The Euclidean quantisation of Kerr-Newman-de Sitter black holes

Abstract

Abstract:We study the family of Einstein-Maxwell instantons associated with the KerrNewman metrics with a positive cosmological constant. This leads to a quantisation condition on the masses, charges, and angular momentum parameters of the resulting Euclidean solutions.

On non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics

“…For this is it is convenient to start with a discussion of the family of the Riemannian Kerr anti-de Sitter metrics with small parameter a. Our presentation follows [14], where Kerr-Newman-de Sitter metrics were considered.…”

Compact singularity-free Kerr–Newman–de Sitter instantons

“…The further condition, imposed in [8], that ϕ is also a globally defined coordinate, requires that x 1 and x 2 are positive integers. However, this is not necessary, and (II.7) suffices to obtain a smooth compact manifold.…”

“…We have not been able to establish directly existence of the desired solutions. Instead, the following strategy, mimicking that of [8], turned out to be successful for all rational pairs (x 1 , x 2 ) which we implemented numerically. We believe that there exists a unique solution of the problem at hand for all pairs (x 1 , x 2 ) ∈ (R + ) 2 satisfying (II.7), so that our condition of rationality of x 1 and x 2 is not a real feature of the set of solutions.…”

“…In recent work [8] we constructed a family of compact Riemannian four-dimensional manifolds, parameterised by four integers and the value of the cosmological constant Λ, obtained by complex substitutions in the Kerr-Newman de Sitter metric. The requirement of 1) smoothness and compactness of the underlying manifold, together with the condition that 2) the original coordinates form a smooth coordinate system away from the axes of rotation, It came as a surprise to us that one can construct a more general class of such compact instantons, which arise if one drops the requirement 2) above.…”

Geodesics on a Kerr–Newman–(anti-)de Sitter instanton

“…We will treat only a 4D instanton (of Kerr type) here, but the differential-geometric question of classifying its geodesics is still well-posed and may have applications to various scattering questions (monopole or otherwise) in instanton backgrounds. Generally speaking, Riemannian solutions to the Einstein equations have emerged as state transition probabilities in Euclidean quantum gravity [2,4,5] and are expected to encode quantum properties of their Lorentzian counterparts. Combining this with the general importance of the Kerr solution in the context of recent developments such as AdS/CFT and Kerr/CFT, understanding the geometric structure of the Riemannian analogue of Kerr is a priority.…”