The long string at the stretched horizon and the entropy of large non-extremal black holes

TL;DR

The paper proposes that long strings localized at the stretched horizon account for black hole entropy by matching the Hagedorn density of states to Hawking thermodynamics. Using the thermal scalar description, it derives an asymptotic density of states in flat and Rindler spaces, showing the Unruh temperature sets the Hagedorn scale near the horizon. A mean-field Shell construction demonstrates that incremental entropy from infalling string shells reproduces the Bekenstein-Hawking area law and, with α'-corrections, Wald entropy, while highlighting a universal $c=6$ sector tied to $T_H=T_{\mathrm{Hawking}}$. The framework provides a general, horizon-wide microstate picture for large non-extremal black holes and offers a controlled path to understanding black hole microstructure through long-string dynamics.

Abstract

We discuss how long strings can arise at the stretched horizon and how they can account for the Bekenstein-Hawking entropy. We use the thermal scalar field theory to derive the asymptotic density of states and corresponding stress tensor of a microcanonical long string gas in Rindler space. We show that the equality of the Hagedorn and Hawking temperatures gives rise to the tree-level entropy of large black holes in accordance with the Bekenstein-Hawking-Wald formula.

Figures (6)

• Figure 1: A cigar-shaped manifold allows winding strings to have arbitrarily small length.
• Figure 2: (a) Energy density $-\left\langle T^{0}_{0}(\mathbf{x})\right\rangle$ as a function of radial distance $\rho$ in units where $\alpha'=1$. (b) Radial pressure $\left\langle T^{\rho}_{\rho}(\mathbf{x})\right\rangle$. (c) Transverse pressure $\left\langle T^{i}_{i}(\mathbf{x})\right\rangle$.
• Figure 3: Left figure: structure of a rotating black hole. $\frac{\partial}{\partial t}$ becomes spacelike in the ergoregion (the gray region), whereas $\chi$ remains timelike in that region. Right figure: most dominant contribution to the grand canonical trace. The Rindler thermal scalar provides the dominant contribution (colored in red) and rotates rigidly along with the hole itself.
• Figure : (a)
• Figure : (b)