NHEG Mechanics: Laws of Near Horizon Extremal Geometry (Thermo)Dynamics
K. Hajian, A. Seraj, M. M. Sheikh-Jabbari
TL;DR
This work develops a universal framework for Near Horizon Extremal Geometries (NHEG) in gravity theories with SL(2,R)×U(1)^N symmetry, defining entropy as a Noether-Wald charge and deriving three global laws: a zeroth law fixing constants k^i and e^p, an entropy law linking S to J_i and q_p with the on-shell Lagrangian, and an entropy perturbation law for linearized probes. The authors compute the relevant Noether charges, including SL(2,R) and non-Abelian densities, and show the entropy is given by a horizon-generated Killing vector’s Noether charge, independent of the chosen horizon surface H. They further connect NHEG to extremal black holes by analyzing near-horizon and near-extremal limits, deriving the entropy perturbation law from the black hole first law and relating NH parameters to temperature derivatives of horizon data. Overall, the results provide a horizon-centric, universal thermodynamic-like structure for NHEG, echoing Wald’s framework but tailored to the enhanced symmetry and absence of a true horizon in NHEG.
Abstract
Near Horizon Extremal Geometries (NHEG) are solutions to gravity theories with $ SL(2,R) \times 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.