide variety of BHs [1][2][3][4][5][6][7][8][9] which have been examined so far without considering any logarithmic correction. For several cases, the product is mass-independent (universal) and in some specific cases the product is not massindependent.…”