NHEG mechanics: laws of near horizon extremal geometry (thermo)dynamics

Abstract

Near Horizon Extremal Geometries (NHEG) are solutions to gravity theories with SL(2, R)×U(1) N (for some N ) symmetry, are smooth geometries and have no event horizon, unlike black holes. Following the ideas by R. M. Wald, we derive laws of NHEG dynamics, the analogs of laws of black hole dynamics for the NHEG. Despite the absence of horizon in the NHEG, one may associate an entropy to the NHEG, as a Noether-Wald conserved charge. We work out "entropy" and "entropy perturbation" laws, which are respectively universal relations between conserved Noether charges corresponding to the NHEG and a system probing the NHEG. Our entropy law is closely related to Sen's entropy function. We also discuss whether the laws of NHEG dynamics can be obtained from the laws of black hole thermodynamics in the extremal limit.

On quantization of AdS3 gravity I: semi-classical analysis

“…As we see the gauge field is specified by a single function B (the sl(2, R) gauge field) which in turn is fixed by the function in the metric f . The sl(2, R) part of the gauge field is along a unit vector n a , which specifies a point on a unit radius AdS 2 space [38]. With the above one can write the explicit solution for (A.5) as…”

Near-horizon extremal geometries: coadjoint orbits and quantization

“…It is smooth for all values of θ; although k does not appear here, it is a measurable physical parameter as its components are related to the angular momentum of the black holesee [26,36,37]. Thus, on the horizon, and of course on the whole NHEG (2.1), one is dealing with an anisotropic torus -a torus with a preferred direction specified by k. In principle, the components of k may take any value; in practice however, the ratios of those components are directly related to ratios of components of angular momentum.…”

“…The near-horizon geometries of extremal Kerr (n = 0) and five-dimensional Myers-Perry black holes (n = 1) fall in this class, 2 while higher-dimensional Myers-Perry solutions (n > 1) do not since they lack U(1) n+1 isometries. In that setting, black hole entropy arises as a Noether charge that also happens to be the central charge of the symplectic symmetry algebra [26,36,37]. It is subject to laws of NHEG mechanics that can be obtained from the zero temperature limit of standard black hole thermodynamics [38].…”

“…Note that under SL(2, R) isometries which keep the t, r parts of the metric intact, rdt transforms by a closed form, rdt → rdt+dξ, where ξ depends on the details of the transformation. Hence SL(2, R) isometries also involve translations of φ i along k i , φ i → φ i + k i ξ, so the vector k also affects the generators (Killing vectors) of the SL(2,R) isometry [36,37]. NHEGs are not black holes (they have no event horizon), but they do have infinitely many bifurcate Killing horizons at constant t, r, all at the same Frolov-Thorne [46] temperature 1/2π [36,37]; see [26, section 2.1] for details.…”

Extremal vanishing horizon Kerr-AdS black holes at ultraspinning limit

“…There are some uniqueness and existence theorems about the near horizon geometry of extremal black holes [18,[22][23][24][25]. Meanwhile, these geometries represent a nice (thermo)dynamic behavior [26,27]. Some features about the near horizon of extremal black holes strongly depend on smoothness of the horizon.…”