Reduced Phase Space Quantization and Quantum Corrected Entropy of Schwarzschild-de Sitter Horizons

TL;DR

The paper applies reduced phase-space quantization to Schwarzschild--de Sitter spacetimes using the Misner--Sharp--Hernandez mass as the internal energy to obtain a two-horizon quantum framework. It shows the BH and cosmological horizon areas and MSH masses are discretized via a pair of decoupled harmonic-oscillator sectors, yielding $M_i=(m_P/\sqrt{2})\sqrt{n_i+1/2}$ and $A_i=8\pi l_P^2(n_i+1/2)$ in the $\Lambda\to0$ limit. Employing the unified first law, it derives a quantum-corrected entropy for each horizon, $S_i = A_i/(4G) + (\pi/2)\ln(A_i/(4G)) + \cdots$, with a non-universal logarithmic coefficient. The results reinforce the universal appearance of logarithmic corrections to horizon entropy across quantum gravity approaches and point to possible observational signatures in SdS thermodynamics and horizon spectroscopy.

Abstract

This paper investigates the quantization of the Schwarzschild--de Sitter (SdS) black hole (BH) using the Misner--Sharp--Hernandez (MSH) mass as the internal energy in a reduced phase space framework. After introducing the canonical variables of the reduced phase space, we derive a discrete spectrum for the surface areas of the BH event horizon (EH) as well as MSH masses. We utilized the MSH mass spectrum to obtain the entropy of the BH. The entropy of the BH and cosmic EHs reveals a logarithmic correction to the Bekenstein--Hawking term. Our results support the robustness of the logarithmic form of quantum corrections in SdS thermodynamics.