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Entanglement entropy of black holes from the first law of thermodynamics

This paper derives a closed, finite expression for the entanglement entropy of quantum matter in spacetimes with Killing horizons by integrating the first law of thermodynamics along a temperature-dependent family of equilibrium backgrounds, thereby reproducing the exact conical entropy (including contact terms) without relying on replica tricks or conical singularities.

Original authors: D. V. Diakonov, Gleb S. Zverev

Published 2026-10-09

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CC BY 4.0

Original authors: D. V. Diakonov, Gleb S. Zverev

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

In the deepest corners of the universe, where gravity is so intense that not even light can escape, black holes behave in ways that seem to defy our understanding of physics. For decades, scientists have treated these cosmic objects as if they were hot, glowing bodies that obey the same rules of heat and energy that govern a cup of coffee or a steam engine. This field, known as black hole thermodynamics, suggests that a black hole has a temperature and an entropy, a measure of disorder or hidden information. The most famous rule in this field states that the entropy of a black hole is directly tied to the size of its surface, the event horizon, rather than the volume of space inside it. This idea has led to a profound question: where does this entropy actually come from? Is it a property of the empty space itself, or does it arise from the quantum particles that swirl around the edge of the abyss?

To answer this, physicists have traditionally used a method that involves imagining the universe in a slightly broken state. They would calculate the entropy by creating a mathematical scenario where the temperature of the black hole does not match its natural, stable temperature. This approach, while useful, requires working with "off-shell" geometries—mathematical shapes that do not actually exist in nature and contain sharp, singular points where the rules of smooth space break down. It is a bit like trying to understand the shape of a perfect sphere by first studying a crumpled ball of paper and then hoping the math works out when you smooth it back down. The results of these calculations have been difficult to interpret, often leaving scientists unsure if the entropy they found was a real physical property or just an artifact of the strange mathematical tools they used.

A new study by Dmitrii Diakonov and Gleb Zverev offers a different path, one that stays strictly within the realm of what actually exists. Instead of forcing the universe into a broken, off-shell state, the researchers decided to calculate the entropy entirely "on-shell," meaning they only looked at black holes that are in perfect thermal equilibrium, exactly as they would appear in reality. They treated the black hole not as a static object, but as a system that can slowly change its temperature, much like a star that slowly cools or heats up over time. By tracking how the energy and pressure of the quantum matter surrounding the black hole change as the temperature shifts, they were able to integrate a fundamental law of thermodynamics to find the total entropy. This method required no imaginary singularities, no broken geometries, and no assumptions about the universe being in a state that cannot physically occur.

The researchers applied this new, cleaner method to two specific types of cosmic environments: a patch of de Sitter space, which resembles a universe expanding with a constant positive energy, and a non-rotating BTZ black hole, a simpler model of a black hole that exists in a universe with a negative curvature. In both cases, they calculated the entropy by looking at the renormalized stress-energy tensor, which is essentially a map of how much energy and pressure the quantum particles exert at every point in space. They then integrated this data along a path of changing temperatures. The result was a precise, finite number for the entropy that did not require any arbitrary cutting off of infinite values, a common problem in other methods.

What makes this finding particularly significant is that it matches the results of the older, more complicated off-shell methods perfectly, but without the mathematical headaches. The study confirms that the entropy calculated through this direct, on-shell approach is identical to the entropy found using the replica trick, a technique that involves creating multiple copies of the universe to count the hidden information. This agreement holds true even when the researchers included a specific correction term, known as a contact term, which appears when the quantum fields interact with the curvature of space in a specific way. This term had previously been a source of confusion, as it seemed to arise from the artificial singularities of the off-shell method. By showing that this term appears naturally in the on-shell calculation, the authors demonstrate that it is a real physical feature of the system, not a mathematical glitch.

The study also clarifies the relationship between two different ways of thinking about entropy: the thermodynamic view, which comes from the laws of heat and energy, and the statistical view, which counts the microscopic arrangements of particles. In many physical systems, these two views give the exact same answer. However, for black holes, there has been a long-standing debate about whether they differ. The authors show that when you account for the way the black hole's horizon shifts slightly due to the presence of quantum matter, the thermodynamic entropy and the statistical entropy align perfectly. This suggests that the "missing" information in a black hole is not a mystery hidden behind a mathematical veil, but a tangible quantity that can be measured by observing how the energy of the surrounding quantum fields responds to changes in temperature.

By avoiding the need for off-shell geometries, this work provides a more robust and physically grounded way to understand black hole entropy. It suggests that the entropy of a black hole is not just a boundary effect, a number attached to the surface of the event horizon, but a quantity that is deeply connected to the volume of space and the energy distribution within it. The researchers found that the entropy can be expressed as a volume integral, a sum of contributions from every point in the space surrounding the hole. This challenges the simple notion that black hole entropy is solely a surface phenomenon and points toward a more complex reality where the interior and exterior of the black hole are inextricably linked.

The implications of this work extend beyond just calculating a number. It offers a new framework for understanding how quantum mechanics and gravity interact. If the entropy can be derived entirely from the behavior of matter in a stable, equilibrium state, it strengthens the idea that black holes are ordinary thermodynamic systems, subject to the same laws that govern the rest of the universe. The study does not claim to solve every mystery of black hole physics, and the authors acknowledge that further work is needed to prove that this method works for all types of black holes, including those that are charged or spinning rapidly. However, for the specific cases they examined, the on-shell method provides a clear, finite, and consistent answer that agrees with the most sophisticated off-shell calculations.

In the end, this research represents a shift in perspective, moving away from the abstract and sometimes confusing mathematical tricks of the past toward a more direct engagement with physical reality. It shows that by respecting the laws of thermodynamics and staying within the bounds of what is physically possible, scientists can uncover the hidden properties of the universe's most extreme objects. The entropy of a black hole, once a source of deep theoretical confusion, begins to look like a natural consequence of the energy and pressure of the quantum fields that surround it, a quantity that can be understood through the simple, steady change of temperature. This approach does not just calculate a value; it offers a clearer picture of what a black hole actually is, grounding one of the most abstract concepts in modern physics in the concrete reality of energy and equilibrium.

Technical Summary: Entanglement Entropy of Black Holes from the First Law of Thermodynamics

Problem Statement
The computation of black hole entanglement entropy for quantum matter has traditionally relied on "off-shell" methods, most notably the replica trick. These techniques require constructing Euclidean manifolds with conical singularities at the horizon, where the temperature $\beta^{-1}$ differs from the Hawking temperature $\beta_H^{-1}$ . This approach introduces several theoretical complications:

Off-shell Geometry: The background metric does not satisfy the vacuum gravitational equations, necessitating the exclusion of regions near the horizon or the use of incomplete solutions.
Divergences: The resulting entropy is typically divergent (proportional to $A/\epsilon^2$ ), requiring renormalization of Newton's constant and higher-curvature couplings.
Contact Terms: For non-minimally coupled fields, the conical method yields a contact term $Q$ (related to the curvature singularity) that lacks a straightforward statistical-mechanical explanation in standard scalar field theory.
Thermodynamic Consistency: There is a persistent question regarding the equivalence between the thermodynamic entropy (derived from the partition function on-shell) and the statistical-mechanical (von Neumann) entropy, particularly when quantum backreaction shifts the horizon area.

The authors pose the fundamental question: Why are off-shell methods necessary to compute black hole entanglement entropy? They aim to establish a direct relation between off-shell calculations and the thermodynamic entropy arising from the on-shell first law of thermodynamics.

Methodology
The paper proposes an entirely on-shell method to compute the entanglement entropy of quantum matter. The core methodology involves:

Integration of the First Law: Instead of introducing conical singularities, the authors consider a one-parameter family of classical equilibrium backgrounds labeled by the Hawking temperature $T_H$ . They integrate the first law of thermodynamics along this family.
Covariant Formulation: The entropy is derived from the variation of the ADM mass formula for a black hole surrounded by classical matter. The matter terms in the mass variation formula are identified with the variation of entanglement entropy ( $T_H \delta S_{ent}$ ).
Key Formula: The resulting expression for the on-shell entanglement entropy of matter is:
$S_{matt}^{on-shell} = \int^{T_H} \frac{dT_H}{T_H} \left( \frac{1}{2} \int \langle T^{\mu\nu} \rangle \frac{\partial g_{\mu\nu}}{\partial T_H} \xi_\rho dV^\rho - \frac{\partial}{\partial T_H} \int \langle T^\mu_\nu \rangle \xi_\nu dV_\mu \right)$
Here, $\langle T^{\mu\nu} \rangle$ is the renormalized expectation value of the stress-energy tensor in the thermal state at the Gibbons–Hawking temperature. Crucially, the background metric $g_{\mu\nu}$ is treated as a function of $T_H$ .
Handling Boundary Terms: The authors address the interchange of integration and differentiation (variation) by fixing the horizon location in rescaled coordinates ( $\bar{r} = r/r_h$ ). They demonstrate that the difference between the covariant integral form and the local thermodynamic form is a boundary term that vanishes under physical conditions (finite energy density for a freely falling observer crossing the horizon).

Key Contributions and Results
The paper provides a closed, finite expression for entanglement entropy that requires no replicas and no conical singularities. The specific contributions and results are:

Equivalence of On-shell and Off-shell Methods:
The authors demonstrate that the on-shell integration method reproduces the results of the off-shell conical singularity method exactly, including the contact term $Q$ for non-minimally coupled fields, in two specific cases:
- De Sitter Space: For the static patch of de Sitter space, the on-shell calculation using the renormalized stress-energy tensor (which is proportional to the metric) reproduces the off-shell result derived via the replica trick. The boundary term vanishes identically because $\rho + p_r = 0$ .
- BTZ Black Hole: For the non-rotating BTZ black hole (asymptotically AdS), the on-shell calculation using the known renormalized stress-energy tensor yields an entropy expression identical to the matter contribution of the off-shell conical entropy, including the specific series terms and the contact term.
  The authors note that while these examples confirm the equivalence, a general proof for arbitrary geometries and field theories remains an open question.
Resolution of the Contact Term $Q$ :
The on-shell method naturally incorporates the contact term $Q$ (defined as $Q = \beta_H \xi \int_\Sigma \langle \phi^2 \rangle$ ) without ad-hoc adjustments. The paper confirms the relation:
$S_{matt}^{on-shell} = S_{matt}^{con} = S_{matt}^{SM} - Q$
where $S_{matt}^{SM}$ is the statistical-mechanical entropy. This validates the interpretation of $Q$ as a necessary component for self-consistent renormalization and its connection to Wald entropy or soft modes.
Thermodynamic Consistency with Backreaction:
In the de Sitter example, the authors demonstrate that when backreaction is included (shifting the horizon area $\bar{A}$ and temperature $\bar{T}_H$ ), the total thermodynamic entropy satisfies:
$S_{tot}^{TD} = \bar{S}_{BH} + S_{matt}^{SM} - Q = \log Z_{total}$
This confirms that the thermodynamic entropy is determined by the free energy of the full system, even at the one-loop level, provided the shift in the horizon area due to quantum matter is accounted for in the gravitational sector.
Extensivity of Entanglement Entropy:
The paper challenges the view that entanglement entropy is purely a boundary term dependent only on the area of the entangling surface. The on-shell formulation expresses the entropy as a volume integral ( $S \sim \int dV$ ), suggesting it is an extensive quantity arising from bulk contributions, even though it can be rewritten as a boundary term via total derivatives in specific off-shell contexts.

Significance and Claims
The authors claim that their work establishes a rigorous link between the statistical-mechanical definition of entanglement entropy and the thermodynamic first law without invoking off-shell geometries, as verified in the specific examples of de Sitter space and the BTZ black hole.

Finite by Construction: The method yields a finite, renormalized entropy directly from the renormalized stress-energy tensor, bypassing the need for explicit UV cutoffs and counterterm matching required in the replica trick.
General Applicability: While the equivalence is explicitly demonstrated for de Sitter space and the BTZ black hole, the formalism is presented as applicable to static, spherically symmetric spacetimes, though a general proof for arbitrary geometries and field theories (including self-interacting fields and higher-derivative gravity) is identified as a direction for future work.
Conceptual Shift: The results suggest that black hole entanglement entropy is fundamentally a thermodynamic quantity that can be computed entirely within the framework of equilibrium states, resolving the apparent necessity of off-shell singularities for its calculation in the tested cases.

The paper concludes by noting that while the equivalence is proven for specific examples, a general proof for arbitrary geometries and field theories (including self-interacting fields and higher-derivative gravity) remains an open question for future work.

Technical Summary: Entanglement Entropy of Black Holes from the First Law of Thermodynamics

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