# Holographic Micro Thermofield Geometries of BTZ Black Holes

## TL;DR

This paper constructs deformed BTZ black hole geometries via holography, demonstrating nonlinear correspondence between boundary states and bulk, and analyzing their relaxation and entropy growth in thermal systems.

## Contribution

It introduces a method to deform BTZ geometries through bulk fields, revealing nonlinear holographic relations and the dynamics of thermal relaxation.

## Key findings

- Deformed geometries have elongated Penrose diagrams.
- Thermofield states can be deformed by operator insertions.
- Relaxation time scale is approximately $eta/2 extpi$.

## Abstract

We find general deformations of BTZ spacetime and identify the corresponding thermofield initial states of the dual CFT. We deform the geometry by introducing bulk fields dual to primary operators and find the back-reacted gravity solutions to the quadratic order of the deformation parameter. The dual thermofield initial states can be deformed by inserting arbitrary linear combination of operators at the mid-point of the Euclidean time evolution that appears in the construction of the thermofield initial states. The deformed geometries are dual to thermofield states without deforming the boundary Hamiltonians in the CFT side. We explicitly demonstrate that the AdS/CFT correspondence is not a linear correspondence in the sense that the linear structure of Hilbert space of the underlying CFT is realized nonlinearly in the gravity side. We also find that their Penrose diagrams are no longer a square but elongated horizontally due to deformation. These geometries describe a relaxation of generic initial perturbation of thermal system while fixing the total energy of the system. The coarse-grained entropy grows and the relaxation time scale is of order $\beta/2\pi$. We clarify that the gravity description involves coarse-graining inevitably missing some information of nonperurbative degrees.