Fourier Analysis of the BTZ Black Hole

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