Black Hole Entropy: Off-Shell vs On-Shell

TL;DR

This work clarifies the distinction between thermodynamical entropy $S^{TD}$ and statistical-mechanical entropy $S^{SM}$ for black holes by juxtaposing on-shell Euclidean methods with several off-shell approaches (brick-wall, conical singularity, blunt cone, volume cut-off). It demonstrates that off-shell entropies generally diverge or differ from the on-shell result, but a universal finite relation emerges: $S^{TD}=S^{BH}(\bar{r}_+)+[S^{SM}-S^{SM}_{Rindler}]$, with a horizon shift ${\bar r}_+$ due to quantum backreaction and a subtraction of Rindler entropy accounting for near-horizon divergences. The conical-singularity and blunt-cone methods yield finite, regulator-independent results for the combination $S^{SM}-S^{SM}_{Rindler}$, while BW and VC introduce regulator-dependent terms that cancel in the on-shell limit. The findings illuminate how quantum corrections modify the background geometry and how renormalization yields finite observable quantities, providing a coherent link between microscopic (statistical) and macroscopic (thermodynamic) entropy notions, including insights into non-equilibrium or dynamical scenarios via off-shell frameworks.

Abstract

Different methods of calculation of quantum corrections to the thermodynamical characteristics of a black hole are discussed and compared. The relation between on-shell and off-shell approaches is established. The off-shell methods are used to explicitly demonstrate that the thermodynamical entropy $S^{TD}$ of a black hole, defined by the first thermodynamical law, differs from the statistical-mechanical entropy $S^{SM}$, determined as $S^{SM}=-\mbox{Tr}(\hatρ^H\ln\hatρ^H)$ for the density matrix $\hatρ^H$ of a black hole. It is shown that the observable thermodynamical black hole entropy can be presented in the form $S^{TD}=π{\bar r}_+^2+S^{SM}-S^{SM}_{Rindler}$. Here ${\bar r}_+$ is the radius of the horizon shifted because of the quantum backreaction effect, and $S^{SM}_{Rindler}$ is the statistical-mechanical entropy calculated in the Rindler space.

Figures (7)

• Figure 1: Embedding diagram for a two-dimensional Gibbons-Hawking instanton. Regularity condition at the Euclidean horizon $r=r_+$ requires $\beta_\infty=\beta_H\equiv 8\pi m$.
• Figure 2: A region $M_B$ of the Gibbons-Hawking instanton with the external boundary $\Sigma_B$ at $r=r_B$. This region is conformal to the 2-D flat unit disk $D^2$.
• Figure 3: Conformal maps of the region $M_{B,\epsilon}$ of the Gibbons-Hawking instanton onto the part $K_{\alpha,\epsilon_x}$ of the cone $C_{\alpha}$, and of the region $K_{\alpha,\epsilon_x}$ onto the cylinder $Q_{\alpha,\epsilon_z}$. $\epsilon$ is the proper distance of the inner boundary $\Sigma_{\epsilon}$ of $M_{B,\epsilon}$ to the horizon. The parameter $\epsilon_x$ is the distance from $\Sigma_B$ to the vertex of the cone along the cone generator, and $\epsilon_z$ is the length of the cylinder generator (both measured in the units of $\mu$). The circumference length of the cylinder, as well as the circumference length of of the external boundary $\Sigma_B$ of the cone, (measured in units $\mu$) is $2\pi\alpha$.
• Figure 4: Conformal map of a singular instanton $M_\beta^\alpha$ onto the standard cone $C_{\alpha}$.
• Figure 5: Blunt instanton and its conformal transformation onto a unit disk $D^2$