Exact Results for the BTZ Black Hole

TL;DR

This work presents a suite of exact results for the BTZ black hole in $AdS_3$, linking its formation, holographic description, and dynamics to precise mathematical structures. By connecting Gott time-machine constructions to BTZ formation, it derives an exact Choptuik scaling exponent $\gamma=1/2$ and identifies the threshold in terms of a hyperbolic isometry. Sullivan's theorem is used to formulate a rigorous No Hair/holography principle, while Brown-Henneaux symmetries combined with Cardy and the Rademacher expansion yield an exact accounting of the Bekenstein-Hawking entropy and its subleading corrections. The dynamical sector is treated by solving the massless scalar wave equation exactly, providing a microscopic CFT interpretation of the decay rate via left-right temperatures $T_L$ and $T_R$, and demonstrating consistency with the semiclassical Hawking decay rate. Together, these results reinforce the BTZ black hole as a powerful, exactly solvable laboratory for holography and quantum aspects of black hole thermodynamics, with implications for higher-dimensional AdS/CFT contexts.

Abstract

In this review, we summarize exact results for the three-dimensional BTZ black hole. We use rigorous mathematical results to clarify the general structure and properties of this black hole spacetime and its microscopic description. In particular, we study the formation of the black hole by point particle collisions, leading to an exact analytic determination of the Choptuik scaling parameter. We also show that a `No Hair Theorem' follows immediately from a mathematical theorem of hyperbolic geometry, due to Sullivan. A microscopic understanding of the Bekenstein-Hawking entropy, and decay rate for massless scalars, is shown to follow from standard results of conformal field theory.