Schwarzschild Black Holes in Various Dimensions from Matrix Theory

TL;DR

The paper investigates Schwarzschild black holes in various dimensions within Matrix theory by extending BFKS to Matrix theory compactified on $T^d$. It leverages an equation of state inferred from near-extremal RR-charged $d$-branes and the DLCQ relation $E = \frac{M^2}{P_-} = \frac{M^2 R}{N}$ to derive the mass–entropy scaling, establishing $N \sim S$ as a key registry. The main result is that the correct scaling $S \sim M^{\frac{D-2}{D-3}} G_D^{\frac{1}{D-3}}$ holds for Schwarzschild black holes in $4 \le D \le 11$, with explicit confirmations for $d=1$ ($D=10$) and $d=4$ ($D=7$) and discussions of special cases ($d=5$, $d=6$). This supports a universal Matrix-theory picture of black hole thermodynamics across dimensions and hints at emergent dimensions in the gauge theory governing thermodynamics.

Abstract

In a recent paper it was shown that the properties of Schwarzschild black holes in 8 dimensions are correctly described up to factors of order unity by Matrix theory compactified on T^3. Here we consider compactifications on tori of general dimension d. Although in general little is known about the relevant d+1 dimensional theories on the dual tori, there are hints from their application to near-extreme parallel Dirichlet d-branes. Using these hints we get the correct mass-entropy scaling for Schwarzschild black holes in (11-d) dimensions.