On Quantum Microstates in the Near Extremal, Near Horizon Kerr Geometry

TL;DR

This paper investigates the quantum aspects of near horizon extremal Kerr geometries using $AdS_2/CFT_1$ correspondence, revealing how horizon shifts affect quantum theories and reproducing black hole entropy and temperature.

Contribution

It introduces a quantum theory for shifted near horizon extremal Kerr geometries via Kaluza-Klein reduction, extending the Kerr/$CFT$ correspondence to include horizon shifts.

Findings

Derived the asymptotic symmetry group of the quantum $AdS_2$ boundary.

Reproduced Bekenstein-Hawking entropy from the Virasoro algebra.

Calculated Hawking temperature consistent with classical results.

Abstract

We study the thermodynamics of near horizon near extremal Kerr (NHNEK) geometry within the framework of $AdS_2/CFT_1$ correspondence. We start by shifting the horizon of near horizon extremal Kerr (NHEK) geometry by a general finite mass. While this shift does not alter the geometry in that the resulting classical solution is still diffeomorphic to the NHEK solution, it does lead to a quantum theory different from that of NHEK. We obtain this quantum theory by means of a Robinson-Wilczek two-dimensional Kaluza-Klein reduction which enables us to introduce a finite regulator on the $AdS_2$ boundary and compute the full asymptotic symmetry group of the two-dimensional quantum conformal field theory on the respective $AdS_2$ boundary. The s-wave contribution of the energy-momentum-tensor of this conformal field theory, together with the asymptotic symmetries, generate a Virasoro algebra with a calculable center, which agrees with the standard Kerr/$CFT$ result, and a non-vanishing lowest Virasoro eigenmode. The central charge and lowest eigenmode produce the Bekenstein-Hawking entropy and Hawking temperature for NHNEK.