The Quantization of a Kerr-AdS Black Hole

Claus Gerhardt

doi:10.1155/2018/4328312

Abstract

We apply our model of quantum gravity to a Kerr-AdS space-time of dimension2m+1,m≥2, where all rotational parameters are equal, resulting in a wave equation in a quantum space-time which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal ones as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur. We also prove that the Kerr-AdS space-time can be maximally extended by replacing in a generalized Boyer-Lindquist coordinate system thervariable byρ=r2such that the extended space-time has a timelike curvature singularity inρ=-a2.

Highlights

In general relativity the Cauchy development of a Cauchy hypersurface S0 is governed by the Einstein equations, where the second fundamental form of S0 has to be specified
In the model of quantum gravity we developed in a series of papers [1,2,3,4,5,6,7] we pick a Cauchy hypersurface, which is only considered to be a complete Riemannian manifold (S0, gij) of dimension n ≥ 3, and define its quantum development to be special solutions of the wave equation
The Laplacian is the Laplacian with respect to gij, R is the scalar curvature of the metric, 0 < t is the time coordinate defined by the derivation process of the equation, and Λ < 0 a cosmological constant

Summary

Introduction

In general relativity the Cauchy development of a Cauchy hypersurface S0 is governed by the Einstein equations, where the second fundamental form of S0 has to be specified. (ii) The stationary eigenfunctions can be looked at as being radiation because they comprise the harmonic oscillator, while we consider the temporal eigenfunctions to be gravitational waves As it is well-known the Schwarzschild black hole or the extended Schwarzschild space has already been analyzed by Hawking [9] and Hartle and Hawking [10]; see the book by Wald [11], using quantum field theory, but not quantum gravity, to prove that the black hole emits radiation. A general solution in all dimension was given in [16] by Gibbons et al and we shall use their metric in odd dimensions, with all rotational parameters supposed to be equal, to define our space-time N, though we shall maximally extend it.

Preparations

The Quantization