Horizon Constraints and Black Hole Entropy

TL;DR

The paper shows that imposing horizon-specific constraints in a 2D dilaton gravity setting modifies the diffeomorphism algebra to include a central extension, enabling a Virasoro description of the horizon. Using Cardy’s formula, it fixes the asymptotic density of states and reproduces the Bekenstein-Hawking entropy S = A / (4 hbar G) from horizon boundary data. The key insight is that black hole entropy arises from would-be gauge degrees of freedom that become physical due to the altered symmetry at the stretched horizon, potentially extending to more general theories via near-horizon conformal symmetry. This approach connects horizon boundary conditions, conformal symmetry, and microscopic state counting in a concrete 2D model with implications for higher-dimensional black holes.

Abstract

To ask a question about a black hole in quantum gravity, one must restrict initial or boundary data to ensure that a black hole is actually present. For two-dimensional dilaton gravity, and probably a much wider class of theories, I show that the imposition of a spacelike ``stretched horizon'' constraint modifies the algebra of symmetries, inducing a central term. Standard conformal field theory techniques then fix the asymptotic density of states, reproducing the Bekenstein-Hawking entropy. The states responsible for black hole entropy can thus be viewed as ``would-be gauge'' states that become physical because the symmetries are altered.

Figures (1)

• Figure 1: $y'=e^{-axy}$ with $y(0)=0$ and $a=10$