⚛️ general relativity

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

This paper employs reduced phase space quantization with the Misner--Sharp--Hernandez mass to derive discrete spectra for Schwarzschild--de Sitter black hole areas and masses, ultimately demonstrating that the resulting entropy for both event and cosmic horizons exhibits a robust logarithmic correction to the Bekenstein--Hawking term.

Original authors: S. Jalalzadeh, H. Moradpour

Published 2026-02-03

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CC BY 4.0

Original authors: S. Jalalzadeh, H. Moradpour

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, complex machine. For a long time, scientists have been trying to figure out how the tiny, quantum world (the world of atoms and particles) fits together with the huge, smooth world of gravity and black holes. This paper is a new attempt to solve that puzzle, specifically for a type of black hole that exists in a universe that is expanding (like our own).

Here is the story of what the authors did, explained simply:

1. The Problem: A Tricky Balance

The authors are studying a Schwarzschild-de Sitter (SdS) black hole. Think of this as a black hole sitting inside a universe that is stretching out.

The Black Hole: It has an "event horizon," a point of no return where gravity is so strong nothing can escape.
The Cosmic Horizon: Because the universe is expanding, there is also a second "horizon" far away. It's like a cosmic fence; things beyond it are moving away so fast we can never reach them.

Usually, when scientists try to measure the energy of a black hole, they use tools designed for empty, flat space. But in this expanding universe, those old tools break. They don't work because there is no "edge" of the universe to measure from.

2. The Solution: A New Ruler (The MSH Mass)

To fix this, the authors used a special tool called the Misner-Sharp-Hernandez (MSH) mass.

The Analogy: Imagine you are trying to weigh a fish inside a swimming pool. If you try to weigh the whole pool, it's messy. But if you use a special net that only weighs the water immediately surrounding the fish, you get a perfect, local measurement.
The MSH mass is that "local net." It measures the energy contained right around the black hole and the cosmic horizon, regardless of how the universe is expanding. It's the perfect ruler for this specific job.

3. The Experiment: Turning the Universe into a Piano

The authors used a method called Reduced Phase Space Quantization.

The Analogy: Imagine the black hole and the cosmic horizon are like two strings on a guitar. In classical physics, these strings can vibrate at any pitch. But in the quantum world, strings can only vibrate at specific, distinct notes (like the keys on a piano).
The authors treated the energy of these horizons as if they were musical notes. They did some complex math (canonical transformations) to show that the energy of the black hole and the cosmic horizon can't be just any number. They must be discrete steps, like climbing a ladder where you can only stand on the rungs, not in between them.

4. The Discovery: The "Logarithmic" Whisper

Once they figured out that the energy comes in these specific steps, they calculated the entropy (a measure of disorder or information) of the black hole.

The Old Rule: For decades, scientists believed entropy was just directly proportional to the size of the black hole's surface area (like the surface of a balloon).
The New Finding: The authors found that there is a tiny "whisper" added to that rule. When you look very closely at the quantum steps, the entropy isn't just the area. It has an extra term that looks like a logarithm.
The Analogy: Imagine you are counting the tiles on a floor. The old rule said, "The number of tiles is exactly the area." The new rule says, "The number of tiles is the area, plus a tiny, subtle correction that depends on how you count them."

5. What This Means

The paper concludes that this "logarithmic correction" is a robust feature. It appears whether you look at the black hole horizon or the cosmic horizon.

The Coefficient: The authors calculated a specific number for this correction (related to $\pi/2$ ). However, they are careful to say this number might change if you use a different mathematical method. It's like getting a slightly different measurement depending on whether you use a ruler or a tape measure, but the fact that there is a correction is the important part.
The Big Picture: This supports the idea that the universe is "pixelated" at the smallest scales. The smooth surface of a black hole is actually made of tiny, discrete quantum bits, and this creates a small, predictable wobble in the entropy formula.

Summary

In short, the authors took a black hole in an expanding universe, used a special local energy meter (MSH mass) to avoid mathematical errors, and discovered that the black hole's energy comes in specific quantum steps. This discovery proves that the entropy of the black hole has a small, logarithmic correction to the standard formula, confirming that quantum mechanics leaves a distinct fingerprint on the thermodynamics of black holes.

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

Problem Statement
While a complete theory of quantum gravity remains elusive, semi-classical approaches have yielded significant insights into black hole (BH) thermodynamics. A central question in this domain is the quantization of BH mass and the resulting corrections to the Bekenstein–Hawking entropy. Previous studies have established that logarithmic corrections to entropy appear across various frameworks (e.g., loop quantum gravity, conformal field theory, string theory), though the precise coefficients vary.

A specific challenge arises in the Schwarzschild–de Sitter (SdS) spacetime, which contains both a BH event horizon (EH) and a cosmological horizon. Standard mass definitions, such as the ADM or Komar masses, are ill-suited for SdS spacetime because it lacks a globally timelike Killing vector and an asymptotically flat region required to anchor these definitions. Consequently, a robust framework is needed to define quasi-local energy and perform reduced phase space quantization in the presence of a positive cosmological constant ( $\Lambda$ ).

Methodology
The authors employ the reduced phase space quantization method, utilizing the Misner–Sharp–Hernandez (MSH) mass as the fundamental internal energy. The methodology proceeds through the following steps:

Definition of Variables: The MSH mass is chosen as the dynamical variable because it provides a coordinate-invariant, quasi-local measure of gravitational energy within a 2-sphere of areal radius $R$ , remaining well-defined even with $\Lambda > 0$ . For the SdS geometry, two MSH masses, $M_1$ and $M_2$ , are associated with the BH and cosmological horizons, respectively.
Thermodynamic Framework: The unified first law of thermodynamics is applied to the SdS spacetime. The authors derive the relationship between the variation of MSH mass ( $dM_i$ ), entropy ( $dS_i$ ), and thermodynamic volume ( $dV_i$ ), incorporating the work density term arising from the cosmological constant.
Canonical Transformation: To facilitate quantization, a canonical transformation is performed. The MSH masses ( $M_i$ ) and their conjugate momenta ( $\Pi_i$ ) are mapped to new variables ( $x_i, p_i$ ) representing a 2D harmonic oscillator phase space. This transformation incorporates the periodicity of the Euclidean time coordinate, which is related to the inverse Hawking temperature of the respective horizons.
Quantization: The system is quantized by promoting the new phase space variables to operators on a Hilbert space. The Hamiltonian operator is constructed, leading to a Schrödinger equation with solutions in terms of Hermite polynomials. This yields a discrete spectrum for the MSH masses and, consequently, the horizon areas.
Entropy Derivation: Using the derived mass spectrum and the unified first law, the authors calculate the entropy. They consider transitions between adjacent quantum states (Hawking radiation processes) to determine the variations in mass and area, from which the quantum-corrected entropy is extracted.

Key Contributions and Results

Discrete Spectra: The paper derives a discrete spectrum for the MSH masses of both the BH and cosmological horizons:
$M_i = \frac{m_P}{\sqrt{2}} \sqrt{n_i + \frac{1}{2}}, \quad n_i = 0, 1, 2, \dots$
where $m_P$ is the Planck mass. This result generalizes the Kuchař–Louko quantization of the Schwarzschild BH to multi-horizon systems.
Area Quantization: The discrete mass spectrum implies a discrete area spectrum for the horizons:
$A_i = 8\pi l_P^2 \left(n_i + \frac{1}{2}\right)$
Logarithmic Entropy Correction: By applying the unified first law to the quantized mass and area variations, the authors derive the entropy for both horizons. The result includes a logarithmic correction to the standard Bekenstein–Hawking term:
$S_i = \frac{A_i}{4G} + \frac{\pi}{2} \ln\left(\frac{A_i}{4G}\right) + \text{const.}$
Coefficient Specificity: The coefficient of the logarithmic term is found to be $\pi/2$ within this specific semiclassical scheme. The authors explicitly note that this value is not universal but depends on the chosen semiclassical approximation and phase space measure.

Significance and Claims
The paper claims that its primary contribution is the successful application of reduced phase space quantization to the SdS spacetime using the MSH mass, thereby overcoming the limitations of ADM/Komar masses in non-asymptotically flat geometries.

The authors assert that their results support the robustness of the logarithmic form of quantum corrections in SdS thermodynamics. They argue that while the numerical coefficient of the logarithmic term varies across different theoretical approaches (e.g., $-3/2$ in loop quantum gravity, $+1/2$ in statistical approaches), the structural presence of the logarithmic term appears to be an intrinsic feature of the gravitational phase space, emerging from the discreteness of geometric degrees of freedom.

The work demonstrates that the MSH mass provides a geometrically invariant foundation for quantization that naturally interpolates between the Schwarzschild ( $\Lambda \to 0$ ) and de Sitter ( $m \to 0$ ) regimes. The authors conclude that their framework offers a transparent route to deriving these corrections directly from quasi-local energy balance, reinforcing the idea that the universality of the logarithmic form stems from the covariance and adiabatic invariance of horizon dynamics, even if the specific prefactors remain model-dependent.