Fourier Analysis of the BTZ Black Hole

TL;DR

This paper extends Fourier analysis techniques to the BTZ black hole within the AdS/CFT framework, revealing that boundary Fourier coefficients encode black hole micro-states and offering insights into black hole entropy.

Contribution

It introduces a Fourier-based bulk to boundary mapping for the BTZ black hole and proposes a toy model linking boundary modes to black hole micro-states and entropy.

Findings

Boundary Fourier coefficients represent black hole micro-states.

The mapping aligns with known AdS/CFT results.

Micro-states correspond to black hole emission modes.

Abstract

In this paper we extend our previous work regarding the role of the Fourier transformation in bulk to boundary mappings to include the BTZ black hole. We follow standard procedures for modifying Fourier Transformations to accommodate quotient spaces and arrive at a bulk to boundary mapping in a black hole background. We show that this mapping is consistent with known results and lends a new insight into the AdS/CFT duality. We find that the micro-states corresponding to the entropy of a bulk scalar field are the Fourier coefficients on the boundary, which transform under the principal series representation of $SL(2,R)$. Building upon this we present a toy model to analyze the implications of this for the origin of black hole entropy. We find that the black hole micro-states live on the boundary and correspond to the possible emission modes of the black hole