Generalized black holes in three-dimensional spacetime

Abstract

Three-dimensional spacetime with a negative cosmological constant has proven to be a remarkably fertile ground for the study of gravity and higher spin fields. The theory is topological and, since there are no propagating field degrees of freedom, the asymptotic symmetries become all the more crucial. For pure (2+1) gravity they consist of two copies of the Virasoro algebra. There exists a black hole which may be endowed with all the corresponding charges. The pure (2+1) gravity theory may be reformulated in terms of two Chern-Simons connections for sl (2, R). This permits an immediate generalization which may be interpreted as containing gravity and a finite number of higher spin fields. The generalization is achieved by replacing sl (2, R) by sl (3, R) or, more generally, by sl (N, R). The asymptotic symmetries are then two copies of the so-called W N algebra, which contains the Virasoro algebra as a subalgebra. The question then arises as to whether there exists a generalization of the standard pure gravity (2+1) black hole which would be endowed with all the W N charges. Since the generalized Chern-Simons theory does not admit a direct metric interpretation, one must define the black hole in Euclidean spacetime through its thermal properties, and then continue to Lorentzian spacetime. The original pioneering proposal of a black hole along this line for N = 3 turns out, as shown in this paper, to actually belong to the so called "diagonal embedding" of sl (2, R) in sl (3, R), and it is therefore endowed with charges of lower rather than higher spins. In contradistinction, we exhibit herein the most general black hole which belongs to the "principal embedding". It is endowed with higher spin charges, and possesses two copies of W 3 as its asymptotic symmetries. The most general diagonal embedding black hole is studied in detail as well, JHEP05 (2014)031 in a way in which its lower spin charges are clearly displayed. The extension to N > 3 is also discussed. A general formula for the entropy of a generalized black hole is obtained in terms of the on-shell holonomies. The relationship between the asymptotic symmetries and the chemical potentials is exhibited, and the equivalence of the different thermodynamical ensembles is discussed. A self-contained account of the background necessary to substantiate the claims made in the paper is included.

Higher-spin charges in Hamiltonian form. I. Bose fields

“…On the one hand, our work fits within a wider program of improving the grasp over the Hamiltonian description of bosonic higher-spin theories [28][29][30]. On the other hand, presenting higher-spin charges in Hamiltonian form may help testing the expected thermodynamical properties of the proposed black hole solutions of Vasiliev's equations [31,32], in analogy with what happened for higher-spin black holes in three space-time dimensions [33][34][35][36]. In that case, the distinction between dynamical variables and Lagrange multipliers has been indeed instrumental in introducing the chemical potentials associated to the higherspin charges carried by generalised black holes [36].…”

“…On the other hand, presenting higher-spin charges in Hamiltonian form may help testing the expected thermodynamical properties of the proposed black hole solutions of Vasiliev's equations [31,32], in analogy with what happened for higher-spin black holes in three space-time dimensions [33][34][35][36]. In that case, the distinction between dynamical variables and Lagrange multipliers has been indeed instrumental in introducing the chemical potentials associated to the higherspin charges carried by generalised black holes [36]. The exact solutions of [31,32] are also expected to be endowed with higher-spin charges [16], 1 but the study of their thermodynamics is still in its infancy (although entropy and thermal equilibrium should be the most striking signatures of a "generalised" black hole).…”

“…The covariant boundary conditions (2.22) would also fix the falloff of the Lagrange multipliers. The resulting conditions, however, would correspond to a particular choice of gauge, while -as noticed in three space-time dimensions [36,52] -the freedom in the choice of the Lagrange multipliers is instrumental in fitting some physically relevant solutions within the boundary conditions (see also [53]). We postpone to future work a detailed analysis of this issue, especially because this freedom does not affect charges.…”

“…In three space-time dimensions, the rewriting (2.43) exhibits the chiral splitting of charges that one obtains in the Chern-Simons formulation with a suitable choice of the boundary value of the Lagrange multipliers [19,20,36]. Introducing the light-cone coordinates x ± = t ± φ, one obtains…”

New entropy formula for Kerr black holes

“…and to situations where Bekenstein-Hawking fails, such as higher spin [27,28] black holes [29,30], where even for the simplest case of spin-3 black holes the entropy looks fairly complicated [31] (L ± 0 and c are the same as in the spin-2 case; W ± 0 denotes the spin-3 zero mode charges with suitable normalization; for W ± 0 = 0 the Cardy formula (8) is recovered)…”