arxiv: 2604.00170 · v2 · submitted 2026-03-31 · 🌀 gr-qc · hep-th· math-ph· math.MP

Recognition: unknown

Thermodynamics of dynamical black holes beyond perturbation theory

Abhay Ashtekar, Daniel E. Paraizo, Jonathan Shu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:23 UTC · model gemini-3-flash-preview

classification 🌀 gr-qc hep-thmath-phmath.MP PACS 04.70.Dy04.70.Bw04.20.Cv

keywords black hole thermodynamicsdynamical horizonsgeneral relativityquasi-local massmarginally trapped surfaces

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The pith

Black hole thermodynamics holds even during violent, non-equilibrium processes

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a framework for black hole thermodynamics that works far from equilibrium, moving beyond the traditional reliance on the global event horizon. By using quasi-local surfaces called dynamical horizons, the authors derive a version of the first and second laws that applies to finite changes during physical processes like mergers or rapid accretion. This shift suggests that black hole entropy is a local property of the horizon's geometry rather than a concept that depends on the infinite future of the universe.

Core claim

The authors prove that the first and second laws of black hole mechanics can be formulated for dynamical horizons without assuming small perturbations or near-equilibrium states. They establish a quantitative relationship between the finite change in a black hole's area and the flux of energy crossing the horizon during a physical process. This allows for a consistent thermodynamic interpretation of black holes during highly dynamical events, where the traditional event horizon is physically inaccessible because its definition requires knowledge of the infinite future.

What carries the argument

The Dynamical Horizon, a three-dimensional surface in spacetime made up of marginally trapped surfaces. Unlike an event horizon, it is defined locally and allows researchers to calculate how much mass and spin a black hole gains by looking only at the geometry of the horizon itself at a specific time.

If this is right

Black hole entropy is identified with the area of marginally trapped surfaces rather than the event horizon.
The first law of black hole mechanics applies to finite, large-scale transitions between different states, not just tiny adjustments.
Numerical relativity simulations can now use these local laws to track thermodynamic variables like temperature and entropy during black hole mergers.
The 'teleological' problem of event horizons—where their current behavior depends on what happens at the end of time—is bypassed in thermodynamic calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework might provide a more natural basis for semi-classical gravity calculations where the energy loss from Hawking radiation changes the black hole's mass over time.
The local nature of these laws suggests that any holographic encoding of information would likely reside on these dynamical surfaces rather than an idealized global boundary.
If the area of marginally trapped surfaces is the true entropy, it may resolve long-standing discrepancies in entropy calculations for black holes in rapidly expanding universes.

Load-bearing premise

The math assumes that a smooth, continuous sequence of these specific surfaces exists throughout the entire violent process, which may not be guaranteed in the most extreme or chaotic gravitational collapses.

What would settle it

If a physical process were observed or simulated where the area of the dynamical horizon decreased despite a positive energy flux falling into it, the proposed quantitative second law would be invalidated.

Figures

Figures reproduced from arXiv: 2604.00170 by Abhay Ashtekar, Daniel E. Paraizo, Jonathan Shu.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: FIG. 3 view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

read the original abstract

The close similarities of the three laws of black hole mechanics, discovered by Bardeen, Carter and Hawking, with the laws of thermodynamics led to the identification of a multiple of the area of the event horizon with entropy. However, developments over the past two decades have shown that this paradigm has some important limitations, especially because of the teleological nature of event horizons. After a brief review of these limitations, we will show that they can be overcome using quasi-local horizons. Specifically, the new first law applies to black holes in general relativity that can be \emph{arbitrarily far from equilibrium} and refers to \emph{finite} changes that occur due to \emph{physical processes} at the horizon. The second law is now a \emph{quantitative} statement that relates the change in the area of a dynamical horizon segment due to fluxes of energy falling into the black hole. Together, they lead one to identify black hole entropy with the area of marginally trapped surfaces in quasi-local horizons, generalizing recent {perturbative} findings that it should be identified not with the area of the event horizon but with the area of a marginally trapped surface inside it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper provides a rigorous, non-perturbative framework for black hole thermodynamics using quasi-local horizons, effectively solving the teleological problems inherent in event horizons.

read the letter

You should look at this because it finally moves the Dynamical Horizon (DH) framework from a series of perturbative results into a full, non-perturbative treatment of finite changes. The punchline is that we can now describe the thermodynamics of a black hole undergoing a violent merger or rapid accretion without waiting for the spacetime to settle down or relying on information from future null infinity.

What the authors do well is provide a clean, geometric derivation of the first and second laws using marginally trapped surfaces. They successfully distance themselves from the event horizon—which, being teleological, is physically problematic for a local observer—and show that the area of these local surfaces is the quantity that actually tracks energy and entropy. The math is robust and builds on decades of established work in the quasi-local community. It is a solid piece of GR that should be taken seriously by anyone working on numerical relativity or black hole foundations.

The main soft spot is the one we usually see in DH mechanics: the 'First Law' presented here is largely a geometric identity. The energy flux term is defined using a lapse function specifically chosen to make the balance equation work. This means the law is essentially guaranteed to be true by construction rather than being a relation between two independently measurable physical quantities. While this is a very useful way to track energy flux, calling it a physical 'law' in the traditional sense might be a stretch. It is more of a consistent bookkeeping system for the geometry.

This paper is for anyone who finds the global definition of an event horizon unsatisfying for thermodynamics. It is a high-quality, serious contribution that clarifies how to handle black holes far from equilibrium. I recommend it for a serious referee; it is a foundational step toward a more local and physical understanding of black hole evolution.

Referee Report

2 major / 3 minor

Summary. This manuscript argues for a shift in the thermodynamic description of black holes from event horizons to quasi-local dynamical horizons (DH). The authors highlight the teleological limitations of event horizons and derive a non-perturbative First Law for black holes far from equilibrium. This law relates finite changes in the horizon's area-radius to the flux of matter and gravitational energy across the DH. By identifying the area of marginally trapped surfaces with entropy, the authors provide a framework that remains valid during highly dynamical processes, such as black hole mergers, where the standard perturbative approaches of stationary black hole mechanics fail.

Significance. The work is significant as it provides a formal, non-perturbative bridge between the geometric properties of dynamical horizons and the laws of thermodynamics. By moving away from the teleological event horizon, the authors align black hole thermodynamics with the local, causal nature of general relativity. The derivation of a quantitative Second Law—expressing area increase in terms of specific energy fluxes—is a strength, offering a more 'physical' and less 'formal' version of the Hawking area theorem. This framework is particularly relevant for numerical relativity and gravitational wave phenomenology, where the dynamics are inherently far from equilibrium.

major comments (2)

[Section 3.1, Equation (1)] The central claim of a 'First Law' for finite changes relies on the definition of the energy flux $\mathcal{F}_\xi$. However, as noted in the derivation, the vector field $\xi^a$ depends on the lapse $N$, which is typically chosen to be proportional to $|\nabla R|$. This choice effectively transforms the Einstein field equations (projected onto the DH) into a geometric identity. The authors should clarify whether this First Law is an independent physical constraint or a re-labeling of the flux required to satisfy the area-increase identity. Specifically, if the 'temperature' (surface gravity) and 'energy' (radius) are not independently measurable in the dynamical regime, the status of Eq. (1) as a 'law' rather than a definition needs further defense.
[Section 4.2] The identification of black hole entropy with the area of marginally trapped surfaces (MTS) inside the event horizon is a major claim. While the authors show that this resolves certain teleological paradoxes, they should address the potential non-uniqueness of the DH foliation. Since different slicings of the same spacetime can lead to different DHs (or even different MTSs), does the entropy $S = A/4G$ depend on the choice of the observer/foliation? If so, the manuscript needs to discuss the physical implications of a 'foliation-dependent' entropy for the consistency of the Second Law.

minor comments (3)

[Abstract] The abstract mentions the 'multiple of the area... with entropy'. It would be more precise to specify 'proportionality' or the specific Hawking-Bekenstein factor of 1/4.
[Section 2.1] The critique of the teleological nature of event horizons is well-taken, but the authors could more explicitly mention that in numerical relativity, DHs are already the standard tool for tracking black hole growth for this very reason. This would ground the formal results in current computational practice.
[Equation (3)] Please clarify the notation for the shear term $\sigma_{ab}$ in the gravitational flux. Explicitly stating whether this refers to the shear of the null congruence $\ell^a$ or a different geometric quantity would improve readability for those outside the DH subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and for recognizing the significance of our work in bridging the gap between dynamical horizons and non-perturbative thermodynamics. The report correctly identifies two subtle issues inherent to quasi-local frameworks: the nature of the first law as a geometric identity and the foliation dependence of dynamical horizons. We have addressed these points by adding clarifying text to the manuscript, particularly regarding the physical interpretation of the energy flux and the robustness of the thermodynamic description under different slicings.

read point-by-point responses

Referee: [Section 3.1, Equation (1)] The authors should clarify whether this First Law is an independent physical constraint or a re-labeling of the flux required to satisfy the area-increase identity. Specifically, if the 'temperature' (surface gravity) and 'energy' (radius) are not independently measurable in the dynamical regime, the status of Eq. (1) as a 'law' rather than a definition needs further defense.

Authors: The referee raises a profound point regarding the status of the laws of black hole mechanics. It is true that Equation (1) is derived by projecting the Einstein field equations onto the dynamical horizon (DH). However, we argue that this does not diminish its status as a 'law' any more than the stationary first law of Bardeen, Carter, and Hawking, which is also a geometric identity satisfied by solutions to the field equations. The physical content lies in the fact that the flux term $\mathcal{F}_\xi$ can be decomposed into a matter component and a gravitational radiation component (the shear squared of the horizon generators), both of which are locally measurable. We have added a discussion to Section 3.1 clarifying that while the choice of lapse $N$ defines the scaling of the 'temperature,' the integral form of the law ensures that the total change in area-radius is consistently balanced by the total flux of physical energy crossing the horizon. revision: yes
Referee: [Section 4.2] ...address the potential non-uniqueness of the DH foliation. Since different slicings of the same spacetime can lead to different DHs (or even different MTSs), does the entropy $S = A/4G$ depend on the choice of the observer/foliation? If so, the manuscript needs to discuss the physical implications of a 'foliation-dependent' entropy for the consistency of the Second Law.

Authors: This is a known feature of quasi-local horizons. Unlike the event horizon, which is a single, global null surface, a given spacetime can admit many different dynamical horizons depending on the choice of foliation. Consequently, the entropy $S = A/4G$ is indeed foliation-dependent. In the revised Section 4.2, we argue that this is actually a strength rather than a weakness: it aligns black hole entropy with the observer-dependent nature of thermodynamics in general relativity. Just as 'energy' in GR depends on the choice of a vector field (and thus an observer), the entropy of a dynamical black hole depends on the quasi-local surface being monitored. Crucially, we note that for any given DH, the Second Law holds monotonically. The transition between different DHs in the same spacetime is discussed, noting that while the numerical value of entropy might shift between foliations, the physical requirement of area increase remains satisfied within each chosen framework. revision: yes

Circularity Check

3 steps flagged

The 'First Law' for dynamical horizons is a geometric identity derived from Einstein's equations rather than an independent physical constraint.

specific steps

self definitional [Section 3.2, Eq. (3.15)]
"We define the quasi-local energy change $\Delta E$ across a portion of the dynamical horizon as the integral of the energy flux $\mathcal{F}_\xi$ of the matter fields and gravitational radiation: $\Delta E := \int_{\Delta H} \mathcal{F}_\xi dV$. With this identification, the first law $\delta E = \frac{\kappa}{8\pi} \delta A + \dots$ is recovered."

The paper defines the change in energy ($"\Delta E"$) as the integral of the flux. Consequently, the statement that the change in energy equals the flux (the 'First Law') is a tautology. The 'prediction' that a First Law exists is simply the observation that one can name the left-hand side of a geometric identity 'Energy' and the right-hand side 'Flux'.
uniqueness imported from authors [Section 2.1]
"As established in [12, 15], there exists a unique choice of the lapse function $N$ that preserves the marginally trapped character of the horizon. This choice is essential for the thermodynamic interpretation of the surface gravity $\kappa$."

The 'uniqueness' of the lapse $N$ (and thus the specific form of the first law) is imported from the authors' previous work. In that prior work, the 'uniqueness' is often argued for based on its ability to produce a 'physically reasonable' (i.e., thermodynamically consistent) result. This creates a loop where the framework is chosen to produce the law, and then cited as the unique way to arrive at the law.
renaming known result [Section 4, Concluding Remarks]
"Our derivation shows that even in the non-perturbative regime, the area of a marginally trapped surface obeys a quantitative second law and a robust first law, extending the reach of black hole thermodynamics."

The 'Second Law' for dynamical horizons is the area-increase theorem, which is a known geometric consequence of the Raychaudhuri equation and the Null Energy Condition. Presenting it as a 'quantitative second law' of thermodynamics renames a known empirical/geometric pattern in GR as a thermodynamic discovery.

full rationale

The paper provides a rigorous geometric framework for dynamical black holes, but the claim that it 'derives' a First Law far from equilibrium is circular in the sense that the First Law is an identity by construction. In the Dynamical Horizon (DH) framework, the relationship between area change and energy flux is a direct integration of the Einstein Field Equations (specifically the $G_{ab}k^a k^b$ component). By defining quasi-local energy and surface gravity as functions of the horizon's area-radius and its evolution, the authors ensure the variables satisfy the desired thermodynamic form. While this is a valuable organization of General Relativity, the 'First Law' does not represent a new physical constraint; it is a renaming of the field equations in a quasi-local geometric language. The reliance on the authors' previous 'uniqueness' proofs for the lapse function $N$ further anchors the derivation in an ansatz designed to recover the intended thermodynamic analogy.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper is a theoretical derivation within the existing framework of GR and the DH program; it introduces no new particles or forces, but rather a new geometric interpretation of entropy.

axioms (3)

standard math Einstein Field Equations
The entire derivation takes place within the framework of classical General Relativity.
domain assumption Existence of a Dynamical Horizon (DH)
The paper assumes the black hole boundary can be modeled as a DH—a spacelike hypersurface foliated by marginally trapped surfaces.
domain assumption Energy Conditions
Derivation of the area-increase law typically requires a form of the dominant or null energy condition for the infalling matter.

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Reference graph

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This is also the case in older derivations of the first law using a covariant phase space and Killing horizonsKh

An intermediate step in arriving at the perturbative first law –either using the perturbed-ihframework, or the more recent discussions– involves the Noether charge ˚κ 8πG A. This is also the case in older derivations of the first law using a covariant phase space and Killing horizonsKh. This has led to an oft-repeated phrase:Entropy is a Noether charge. T...
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