Near Horizon Extremal Geometry Perturbations: Dynamical Field Perturbations vs. Parametric Variations
K. Hajian, A. Seraj, M. M. Sheikh-Jabbari
TL;DR
This work extends the Noether-Wald framework to Near Horizon Extremal Geometries (NHEG) and investigates which perturbations preserve the Entropy Perturbation Law (EPL). By enforcing linearized equations of motion, a pair of SL$(2,\mathbb{R})$-invariance conditions, and asymptotic isometries, the authors show that dynamical perturbations are highly constrained and must arise from variations within the NHEG parameter space. They introduce and analyze two perturbation classes: dynamical field perturbations $\delta\Phi$ and parametric perturbations $\hat{\delta}\Phi$, proving EPL holds for both. A central result is the NHEG perturbation uniqueness theorem: under the stated conditions, all perturbations are exactly the parametric ones, implying no local dynamics beyond nearby NHEG solutions; this supports a no-dynamics perspective near the NHEG horizon and has implications for Kerr/CFT and NHEG microstate considerations.
Abstract
In arXiv:1310.3727 we formulated and derived the three universal laws governing Near Horizon Extremal Geometries (NHEG). In this work we focus on the Entropy Perturbation Law (EPL) which, similarly to the first law of black hole thermodynamics, relates perturbations of the charges labeling perturbations around a given NHEG to the corresponding entropy perturbation. We show that field perturbations governed by the linearized equations of motion and symmetry conditions which we carefully specify, satisfy the EPL. We also show that these perturbations are limited to those coming from difference of two NHEG solutions (i.e. variations on the NHEG solution parameter space). Our analysis and discussions shed light on the "no-dynamics" statements of arXiv:0906.2380 and arXiv:0906.2376.