Entropy and Temperature of Black 3-Branes

TL;DR

The paper probes the microscopic origin of black 3-brane entropy by comparing slightly non-extremal supergravity solutions to a D3-brane microstate count, modeling non-extremal excitations as a 3+1 dimensional gas of massless open-string states. It finds that the statistical entropy scales as $(\delta M/M_0)^{3/4}$ and nearly matches the Bekenstein-Hawking entropy, with exact agreement obtained when counting only transverse modes (N=6). For extremal, momentum-carrying 3-branes the horizon area vanishes, reflecting the non-extensivity of BPS entropy, and a finite-temperature interpretation links the gas temperature to the Hawking temperature. The results provide strong support for D-brane microstate counting while raising questions about a precise numerical factor and the role of worldvolume dynamics or boundary conditions in achieving exact entropy matching.

Abstract

We consider slightly non-extremal black 3-branes of type IIB supergravity and show that their Bekenstein-Hawking entropy agrees, up to a mysterious factor, with an entropy derived by counting non-BPS excitations of the Dirichlet 3-brane. These excitations are described in terms of the statistical mechanics of a 3+1 dimensional gas of massless open string states. This is essentially the classic problem of blackbody radiation. The blackbody temperature is related to the temperature of the Hawking radiation. We also construct a solution of type IIB supergravity describing a 3-brane with a finite density of longitudinal momentum. For extremal momentum-carrying 3-branes the horizon area vanishes. This is in agreement with the fact that the BPS entropy of the momentum-carrying Dirichlet 3-branes is not an extensive quantity.