Einstein - Yang - Mills black hole solutions in three dimensions

Abstract

In this paper, a procedure which gives Euclidean solutions of 3-dimensional Einstein-Yang-Mills equations when one has solutions of the Einstein equations is proposed. The method is based on reformulating Yang-Mills theory in such a way that it becomes a gravity. It is applied to find black hole solutions of the coupled Einstein Yang-Mills equations. 1 Introduction.Solutions of Einstein-Yang-Mills (EYM) equations have already been the motivation for several studies [1]. For most of them the ansatz used is a sophisticated version of the Reissner-Nordstrm solution. This solution teaches a lot about the properties of EYM black holes, but it requires a numerical evaluation in the final stage. Here, a method is proposed to build analytically a family of Euclidean solutions to these equations. In this report, the method is restricted to three dimensions and SU (2) is chosen as the Yang-Mills (YM) group.The study of 3 dimensional gravity is important because it provides a compromise between the triviality of 2-d gravity and the intricacy of the 4-d one. Classically, the Riemann tensor is entirely determined by the Ricci tensor and by the scalar curvature, thus the solutions of the Einstein equations can be studied almost systematically [2]. From the quantum point of view, the formalism proposed by Achùcarro, Townsend, Witten [3,4] shows that 3-d quantum gravity is, in principle, integrable [4] albeit not trivial. It can serve as a good model for exploring the complexity of quantum gravity [5] or at least of quantum field theory in curved space-time.The study of quantum properties by a functional integral means that one has to look for Euclidean solutions that will provide a starting point for a saddle point approximation. This method has shown its efficiency in the development of black hole thermodynamical properties and in quantum cosmology [6]. An important step is to introduce non-trivial matter fields such as the YM field in this procedure.In the present work, the key to finding Euclidean solutions for the EYM equations is based on describing the 3-d YM theory in terms of gauge invariant variables [7]. In this procedure, YM theory takes the form of a gravity. Thus, the EYM equations reduce to those of two coupled gravities. The similarity between the two sets of equations leads to a very simple ansatz which reduces the two coupled equations to one simple Einstein equation.We apply our ansatz to the Euclidean continuation of BTZ black hole which is a particularly interesting three dimensional solution [8] of the Einstein equations. The mass and the energy of the solution are calculated with the help of the quasi-local formalism developed in [9] and [10].

Gravitating non-Abelian solitons and black holes with Yang–Mills fields

“…Relatively little is known about solutions with symmetries other than spherical. At the same time, we do not discuss here solutions for D = 4, since a separate review would be necessary for this (see [106,216,65] for some results in D = 3). A large number of such solutions can be obtained from the ten-dimensional heterotic five-brane via various compactifications [118].…”

Cold dark matter: Bose-Einstein condensation of gluons in Anti-de Sitter space time

“…Our idea is thus to interpret ρ ind AdS as a comoving density that, when evaluated at point δ, would plays the role of the anti-desitterian world matter, induced by QCD, to cancel the contribution of the comoving CC at point δ. Now, it turns out that following an idea of Sakharov [29] and the work of Adler [30] such a contribution can be rigorously evaluated (or at least estimated) [31,32]. The idea of Sakharov was that the non-renormalizable Einstein-Hilbert action would be an effective theory resulting from the coupling of a renormalizable gauge theory to a renormalizable gravitational theory quadratic in the curvature.…”

“…1/3 (32) to become the currently observed dark matter. The above formula involving the value 449 ζ(3) ≈ 1.2 of the Riemann function is standard for all isotropic harmonic traps (see for instance [39]).…”

Three dimensional nonlinear magnetic AdS solutions through topological defects

“…Due to these facts, investigation of (2+1)-dimensional spacetimes is important. Three dimensional solutions of black holes and magnetic solutions have been studied by many authors [50][51][52][53][54][55][87][88][89][90][91][92][93]. This tremendous interest in these solutions is due to fact that three dimensional solutions contribute fundamentally to conceptual issues of astrophysical subjects such as black holes thermodynamics [94][95][96].…”