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Heat and work in black hole thermodynamics via holography

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Holography gives black hole thermodynamics a general notion of heat and work, with a horizon-area second law that needs no energy conditions.

desk verdict A genuinely new heat/work split for composite black holes with a clean boundary second law, but the advertised bulk area law is not yet proven in part of the parameter range because of a renormalization-scheme gap the authors themselves identify. read the letter →

arxiv 2412.15697 v1 pith:X2TZ35WY submitted 2024-12-20 hep-th cond-mat.stat-mechgr-qc

Tomohiro Shigemura , Keito Shimizu , Sotaro Sugishita , Daichi Takeda , Takuya Yoda This is my paper · ORCID
classification hep-thcond-mat.stat-mechgr-qc
keywords blackholethermodynamicsheatandworkAdS/CFTcorrespondencemaximumentropycoarsegrainingsecondlawdouble-tracedeformationholographicrenormalizationrelative
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to complete black hole thermodynamics by giving it two things it has mostly lacked: a general distinction between heat and work, and a second law for dynamical black holes. The strategy is to define thermodynamics first in the dual field theory: a target holographic CFT coupled to bath holographic CFTs by double-trace deformations is coarse-grained through the maximum-entropy principle, keeping only the expectation values of the Hamiltonians and one chosen operator. Because the true state and the coarse-grained state are separated by a nonnegative relative entropy, the coarse-grained total entropy never decreases, and the AdS/CFT dictionary turns this into a bulk statement: the sum of the areas (divided by $4G$) of the auxiliary Euclidean black holes that coarse-grain the system at time $t$ is at least its value at $t=0$. This gives a first law $\dot E = \delta W + \delta \tilde Q$ in which work is produced by external sources, and it does so without assuming energy conditions. If correct, the framework turns the black hole second law into a test that a gravitational model must pass to be UV-complete; the paper also shows that the standard holographic renormalization scheme can fail this test for boundary operators with $1<\Delta<2$.

What carries the argument

The load-bearing object is the maximum-entropy coarse-grained state, together with the relative-entropy inequality that controls its thermodynamic arrow. For a target system and a bath, the coarse-grained state is $\bar\rho(t)=Z^{(s)}(t)^{-1}e^{-\beta(t)(H_*^{(s)}-\mu(t)O)}\otimes Z^{(b)}(t)^{-1}e^{-B(t)H_*^{(b)}}$, with the Lagrange multipliers $\beta(t),\mu(t),B(t)$ fixed by matching the observed expectation values. The second law $S(t)\ge S(0)$ follows from $0\le S(\rho(t)\|\bar\rho(t))$ together with the initial condition $\rho(0)=\bar\rho(0)$. The translation to gravity uses the path-integral form of the double-trace deformation: the coupling $v\,O^{(s)}O^{(b)}$ and the source $w\,O^{(s)}$ become mixed boundary conditions $\hat\Phi^{(s)}+w+v\Pi^{(b)}=0$ and $\hat\Phi^{(b)}+v\Pi^{(s)}=0$ on two asymptotically AdS spacetimes glued along the boundary, with $\Pi$ the renormalized one-point function. Each coarse-grained partition function is a stationary Euclidean black hole, so its horizon area computes the coarse-grained entropy, and the nonnegativity of relative entropy becomes the bulk area inequality. The examples use the retarded Green's function of a two-dimensional CFT and its bulk counterpart to compute the second-order entropy production.

What would settle it

A concrete check is to compute the entropy production density $-\int_0^t dt'\,w(t')\dot\Pi(t')$ for the source (4.21) at $\Delta=1.5$ using the bulk expression (4.11) and, separately, using the CFT retarded Green's function (B.30) with the finite regulator (B.31). The paper predicts the bulk quantity is negative at early times while the regulated CFT quantity stays nonnegative; if both became negative for the same $t$, the violation would not be a renormalization artifact and the claimed second law would fail. Conversely, finding a renormalization scheme where the bulk one-point function equals the CFT expectation value even while the source is on would confirm the paper's central claim in the problematic window.

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Extended reading notes

Core claim

The central claim is that a composite system of AdS black holes, a target black hole plus bath black holes interacting only through their common conformal boundary, has a well-defined thermodynamics whose second law is inherited from quantum mechanics. On the boundary, the state is replaced at each time $t$ by a maximum-entropy state $\bar\rho(t)$ that reproduces $\langle H_*^{(s)}\rangle_t$, $\langle O\rangle_t$, and $\langle H_*^{(b)}\rangle_t$; the coarse-grained entropy is $S(t)=S^{(s)}(t)+S^{(b)}(t)$. Since relative entropy is nonnegative and the initial state is assumed to already be of the coarse-grained form, $S(t)\ge S(0)$ for any unitary evolution. The AdS/CFT dictionary identifies the partition function of $\bar\rho(t)$ with a stationary Euclidean black hole; for Einstein gravity the coarse-grained entropy is the horizon area over $4G$, so the boundary second law becomes $\frac{A^{(s)}(t)}{4G^{(s)}}+\frac{A^{(b)}(t)}{4G^{(b)}}\ge\frac{A^{(s)}(0)}{4G^{(s)}}+\frac{A^{(b)}(0)}{4G^{(b)}}$. The first law is $\dot E=\delta W+\delta\tilde Q$ with $\delta W=\int d^{d-1}x\sqrt{\sigma}\,\dot w\,\Pi^{(s)}$, so work is the energy change driven by the external source and heat is the remainder. The paper verifies the second law perturbatively in a three-dimensional Einstein-scalar model around BTZ: it holds for $0<\Delta<1$, while for $1<\Delta<2$ the conventional counterterm renormalization of [68] makes the bulk entropy production negative, although the CFT with a finite regulator satisfies the second law; the paper concludes that the standard scheme, which acts like dimensional regularization, is incompatible with the non-negativity of relative entropy, and formulates the bulk area inequality as a necessary criterion for UV completeness. Section 5 also records a caveat: the dictionary (3.20) for a $U(1)$ current appears to require dropping the source term in the stress tensor for charged static black holes.

Load-bearing premise

The argument presumes that the bulk value computed for the field dual to $O$ agrees with the quantum expectation value of $O$ at every time entering the entropy production, including the instants when the external source is active. The paper itself finds that for $1<\Delta<2$ this equality is guaranteed only after the source is switched off, so if the equality fails at coincident times the bulk area inequality is not the relative-entropy second law in that regime.

Editorial extensions

If this is right

  • Target and bath black holes exchanging energy through the glued asymptotic boundaries obey a second law on the summed horizon areas of their coarse-graining Euclidean black holes, Eq. (3.34), with no energy condition assumed.
  • Work and heat acquire operational meanings in black hole thermodynamics: work is $\delta W=\int d^{d-1}x\sqrt{\sigma}\,\dot w(t,\vec x)\Pi^{(s)}(t,\vec x)$ from the external source, and the first law $\dot E=\delta W+\delta\tilde Q$ holds for the composite system.
  • The bulk area inequality becomes a necessary criterion for a gravitational model to be UV-complete: a holographic model whose renormalized boundary one-point function violates the second law is incompatible with unitarity.
  • In the Einstein-scalar example the entropy production is nonnegative for $0<\Delta<1$, while the standard holographic renormalization of [68] fails for $1<\Delta<2$; the paper therefore calls for a renormalization scheme that respects relative entropy.
  • The framework extends to multiple interacting CFTs and to respecting additional conserved charges such as angular momentum, giving generalized Gibbs states dual to more general black holes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If a renormalization scheme that respects relative entropy is found, the same area inequality could be used as a practical scan over effective gravitational theories: any model whose Euclidean coarse-graining entropy decreases under a unitary boundary quench would be discarded as not UV-complete.
  • The framework leaves open the more striking regime in which the target black hole's own entropy decreases while the bath absorbs heat; a time-dependent coupling $v(t)$ or a non-perturbative quench could realize such a black hole engine, which the paper lists as a future direction.
  • The $1<\Delta<2$ breakdown is likely a symptom of a general operator-smearing problem in real-time holography: bulk computations automatically implement dimensional or analytic continuation, while the CFT needs a finite regulator at lightlike coincidences; tests in higher-dimensional CFTs or with non-uniform sources could show whether this scheme-dependence is universal.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a thermodynamic framework for composite black hole systems using holography. On the boundary, a target CFT and bath CFTs are coupled by double-trace deformations; the authors define a coarse-grained state at each time by maximizing entropy subject to expectation values of the Hamiltonians and selected operators. They derive a second law S(t) ≥ S(0) from unitary evolution and non-negativity of relative entropy, and a first law E-dot = δW + δQ-tilde with work δW from the protocol w(t). They then translate the construction to the bulk using path-integral manipulations, obtaining a composite system of asymptotically AdS spacetimes with interacting boundary conditions, and identify the coarse-grained entropy with horizon areas of auxiliary Euclidean black holes, yielding the area inequality (3.34) and the work formula (3.38). As a test, they compute perturbative entropy production for a scalar source on a BTZ black hole: the bulk result satisfies the second law for 0<Δ<1 but violates it for 1<Δ<2. They argue the violation is a renormalization-scheme artifact: the standard holographic renormalization [68] corresponds to dimensional regularization, and Appendix C shows the bulk one-point function equals the CFT expectation value only after the source is switched off. The paper concludes that the second law can serve as a UV-completeness criterion and that a new renormalization scheme is needed.

Significance. If the central translation were fully justified, the framework would be a significant conceptual advance: it provides a microscopic (boundary) derivation of the second law for composite black holes with heat and work, without invoking energy conditions, and it gives a concrete falsifiable diagnostic (positivity of entropy production) for gravitational models. The boundary maximum-entropy argument is concise, and the path-integral derivation of the double-trace gluing is explicit. The manuscript is also unusually candid: Appendix C and Section 5 openly state the coincident-time dictionary gap and the ambiguity in Eq. (3.20). However, because that gap is load-bearing for the advertised bulk second law in a range of conformal dimensions, the significance is at present conditional. The relative-entropy step is not circular: it uses an external theorem, and the bulk inequality is then a consequence of the dictionary. No parameters are fitted to make the example pass; the boundary computation is an independent check.

major comments (3)
  1. [Sec. 4.1, Appendix C] The central translation from the boundary second law to the bulk area inequality (3.34) uses (3.16), i.e. Π(t) = ⟨O⟩_t, at all times appearing in the entropy production. Appendix C proves this equality only after the source w(t) has been switched off: the retarded Green's function is a distribution whose admissible test functions vanish at the coincident point, and the entropy production (4.20) integrates w(t′) times dΠ/dt′ over the whole source-active interval. Consequently, for 1<Δ<2 the quantity shown in Fig. 3 is not the CFT entropy production, and the apparent violation cannot be read as a failure of relative entropy; it is evidence that the holographic renormalization scheme [68] does not implement the boundary dictionary at coincident times. Since this step is load-bearing, the bulk second law (3.34) is not established in that regime, and the two-system inequality (4.32) inherits the same gap. The authors acknowledge this in Section 5, but the abstract and introduction state the result without this restriction.
  2. [Sec. 2.3, Sec. 3.2, Eq. (2.20)] The paper defines two heat notions, δQ in the second law and δQ-tilde in the first law, and identifies them only when [H^(s)+H^(b)_*, V] is negligible, referred to as a resonant interaction. No argument is given that the holographic double-trace interaction V = ∫√σ v O^(s)O^(b) satisfies this condition. In the bulk translation, δQ-tilde is then identified with the bath mass change M^(b)(t)-M^(b)(0) after Eq. (3.38). If the resonant condition is not met, the heat appearing in the first law differs from the energy lost by the bath, so the proposed notion of heat is not uniquely defined outside that regime. This should be either promoted to an explicit assumption on the allowed bulk couplings or separated into two named quantities.
  3. [Sec. 5, Eq. (3.20)] The paper itself flags an unresolved ambiguity in the dictionary (3.20) for U(1) currents and charged static black holes, concerning whether the source term should appear in ⟨T^(s)⟩. Equation (3.20) underlies the Hamiltonian definitions (3.22), the masses M(t), and the bulk work formula (3.38); therefore the advertised generality of the bulk thermodynamic framework for generic composite black holes is conditional. The scalar examples do not test this ambiguity, so the issue must be resolved or the claims restricted to neutral cases.
minor comments (4)
  1. [Eq. (4.21)] The displayed source is constant, λ e^4, on 0<t<1 because -1/(t-1) = 1/(1-t); presumably the intended bump function is exp(-1/t - 1/(1-t) + 4). This typo makes Fig. 3 non-reproducible as written.
  2. [Fig. 3 caption] The caption says the vertical axis is only specified 'up to a positive overall factor'; this is acceptable for a sign check, but the source amplitude λ and the scalar normalization C used in the numerical evaluation are not stated, so the plots cannot be reproduced quantitatively.
  3. [Eq. (B.30)] The branch of the complex powers and the precise iϵ prescription in the retarded Green's function should be specified; for non-integer Δ the expression is otherwise ambiguous.
  4. [Sec. 3.2, Eq. (3.33)] Footnote 10 acknowledges that matter entropy contributions can be needed; the main text should state explicitly that Eq. (3.34) is the classical Einstein-gravity special case rather than the general statement of the proposed framework.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: boundary second law follows from an external relative-entropy theorem; the admitted bulk dictionary gaps are correctness caveats, not circular reductions.

full rationale

The derivation chain is not circular. The boundary second law Eq. (2.8) follows from the non-negativity of relative entropy Eq. (2.9) and unitary evolution, both external to this paper; the max-entropy coarse-graining is a definitional procedure, not fitted to any target bulk quantity. The bulk translation uses the standard AdS/CFT dictionary (3.16), (3.20) and the entropy-area identification (3.33), which cites the authors' prior work [32]; however, [32] supplies an independent, parameter-free derivation whose assumptions do not include the composite-bath setup, so this self-citation carries real evidence. The work formula (3.38) is just the dictionary applied to the boundary definition, and the first law (2.19) holds as a definitional accounting identity because heat is defined as the remainder; this is conventional in quantum thermodynamics, not a hidden prediction. The paper's own Appendix C admits that the bulk one-point function Pi(t) equals the CFT expectation value <O>_t only after the source is switched off, so the bulk inequality (4.20) is violated for 1 < Delta < 2; and Section 5 records an unresolved ambiguity in dictionary (3.20) from the erratum of [32]. These are honest validity gaps in a parameter regime, but they do not reduce the central claim to its own inputs. The score of 2 reflects minor self-citations and these admitted dictionary caveats, not a circular derivation.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard holographic dictionary assumptions rather than fitted parameters; the only hand-chosen numbers are regulators and unit choices. The singular input is an as-yet-unspecified renormalization scheme for Delta > 1, which the paper itself flags.

free parameters (5)
  • Source amplitude lambda in w(t) = small, O(lambda), not specified numerically
    Eq. (4.21): the numerical check assumes a specific bump source with small amplitude; no data fitting is performed.
  • Scalar normalization constant C = positive, arbitrary
    Eq. (4.2): called out by the authors as an ambiguity in normalization of Phi; it cancels or is absorbed into the definition of the operator.
  • Bath coupling v in the two-system example = constant in Section 4.2
    Set constant for simplicity; entropy production depends on v through Pi^(2), but v is not fitted to data.
  • BTZ horizon ratio r_+/L^2 = set to 1 in Fig. 3
    Unit choice in the numerical check, not a fit to data.
  • UV regulator epsilon = varies from 10^-1 to 10^-10 in Figs. 4-6
    Kept finite to regularize divergences for Delta >= 1; the results depend on it for Delta > 1.
assumptions (8)
  • domain assumption AdS/CFT correspondence provides a microscopic definition of the bulk quantum gravity and equates boundary and bulk partition functions.
    Invoked throughout Sections 3 and 4 as the translation tool; without it the bulk second law does not follow.
  • domain assumption Large N limit: the bulk theory is classical and the generating functional is dominated by a saddle point.
    Section 3.1: 'In the large N limit, we can evaluate the generating functional at the saddle point'; this justifies classical bulk equations and factorization in (3.17).
  • domain assumption For Einstein gravity, the coarse-grained entropy of the boundary state equals the horizon area of the auxiliary Euclidean black hole, S = A/(4G).
    Section 3.2, Eqs. (3.32)-(3.33); standard holographic dictionary but load-bearing for the bulk form of the second law.
  • domain assumption Initial state is of generalized Gibbs form rho(0) = rho-bar(0).
    Eq. (2.10); required for the second law proof and corresponds to a steady state for t < 0.
  • standard math Non-negativity of relative entropy S(rho || rho-bar) >= 0.
    Used in Eq. (2.9) to prove the boundary second law; a theorem of quantum information theory.
  • ad hoc to paper Resonant interaction: [H^(s) + H^(b), V] is sufficiently small to identify delta-Q with delta-Q-tilde.
    Eq. (2.20); the paper states this condition is sufficient and does not verify it in the examples.
  • domain assumption Spatially uniform source makes the thermal two-point function universal in 2D CFT and ties the bulk and boundary calculations.
    Section 4.1 and Appendix B; the numerical check is restricted to this case.
  • ad hoc to paper The class of test functions in Appendix C (decaying as e^{-1/t}, Schwartz) defines the retarded Green's function as a distribution.
    Appendix C; this choice is required for the claimed CFT/bulk equivalence of the retarded Green's function and excludes coincident-time behavior that drives the Delta > 1 violation.

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Pith. "Pith review of Heat and work in black hole thermodynamics via holography." pith.science (2026). https://pith.science/paper/X2TZ35WY

@misc{pith2026241215697,
  author       = {Pith},
  title        = {Pith review of: Heat and work in black hole thermodynamics via holography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2TZ35WY}},
  note         = {Machine review of arXiv:2412.15697}
}
read the original abstract

We propose a formulation of black hole thermodynamics that incorporates the notions of heat and work, based on the thermodynamics in quantum theory and the AdS/CFT correspondence. First, for coupled holographic CFTs, we define a coarse-graining procedure adopting the principle of maximum entropy. Employing this approach, when the system is divided into a target system and thermal baths, we formulate the first and second laws, as well as the fundamental thermodynamic relation. Then, by translating the resulting thermodynamics into the AdS gravity language, we construct a thermodynamic framework for composite black hole systems that encompasses both heat and work. This formulation relies on holography, but not on energy conditions on the gravity side. We also argue that the second law serves as a necessary criterion for the UV completeness of gravitational theories.

Figures

Figures reproduced from arXiv: 2412.15697 by the authors.

Figure 1
Figure 1. The system and baths consisting of single-sided asymptotically AdS black [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The system and baths on both boundary and bulk theories. We construct [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. The numerical check of the second law. The horizontal axis is the time [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Time-dependence of δE for ∆ > 1 with ϵ = 10−1 (left) and ϵ = 10−3 (right). The source w(t) is given as (B.36). The vertical axis represents ϵ 2∆−2 δE(t), and the horizontal axis is t with β = π. 0.2 0.4 0.6 0.8 1.0 10 20 30 40 const linear sin 2 4 6 8 10 20 40 60 80 10…
Figure 5
Figure 5. Figure 5: (Left) Time-dependence of δE for ∆ = 1 with various sources w(t) = 1, w(t) = t, w(t) = sin(πt/2) in the interval 0 ≤ t ≤ 1. The parameters are set as ϵ = 10−3 and β = π. (Right) ϵ-dependence of δE for ∆ = 1. The vertical axis represents δE(1/2) with the constant source…
Figure 6
Figure 6. Figure 6: (Left) Time-dependence of δE for ∆ < 1 with ϵ = 10−3 . The source takes the constant value w(t) = 1 (only) in the interval 0 ≤ t ≤ 1. The vertical axis represents δE(t), and the horizontal axis is t with β = π. Note that δE(t) is not rescaled by ϵ unlike the plots for …

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Cited by 1 Pith paper

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  1. Work distribution and fluctuation theorem in AdS/CFT

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    The work distribution of a two-point measurement in a holographic CFT is expressed as a Schwinger-Keldysh bulk path integral, giving a bulk Tasaki-Crooks fluctuation theorem that is verified for a scalar probe on BTZ.

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