Holographic Central Charge Effects on Black Hole Thermodynamics and Quantum Information

TL;DR

The paper investigates how the holographic boundary central charge $C$ governs the nature of bulk gravity in AdS/CFT, spanning classical and quantum regimes via AdS–Schwarzschild black holes. It combines Loop Quantum Gravity (for small $C$) with classical GR (for large $C$) to reveal contrasting thermodynamic behaviors: quantum black holes remain thermodynamically stable with entropy reduced relative to the classical Bekenstein–Hawking value, while classical black holes exhibit a two-phase structure with unstable small and stable large branches. The entanglement entropy of Hawking radiation is analyzed through the island formula, showing a $C$-dependent pre-Page-time growth and a post-Page-time entropy consistent with unitarity, including a logarithmic correction in $C$. Overall, the boundary central charge acts as a crucial holographic parameter linking boundary degrees of freedom to bulk geometry, black hole thermodynamics, and information recovery in evaporation.

Abstract

In this paper, based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, we highlight the fundamental role of the holographic central charge in connecting the boundary theory to quantum information, black hole thermodynamics, and the nature of gravity in the bulk. We establish that the large central charge of the boundary conformal field theory corresponds to classical gravity, while a small central charge corresponds to quantum gravity described by Loop Quantum Gravity. In addition, we study the thermodynamic behavior of AdS-Schwarzschild black holes for both large and small central charges. For large central charge, the classical AdS-Schwarzschild black holes have two phases: unstable small black holes and stable large black holes. Conversely, for small central charge, black holes are stable, and their entropy is smaller than that of classical black holes. To explore the influence of the boundary central charge on the information loss paradox, we use the island formula to recover the Page curve. We find that before the Page time, the entanglement entropy of Hawking radiation increases with time, and its slope is determined by the central charge of the boundary theory. After the Page time, the island inside black holes emerges, and the unitarity of black hole evaporation is restored, yielding a constant entropy consistent with the Page curve. This entanglement entropy, i.e. after the Page time, depends on the Bekenstein-Hawking entropy and includes a logarithmic correction related to the central charge.

Figures (3)

• Figure 1: Heat capacity normalized as a function of entropy for various values of the central charge.
• Figure 2: Penrose diagrams of classical and quantum black holes without island.
• Figure 3: Penrose diagrams of classical and quantum black holes with island, represented by the green curve.