Spacetime geometry in higher spin gravity

Abstract

Higher spin gravity is an interesting toy model of stringy geometry. Particularly intriguing is the presence of higher spin gauge transformations that redefine notions of invariance in gravity: the existence of event horizons and singularities in the metric become gauge dependent. In previous work, solutions of spin-3 gravity in the SL(3,R) × SL(3,R) Chern-Simons formulation were found, and were proposed to play the role of black holes. However, in the gauge employed there, the spacetime metric describes a traversable wormhole connecting two asymptotic regions, rather than a black hole. In this paper, we show explicitly that under a higher spin gauge transformation these solutions can be transformed to describe black holes with manifestly smooth event horizons, thereby changing the spacetime causal structure. A related aspect is that the Chern-Simons theory admits two distinct AdS 3 vacua with different asymptotic W -algebra symmetries and central charges. We show that these vacua are connected by an explicit, Lorentz symmetry-breaking RG flow, of which our solutions represent finite temperature generalizations. These features will be present in any SL(N,R) × SL(N,R) Chern-Simons theory of higher spins.

Thermodynamics of higher spin black holes in 3D

“…Our analysis reveals that the black hole gauge metric of [38] applies only to the BTZ-branch and the unstable branch-II (which merge at a certain temperature). In fact the unstable branch-II attains its maximal value of spin-3 charge at zero temperature where it has non-vanishing entropy, and precisely matches the extremal solution discussed in [38,40]. Branch III which has sensible thermodynamics and dominates the high temperature physics does not appear to be smooth in this gauge.…”

“…The latter are algebraic constraints on the mass and charge, and possess multiple solutions. We find that for general nonvanishing spin-3 chemical potential µ and temperature T , there are four possible branches of solutions of which only one, which we refer to as the "BTZ-branch", smoothly connects to the BTZ black hole at zero spin-3 chemical potential and which was the subject of the recent works [33,38,39,40,41] (see [42] for a black hole solution with a non-vanishing spin-4 field ). Of the three additional branches, one (branch-II) is thermodynamically unstable (similarly to the small black hole in AdS 5 ), the second is thermodynamically stable (branch-III), and the last one is thermodynamically disfavoured for all temperatures and chemical potentials.…”

“…The original work of [33] and subsequent papers on the topic [38,37] were focussed on solving the holonomy conditions (2.12) subject to the requirement that the higher spin charge W vanishes with the chemical potential µ. However, the equations (2.12) admit three additional solutions.…”

“…With non-vanishing spin-3 charge and energy density, the spacetime metric generically acquires the form of a wormhole geometry wherein both spatial and temporal circles remain finite everywhere. It was, however, shown in [38] that one can find an SL(3, R) transformation which turns the wormhole gauge metrics to black hole spacetimes with smooth horizons. We refer the reader to [38,40] for the details of the construction of these gauge transformations.…”

“…This is, of course, a gauge dependent description and it was demonstrated explicitly in [38] that the wormhole geometries can be gauge transformed to the so-called black hole gauge. In this gauge the metric has a horizon where the Euclidean time circle shrinks smoothly to zero.…”