On thermal holographic RG flows

TL;DR

This work extends holographic RG flow analysis to thermal states by formulating a finite-temperature Hamilton-Jacobi framework for Einstein gravity coupled to a scalar. It derives a thermal Callan-Symanzik equation for the boundary CFT from the bulk Hamiltonian constraint and identifies a Ward identity for broken dilatation, linking temperature to RG running. Exterior dynamics reproduce a modified dilatation Ward identity at finite temperature, while interior dynamics near the black hole singularity are described by a Kasner/BKL-like structure encoded in a Wheeler–DeWitt-type equation for the on-shell action. Together, these results illuminate how finite temperature reshapes holographic RG structure and offer a boundary perspective on interior black hole dynamics with potential ties to Carrollian gravity and quantum cosmology.

Abstract

Holographic Renormalization Group (RG) flows, described by Einstein gravity coupled to matter fields, have been thoroughly explored in the context of vacuum states. In this work, we shift the focus to thermal states. Using the Hamilton-Jacobi formalism for the coupled system, we derive the Ward identity associated with broken dilatation symmetry in thermal correlators and obtain the modified Callan-Symanzik equation for the dual thermal CFT. We then employ the Hamiltonian approach to analyze the black hole interior and comment on the near-singularity behavior from this perspective.

Figures (1)

• Figure 1: In this Penrose diagram of an eternal black hole, the right exterior is shaded in black and the interior is shaded in red. The black arrow in the exterior shows an inwards pointing normal vector, which is space-like. Various slices shown by black lines are constant radial surfaces. The radial Hamiltonian evolves the slices in the direction of the normal vector. Similarly, in the interior, the red arrow shows the normal vector, which is time-like in nature. We have found the Hamiltonian that does the evolution for these time-like slices. The singularity is marked with zagged lines. In section \ref{['interior']}, we explore the implications of Hamiltonian constraints near the singularity.