Black hole free energy landscapes from constrained instantons

Jia-Zhou Liu; Shan-Ping Wu; Shao-Wen Wei; Yu-Xiao Liu

REVIEW

Black hole free energy landscapes from constrained instantons

Shan-Ping Wu, Jia-Zhou Liu, Yu-Xiao Liu, Shao-Wen Wei

General Relativity and Quantum Cosmology · Submitted 19 Oct 2026

Pith's verdict: Provides the first systematic path integral derivation of black hole free energy landscapes using constrained instantons, but the analysis remains formal on fluctuation determinants and complex contours.

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arXiv ID2610.26094 (version 2)

Pith IDHLJ2V6RO, Pith's permanent record of this paper and its review

CitationJia-Zhou Liu, Shan-Ping Wu, Shao-Wen Wei, Yu-Xiao Liu. "Black hole free energy landscapes from constrained instantons." arXiv:2610.26094 (2026). https://pith.science/paper/HLJ2V6RO

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Editor's note

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The paper does something genuinely new: it derives the generalized free energy landscape for black holes from first principles in the Euclidean path integral, using factorization across an auxiliary interface with partial-Neumann data and constrained instantons. The result reproduces the expected form F = E - ΩJ - TS with Wald entropy for f(Riemann) theories, bypassing the small-deficit expansion that has always felt ad hoc. The construction is internally consistent and the examples (Schwarzschild, Kerr, EGB, NMG) verify the sewing conditions explicitly.

What it does well: the formal machinery is clean, the connection between jumps in lapse/shift and Lagrange multipliers is explicit, and the paper honestly marks every assumption and limitation. The extension to higher-curvature theories using the auxiliary formulation is nontrivial and convincingly demonstrated for the checked cases.

The soft spots are real but proportional. The path integral manipulations are entirely formal: no functional measure, no gauge fixing, no fluctuation determinant. That is a gap for any claim about a full semiclassical story. The rotating examples require complex contours that are not specified. The injectivity condition on the map from collective coordinates to interface data is stated as a condition to check per theory, but is only verified for the examples; a general proof is missing. These are acknowledged in the conclusions (Sec. 7). The paper does not oversell itself.

Who this is for: anyone working on black hole thermodynamics, free energy landscapes, or path integral methods in quantum gravity. It will be cited and used. A serious referee can handle it: the central algebraic derivation is correct under the stated assumptions, and the open issues are clear targets for future work.

Recommendation: accept with minor revision. The authors should add a brief remark on what a complete semiclassical evaluation would require (fluctuation prefactors, contour prescription) to manage reader expectations, but the paper is publishable as is.

What would settle it

For a stationary black hole family where the map $\lambda^m \mapsto (S, J_{B,I})$ is not surjective onto the tangent space (e.g., an extremal limit where entropy and angular momentum variations become dependent), the stationarity condition (4.18) would predict equilibrium at $T \neq T_H$ or $\Omega_B \neq \Omega_H$, contradicting the smooth equilibrium geometry. This can be tested by constructing a family of near-extremal Kerr black holes with fixed external $T$ and $\Omega$ and checking whether the stationary point of $F_{\text{gen}}$ coincides with the smooth black hole only when the rank condition holds. A second test: for a general $f(\text{Riemann})$ theory not of the EGB or NMG type, one can attempt to explicitly verify the sewing conditions (6.18) for a stationary black hole and see if they fail for certain auxiliary projections, thereby revealing a limitation of the construction.

Referee report

Significance

If the result is correct, the paper provides a principled path-integral derivation of the off-shell free energy landscapes that are widely used in studies of black hole phase transitions, stochastic dynamics, and thermodynamic topology. The derivation avoids small-deficit expansions and extends cleanly to higher-curvature theories, which is important for gravitational effective field theories and for understanding black hole thermodynamics beyond Einstein gravity. The explicit verification of the sewing data in the Gauss-Bonnet and new massive gravity examples adds concrete support. The paper also gives a clear discussion of the conditions (injectivity of the collective-coordinate map to the interface data) under which stationary points of the landscape correspond to equilibrium saddles.

Major objections (3)

[Sec. 3 and Appendix A.1] The path integral manipulations (delta-functionals, boundary transforms, and the duality relation (A.1)) are entirely formal. While the paper states that functional measures, gauge-fixing, and fluctuation determinants are subleading in the semiclassical expansion, the construction relies on the existence of well-defined integration contours for the Lagrange multiplier fields (Eq. (3.12)) and on the assumed duality between Dirichlet and partial-Neumann representations. The authors should clarify the assumptions required for these formal steps to hold, or explicitly restrict the claims to the leading semiclassical order where only the classical action is used. The authors acknowledge this: “For the rotating configurations considered here, admis- sible complex integration contours also remain to be determined.”
[Sec. 4.3 and 6.3] The condition for stationarity (T = T_H and Ω^I_B = Ω^I_H) is derived using the first law (Eq. (4.20)) under the assumption that the retained variations independently span the entropy and angular momentum directions. The paper acknowledges the possibility of degeneracy (rank condition), but this condition is load-bearing for the physical interpretation of the landscape. The examples considered satisfy the condition, but for generic f(Riemann) families the Jacobian of λ^m → (S, J_{B,I}) should be checked. The paper should discuss whether this condition might fail in interesting scenarios and what that would imply.
[Sec. 6 and Appendix B.3] The extension to f(Riemann) theories relies on the auxiliary field formulation and the claim that the complete partial-Neumann data match under the Killing reparametrization (4.3). The general argument in Appendix B.3 uses the fact that the auxiliary configuration shares the Killing symmetries of the metric. While this is plausible, the paper does not provide an explicit coordinate-independent proof that the auxiliary fields (θ_{∥AB}, χ_{∥A}, Ψ_⊥) transform as required for an arbitrary parity-even f(Riemann) Lagrangian. The examples (EGB, NMG) verify the sewing, but a more general demonstration (or a statement of the necessary conditions on the Lagrangian) would strengthen the claim.

Minor objections (5)

[Eqs. (2.12)–(2.14)] The boundary terms for the lapse and shift are both denoted B_N and B_N with an indistinguishable overbar. Use distinct symbols (e.g., B_N and B_{\bar{N}}) to avoid ambiguity.
[Sec. 5.3 (Kerr–de Sitter)] The rotating de Sitter construction is presented only schematically. The authors state that the momentum response and cross-section geometry match, but they do not explicitly verify these matching conditions for the Kerr–de Sitter metric. A brief check or a reference to a fuller treatment would be helpful.
[Eq. (6.13) and Eq. (B.19)] The notation χ_{∥(AB)2} is used in the text but the subscript '2' is not explained. It is defined in Eq. (B.10) as χ_{∥AB}^2, but the placement of indices is confusing. Clarify the notation.
[Sec. 7, Conclusions] The discussion of future work (fluctuation prefactors, complex contours) is welcome, but the paper could also mention that the present construction assumes the existence of a regular constraint surface after gauge reduction (as noted in Sec. 3.3), and that this may fail in the presence of gauge anomalies or topological obstructions.
[Throughout] Several equations contain the symbol '[X]' for the interface jump, but the convention is only defined in Sec. 2.1. It would be helpful to restate the jump definition at the start of Secs. 3 and 6 for readers who skip forward.

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The paper

The paper claims

Gravitational path integral factorization derives black hole free energy landscapes from constrained instantons, giving a semiclassical foundation for the generalized free energy F_gen = E - Ω J - T S that holds for Einstein gravity and…

The reading

The paper provides a semiclassical path integral derivation of generalized black hole free energy landscapes, which until now were mostly justified by conical singularity methods or by flat. Using gravitational path integral factorization, the authors cut the Euclidean spacetime at an auxiliary hypersurface and fix partial-Neumann data there. The resulting constrained stationary configurations—called constrained instantons—allow the lapse and tangential shift to jump across the interface, encoding thermal and rotational mismatches between the black hole horizon and an external heat bath. For stationary black holes with an asymptotic boundary, the constrained saddle action takes the form $I_{\text{constr}} = \beta (E_{\mathcal{B}} - \Omega_{\mathcal{B}}^I J_{\mathcal{B},I}) - S$, where the entropy $S$ comes entirely from the regular horizon cap. Dividing by $\beta$ gives the generalized free energy $F_{\text{gen}} = E_{\mathcal{B}} - \Omega_{\mathcal{B}}^I J_{\mathcal{B},I} - T S$, whose interior stationary points match the smooth equilibrium black hole saddles when $T = T_H$ and $\Omega = \Omega_H$. The construction extends to local parity-even $f(\text{Riemann})$ theories without curvature derivatives, where the same form holds with Wald entropy, and is explicitly checked for Einstein–Gauss–Bonnet and new massive gravity. This puts free energy landscape methods on a firm path integral footing, applicable beyond Einstein gravity.

What carries the argument

The central mechanism is the constrained instanton construction in the gravitational path integral: the spacetime is cut along an auxiliary codimension-one hypersurface $T_r$, and the two subregions are sewn at fixed partial-Neumann (pN) data $(E, J^A; \sigma_{AB})$ on $T_r$, where $E$ is the densitized Brown–York energy, $J^A$ the momentum density, and $\sigma_{AB}$ the cross-section metric. This pN data is held fixed across the interface, while the lapse $N$ and tangential shift $N^A$ are allowed to jump. The jumps are supported by Lagrange multipliers arising from a delta-functional that enforces the fixed interface data. The resulting paired subregion solutions (a regular horizon cap and an exterior extending to the asymptotic boundary) are constrained instantons: they are stationary only under variations that preserve the pN data. The action evaluates exactly to $\beta(E - \Omega J) - S$, with no conical defect or deficit expansion. For local parity-even $f(\text{Riemann})$ theories without curvature derivatives, the same construction works with a larger set of auxiliary interface data, and the entropy becomes the Wald entropy.

Extended reading notes

Core claim

The paper shows that the leading semiclassical weight of a black hole free energy landscape can be derived directly from the gravitational path integral by factorizing the spacetime across an auxiliary hypersurface and fixing partial-Neumann data there. For a stationary black hole family, the constrained action equals $\beta (E_{\mathcal{B}} - \Omega_{\mathcal{B}}^I J_{\mathcal{B},I}) - S$, where $E_{\mathcal{B}}$ and $J_{\mathcal{B},I}$ are the renormalized Hamiltonian charges at the asymptotic boundary, $S$ is the black hole entropy (Bekenstein–Hawking in Einstein gravity, Wald in higher-curvature theories), and $\beta = 1/T$ is the prescribed inverse temperature. This result holds for regular subregion geometries—no conical singularities or small-deficit expansions are needed. The entropy originates from the regular horizon cap, while the exterior region provides the $\beta(E - \Omega J)$ combination. The partial-Neumann boundary conditions on the auxiliary interface allow the lapse and tangential shift to jump, accommodating mismatches between the external thermodynamic sources and the horizon values. At equilibrium, the jumps vanish and one recovers the smooth black hole saddle. The construction is explicitly verified for Schwarzschild, Kerr, Schwarzschild–de Sitter, Kerr–de Sitter, five-dimensional static AdS black holes in Einstein–Gauss–Bonnet gravity, and the rotating BTZ branch of new massive gravity.

Load-bearing premise

The map from the finite set of collective coordinates (e.g., horizon radius, rotation parameter) to the complete local partial-Neumann interface data is injective, and the retained variations independently span the entropy and angular momentum directions. If this fails, stationarity of the generalized free energy does not imply $T = T_H$ and $\Omega_B = \Omega_H$. The paper acknowledges this rank condition in Sec. 4.3. A second weak assumption, stated in Sec. 6.2, is that the paired subregion solutions constructed from the same stationary black hole solution satisfy the complete partial-Neumann sewing conditions for the higher-curvature auxiliary field data, which is verified only for the explicit examples (EGB, NMG) and not proven for general $f(\text{Riemann})$ theories.

Editorial extensions

If this is right

The path integral derivation provides a first-principles semiclassical basis for the widely used generalized free energy landscape approach to black hole phase transitions and stochastic dynamics.
The result extends naturally to higher-curvature theories of gravity, such as Einstein–Gauss–Bonnet and new massive gravity, indicating that the generalized free energy form $F_{\text{gen}} = E - \Omega J - T S$ with Wald entropy is universal for local parity-even $f(\text{Riemann})$ theories.
The construction avoids the ambiguities of conical deficit methods at higher order in the deficit angle, offering a regular framework for computing subleading quantum corrections.
The interface picture suggests a thermodynamic interpretation where the two sides of the cut behave as constrained subsystems with matched quasilocal responses but generally different conjugate potentials, potentially connecting to stochastic descriptions of black hole thermalization.

Reading between the lines

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

If the constrained instanton construction is correct, then for any stationary black hole in a local parity-even gravitational theory, the off-shell generalized free energy at fixed external temperature and angular velocities is always $F_{\text{gen}} = E - \Omega J - T S$, with the entropy defined by the Wald formula. This universality suggests that the free energy landscape is not merely a phenom
The paper leaves open the calculation of fluctuation determinants and the measure in the reduced integral over collective coordinates. Completing this would yield the full thermodynamic potential $-T \ln Z_{\text{red}}$ and normalized averages, potentially revealing subleading corrections to black hole thermodynamics and phase transition rates.
The interface multipliers that support the lapse and shift jumps could be interpreted as chemical potentials for the thermal and angular mismatches. In a more complete quantum treatment, these might fluctuate, giving a dynamical meaning to the constrained sectors beyond the saddle approximation.
For compact geometries like Schwarzschild–de Sitter, the same framework gives $I_{\text{constr}} = -(S_b + S_c)$ with no $\beta E$ term, suggesting that the free energy landscape for de Sitter black holes should be defined with respect to a reference period, not an external heat bath. This could clarify the thermodynamic interpretation of de Sitter horizons.

Original abstract

We apply gravitational path integral factorization to provide a semiclassical interpretation of generalized black hole free energy landscapes at fixed external temperature and angular velocities. Boundary integral transforms express the formal partition function as an integral over constrained sectors labeled by complete local partial-Neumann data on an auxiliary hypersurface. Independent variation establishes paired regular subregion solutions as constrained instantons, with Lagrange multipliers supporting lapse and tangential shift jumps that accommodate thermal and rotational mismatches. For selected nonextremal stationary families with an asymptotic boundary, the constrained saddle action takes the form $I_{\rm constr}=\beta\mathcal F_{\rm gen}=\beta(E_{\mathcal B}-\Omega_{\mathcal B}^{I}J_{\mathcal B,I})-S$, with $\beta=T^{-1}$. Here $E_{\mathcal B}$ and $J_{\mathcal B,I}$ are renormalized Hamiltonian charges and $S$ is the black hole entropy, whose contribution to the action comes from the regular horizon cap. Interior stationary points of the landscape correspond to smooth equilibrium saddles when the retained variations independently span the entropy and angular momentum directions. We extend the construction to local, parity-even $f(\mathrm{Riemann})$ theories without curvature derivatives by retaining the complete auxiliary interface data. The same thermodynamic form follows with Wald entropy, without a small-deficit expansion. Schwarzschild and Kerr black holes illustrate the Einstein construction, while five-dimensional static AdS black holes in Einstein--Gauss--Bonnet gravity and the rotating BTZ branch of new massive gravity provide explicit checks of the higher-curvature extension.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 7,141 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.37 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.22 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We apply gravitational path integral factorization to provide a semiclassical interpretation of generalized black hole free energy landscapes at fixed external temperature and angular velocities. Boundary integral transforms express the formal partition function as an integral over constrained sectors labeled by complete local partial-Neumann data on an auxiliary hypersurface. Independent variation establishes paired regular subregion solutions as constrained instantons, with Lagrange multipliers supporting lapse and tangential shift jumps that accommodate thermal and rotational mismatches. For selected nonextremal stationary families with an asymptotic boundary, the constrained saddle action takes the form $I_{\rm constr}=\beta\mathcal F_{\rm gen}=\beta(E_{\mathcal B}-\Omega_{\mathcal B}^{I}J_{\mathcal B,I})-S$, with $\beta=T^{-1}$. Here $E_{\mathcal B}$ and $J_{\mathcal B,I}$ are renormalized Hamiltonian charges and $S$ is the black hole entropy, whose contribution to the action comes from the regular horizon cap. Interior stationary points of the landscape correspond to smooth equilibrium saddles when the retained variations independently span the entropy and angular momentum directions. We extend the construction to local, parity-even $f(\mathrm{Riemann})$ theories without curvature derivatives by retaining the complete auxiliary interface data. The same thermodynamic form follows with Wald entropy, without a small-deficit expansion. Schwarzschild and Kerr black holes illustrate the Einstein construction, while five-dimensional static AdS black holes in Einstein--Gauss--Bonnet gravity and the rotating BTZ branch of new massive gravity provide explicit checks of the higher-curvature extension.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We apply gravitational path integral factorization to provide a semiclassical interpretation of generalized black hole free energy landscapes at fixed external temperature and angular velocities.	contribution 0.92	0.94	0.31	asserted 1.00
s1	Boundary integral transforms express the formal partition function as an integral over constrained sectors labeled by complete local partial-Neumann data on an auxiliary hypersurface.	method 0.94	0.81	0.07	asserted 1.00
s2	Independent variation establishes paired regular subregion solutions as constrained instantons, with Lagrange multipliers supporting lapse and tangential shift jumps that accommodate thermal and rotational mismatches.	method 0.85	0.87	0.10	asserted 1.00
s3	For selected nonextremal stationary families with an asymptotic boundary, the constrained saddle action takes the form $I_{\rm constr}=\beta\mathcal F_{\rm gen}=\beta(E_{\mathcal B}-\Omega_{\mathcal B}^{I}J_{\mathcal B,I})-S$, with $\beta=T^{-1}$.	result 0.98	0.87	0.32	asserted 1.00
s4	Here $E_{\mathcal B}$ and $J_{\mathcal B,I}$ are renormalized Hamiltonian charges and $S$ is the black hole entropy, whose contribution to the action comes from the regular horizon cap.	interpretation 0.53	0.73	0.08	asserted 1.00
s5	Interior stationary points of the landscape correspond to smooth equilibrium saddles when the retained variations independently span the entropy and angular momentum directions.	interpretation 0.80	0.79	0.15	asserted 0.98
s6	We extend the construction to local, parity-even $f(\mathrm{Riemann})$ theories without curvature derivatives by retaining the complete auxiliary interface data.	contribution 0.92	0.91	0.07	asserted 1.00
s7	The same thermodynamic form follows with Wald entropy, without a small-deficit expansion.	result 0.93	0.85	0.13	asserted 1.00
s8	Schwarzschild and Kerr black holes illustrate the Einstein construction, while five-dimensional static AdS black holes in Einstein--Gauss--Bonnet gravity and the rotating BTZ branch of new massive gravity provide explicit checks of the higher-curvature extension.	contribution 0.49	0.81	0.09	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Schematic free energy landscape at fixed external thermodynamic data. Along the retained collective direction, local minima (blue) represent stable or metastable equilibrium black holes, local maxima (red) represent unstable equilibrium black holes, and generic points on the curve are off shell with respect to these data. Motivated by this conical realization, we seek a complementary path integral derivation of generalized free energy in which the black hole parameters can vary at fixed ensemble temperature without imposing T = TH. Our aim is to determine the corresponding leading…

**Figure 2.** Schematic of the orthogonal temporal–radial foliation at a finite outer radius R. The outer solid cylinder denotes B = TR for a finite physical boundary, and the inner dashed cylinder is the auxiliary hypersurface Tr. The blue disk represents the portion of Στ enclosed by TR; its part inside the red curve is Σ (r) τ . The orange and red curves are the outer cross-section S (R) τ = Στ ∩ TR and the interface cross-section S (r) τ = Στ ∩ Tr, respectively. The arrows indicate the unit normals u and n, whose dual one-forms are u = N dτ and n = P dρ, respectively. The upper and lower end

**Figure 3.** Schematic of the two subregions obtained by cutting the manifold along the auxiliary hypersurface Tr. The displayed boundary components T − r ⊂ ∂M1 and T + r ⊂ ∂M2 are two copies of the same interface. For the initial smooth geometry, choosing n to point from M1 toward M2 gives the outward normals n out 1 = n and n out 2 = −n. The superscripts − and + label the two copies only. The gap is drawn solely to distinguish the copies after the cut; the Dirichlet or pN sewing conditions specified below determine which boundary data are identified. is the value on the + side minus that on…

**Figure 4.** A constrained black hole sector obtained by cutting along the auxiliary surface Tr. Its two copies bound the inner region M1 and the outer region M2. The pN sewing conditions identify (E,JA; σAB), equivalently (ε, jA, σAB), with responses expressed in the common-normal convention. The lapse and tangential shift on T − r and T + r need not agree. The outer region extends from T + r to the physical asymptotic boundary B. angular coordinates with the helical identification in Eq. (4.4). Unless stated otherwise, N, N A, jA, and JA denote Euclidean fields and responses, which may be…

Figures reproduced from arXiv:2610.26094 by the authors.

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