arxiv: 2604.20153 · v1 · submitted 2026-04-22 · 🌀 gr-qc

Recognition: unknown

Thermodynamics and phase transitions of nonlinearly scalarized black holes in Einstein-scalar-Gauss-Bonnet theory

De-Cheng Zou , Xu Yang , Meng-Yun Lai , Hyat Huang , Yun Soo Myung

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:26 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black hole thermodynamicsscalarized black holesEinstein-scalar-Gauss-Bonnet theoryphase transitionsfirst-order transitionslatent heat

0 comments

The pith

Nonlinearly scalarized black holes undergo a first-order phase transition from the Schwarzschild solution with non-zero latent heat.

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

The paper computes thermodynamic quantities including mass, temperature, entropy, and Gibbs free energy for static nonlinearly scalarized black holes in Einstein-scalar-Gauss-Bonnet theory with polynomial coupling functions. It verifies that these solutions satisfy the first law of black hole thermodynamics in its standard form. The central finding is that the phase transition between Schwarzschild black holes and scalarized ones is first-order, marked by a discontinuous change in thermodynamic variables and a non-zero latent heat. A reader would care because this provides an explicit example of how scalar fields coupled to curvature terms alter the thermodynamic stability and energy balance of black holes, analogous to phase changes in ordinary matter but in a gravitational setting.

Core claim

Based on previously constructed solutions, the nonlinearly scalarized black holes possess well-defined thermodynamic quantities that obey the first law. The phase transition from the Schwarzschild black hole to the scalarized black hole is first-order and involves non-zero latent heat, as determined by comparing the free energies and identifying the jump in entropy or other quantities at the transition point.

What carries the argument

The comparison of Gibbs free energy between the Schwarzschild and scalarized branches as a function of temperature or coupling parameters to determine transition order and compute latent heat.

If this is right

Scalarized black holes become thermodynamically favored over Schwarzschild ones beyond a critical coupling strength.
The transition releases or absorbs a finite amount of energy corresponding to the latent heat.
The first law holds in its usual differential form for these solutions.
Phase coexistence occurs at specific values of the polynomial coupling parameters.

Where Pith is reading between the lines

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

Similar first-order transitions might appear in other higher-curvature theories with scalar couplings if the free-energy comparison is repeated.
The latent heat could be related to the energy stored in the nontrivial scalar field profile outside the horizon.
If such transitions occur in realistic astrophysical settings, they might leave imprints in the final state of collapsing stars.

Load-bearing premise

The previously constructed scalarized black hole solutions are valid, stable, and obey standard thermodynamic relations without additional corrections arising from the Gauss-Bonnet term.

What would settle it

An explicit computation showing continuous entropy and vanishing latent heat across the critical coupling value where the branches meet would show the transition is not first-order.

Figures

Figures reproduced from arXiv: 2604.20153 by De-Cheng Zou, Hyat Huang, Meng-Yun Lai, Xu Yang, Yun Soo Myung.

**Figure 1.** Figure 1: Scalar hair ϕH at the horizon, scalar charge Qs, and regularity parameter ∆ as functions of the mass M for coupling function ζ1(ϕ) with α = 1/4 and β = 25/8. We find three branches (solid, dashed, dotted) of scalarized solutions 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: Scalar hair ϕH at the horizon, scalar charge Qs and regularity parameter ∆ as a function of mass M for coupling function ζ1(ϕ) with α = 1/4 and β = 1000/8. A representative scalarized solution for ζ1(ϕ) is shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Graph for scalarized BH with rH = 0.03. Here, the coupling function is ζ1(ϕ) with α = 1/4 and β = 1000/8. 3 Thermodynamics and Phase Transition In this section, we study the thermodynamic properties of the nonlinearly scalarized black holes in EsGB theory. Based on the numerical solutions obtained above, we compute the thermodynamical quantities of scalarized black holes. The Hawking temperature is given b… view at source ↗

**Figure 4.** Figure 4: The entropy S as a function of mass M for polynomial coupling function ζ1(ϕ) with four different β. Here, δS = S − S0 with S0 for Schwarzschild black hole. The global thermodynamic preference is determined by the Helmholtz free energy. In [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Difference of free energy δF = Fscal−FSBH as a function of Hawking temperature TH for ζ1(ϕ) with four different β. A key point is that the conditions δS = 0 and δF = 0 do not occur at the same state. Therefore, the phase transition is not determined simply by the entropy comparison, but by the free-energy competition between the two phases. At T = Tc, the two phases have the same Helmholtz free energy but … view at source ↗

**Figure 6.** Figure 6: Two differences of δS and δF for exponential coupling function ζe(ϕ) = 1 4β 1 − e −βϕ4 with three different curves of β = 25 (blue), 100(red) and 1000(green) [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Twenty random solutions are distributed for each branch to check the first-law [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: (b) shows the regularity parameter ∆, which is required by the horizon regularity condition. Starting from ∆ = 1 in the Schwarzschild limit, ∆ decreases monotonically with increasing mass and approaches zero at the endpoint of the branch. The condition ∆ ≥ 0 therefore determines the domain of regular scalarized solutions. The corresponding thermodynamic behavior is displayed in [PITH_FULL_IMAGE:figures/fu… view at source ↗

**Figure 9.** Figure 9: Entropy S and difference of free energy δF with ζ3(ϕ) = αϕ4 . References [1] J. D. Bekenstein, “Exact solutions of Einstein conformal scalar equations,” Annals Phys. 82, 535 (1974). [2] J. D. Bekenstein, “Black Holes with Scalar Charge,” Annals Phys. 91, 75 (1975). [3] K. A. Bronnikov and Y. .N. Kireev, “Instability of Black Holes with Scalar Charge,” Phys. Lett. A 67, 95 (1978). [4] J. D. Bekenstein, “Nov… view at source ↗

read the original abstract

We investigate the thermodynamic properties of static nonlinearly scalarized black holes in Einstein-scalar-Gauss-Bonnet theory with polynomial coupling functions. Based on the scalarized solutions constructed previously, we compute thermodynamical quantities of these scalarized black holes. Moreover, we examine the first law of black hole thermodynamics and consider the phase transitions between Schwarzschild and scalarized black holes. It shows that a phase transition from Schwarzschild black hole to scalarized black hole is a first-order with non-zero latent heat.

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 adds thermodynamic quantities and a first-order phase transition claim with latent heat to previously known nonlinear scalarized black holes in ESGB theory, but the entropy used for the free-energy difference needs checking against Wald corrections.

read the letter

The core new piece here is the thermodynamic analysis layered on top of existing scalarized solutions. The authors take the polynomial-coupling black holes from prior work, compute mass, temperature, entropy, and free energy, verify the first law holds, and then compare potentials to argue for a first-order transition from Schwarzschild to scalarized with nonzero latent heat. That specific calculation and the latent-heat number are not in the earlier papers they cite, so the extension is real even if the solutions themselves are not.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates the thermodynamic properties of static nonlinearly scalarized black holes in Einstein-scalar-Gauss-Bonnet theory with polynomial coupling functions. Using previously constructed scalarized solutions, it computes thermodynamic quantities, verifies the first law, and analyzes phase transitions between Schwarzschild and scalarized black holes, concluding that the transition is first-order with non-zero latent heat.

Significance. If the thermodynamic framework is correctly implemented, the work supplies a concrete demonstration of first-order phase transitions with explicit latent heat in a modified-gravity scalarization model, which may bear on stability criteria and potential observational signatures.

major comments (1)

In the section computing thermodynamic potentials and latent heat (following the first-law verification), the entropy appears to be taken as the area law S = A/4. In ESGB theory the correct entropy is the Wald entropy, which receives explicit additive contributions from the Gauss-Bonnet term evaluated on the horizon together with the scalar coupling function. Using the area law instead can change the free-energy difference that defines the latent heat; the sign or magnitude of the reported non-zero latent heat may therefore be altered once the Wald correction is restored. Please state the entropy formula explicitly and recompute the latent heat if the Wald expression is not already employed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the important issue concerning the entropy formula in our thermodynamic analysis. We address the comment in detail below.

read point-by-point responses

Referee: In the section computing thermodynamic potentials and latent heat (following the first-law verification), the entropy appears to be taken as the area law S = A/4. In ESGB theory the correct entropy is the Wald entropy, which receives explicit additive contributions from the Gauss-Bonnet term evaluated on the horizon together with the scalar coupling function. Using the area law instead can change the free-energy difference that defines the latent heat; the sign or magnitude of the reported non-zero latent heat may therefore be altered once the Wald correction is restored. Please state the entropy formula explicitly and recompute the latent heat if the Wald expression is not already employed.

Authors: We thank the referee for this observation. Upon re-examination of our calculations, we confirm that the entropy was evaluated using the area law S = A/4. We fully agree that the correct entropy in Einstein-scalar-Gauss-Bonnet theory is the Wald entropy, which includes an explicit correction arising from the Gauss-Bonnet term and the scalar coupling function at the horizon. This correction can indeed affect the free-energy differences and the value of the latent heat. In the revised manuscript we will (i) explicitly state the Wald entropy formula, (ii) recompute all thermodynamic potentials and the latent heat with the corrected entropy, and (iii) update the discussion of the first-order phase transition accordingly. We expect these changes to strengthen the thermodynamic analysis without altering the overall conclusion that the transition is first-order. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analysis relies on external prior solutions

full rationale

The paper computes thermodynamic quantities and phase transitions for scalarized black holes by taking previously constructed solutions as given inputs. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim (first-order transition with latent heat) to tautology are present. The thermodynamic comparison uses standard methods on those inputs without internal fitting or redefinition that would force the result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all details are deferred to prior work on the solutions.

pith-pipeline@v0.9.0 · 5392 in / 1071 out tokens · 43240 ms · 2026-05-10T00:26:31.926442+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 27 canonical work pages · 1 internal anchor

[1]

Exact solutions of Einstein conformal scalar equations,

J. D. Bekenstein, “Exact solutions of Einstein conformal scalar equations,” Annals Phys.82, 535 (1974)

1974
[2]

Black Holes with Scalar Charge,

J. D. Bekenstein, “Black Holes with Scalar Charge,” Annals Phys.91, 75 (1975)

1975
[3]

Instability of Black Holes with Scalar Charge,

K. A. Bronnikov and Y. .N. Kireev, “Instability of Black Holes with Scalar Charge,” Phys. Lett. A67, 95 (1978)

1978
[4]

Novel ‘‘no-scalar-hair’’ theorem for black holes,

J. D. Bekenstein, “Novel ‘‘no-scalar-hair’’ theorem for black holes,” Phys. Rev. D51 (1995) no.12, R6608

1995
[5]

Nonperturbative strong field effects in tensor - scalar theories of gravitation,

T. Damour and G. Esposito-Farese, “Nonperturbative strong field effects in tensor - scalar theories of gravitation,” Phys. Rev. Lett.70(1993), 2220-2223

1993
[6]

New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories

D. D. Doneva and S. S. Yazadjiev, “New Gauss-Bonnet Black Holes with Curvature- Induced Scalarization in Extended Scalar-Tensor Theories,” Phys. Rev. Lett.120, no.13, 131103 (2018) [arXiv:1711.01187 [gr-qc]]

work page Pith review arXiv 2018
[7]

Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling

H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, “Spontaneous scalar- ization of black holes and compact stars from a Gauss-Bonnet coupling,” Phys. Rev. Lett.120, no.13, 131104 (2018) [arXiv:1711.02080 [gr-qc]]. 13

work page Pith review arXiv 2018
[8]

Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories

G. Antoniou, A. Bakopoulos and P. Kanti, “Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories,” Phys. Rev. Lett.120, no.13, 131102 (2018) [arXiv:1711.03390 [hep-th]]

work page Pith review arXiv 2018
[9]

Antoniou, A

G. Antoniou, A. Bakopoulos and P. Kanti, “Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories,” Phys. Rev. D97(2018) no.8, 084037 [arXiv:1711.07431 [hep-th]]

work page arXiv 2018
[10]

Gregory-Laflamme instability of black hole in Einstein-scalar-Gauss-Bonnet theories,

Y. S. Myung and D. C. Zou, “Gregory-Laflamme instability of black hole in Einstein-scalar-Gauss-Bonnet theories,” Phys. Rev. D98, no. 2, 024030 (2018) [arXiv:1805.05023 [gr-qc]]

work page arXiv 2018
[11]

Stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity,

H. O. Silva, C. F. B. Macedo, T. P. Sotiriou, L. Gualtieri, J. Sakstein and E. Berti, “Stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity,” Phys. Rev. D99(2019) no.6, 064011 [arXiv:1812.05590 [gr-qc]]

work page arXiv 2019
[12]

Axial perturbations of the scalarized Einstein-Gauss-Bonnet black holes,

J. L. Bl´ azquez-Salcedo, D. D. Doneva, S. Kahlen, J. Kunz, P. Nedkova and S. S. Yazad- jiev, “Axial perturbations of the scalarized Einstein-Gauss-Bonnet black holes,” Phys. Rev. D101(2020) no.10, 104006 [arXiv:2003.02862 [gr-qc]]

work page arXiv 2020
[13]

Axial perturbations of hairy Gauss-Bonnet black holes with a massive self-interacting scalar field,

K. V. Staykov, J. L. Bl´ azquez-Salcedo, D. D. Doneva, J. Kunz, P. Nedkova and S. S. Yazadjiev, “Axial perturbations of hairy Gauss-Bonnet black holes with a massive self-interacting scalar field,” Phys. Rev. D105(2022) no.4, 044040 [arXiv:2112.00703 [gr-qc]]

work page arXiv 2022
[14]

Phase structure of holographic superconductors in an Einstein-scalar-Gauss-Bonnet theory with spontaneous scalarization,

H. Guo, W. L. Qian and B. Wang, “Phase structure of holographic superconductors in an Einstein-scalar-Gauss-Bonnet theory with spontaneous scalarization,” Phys. Rev. D109(2024) no.12, 124038 [arXiv:2401.09846 [gr-qc]]

work page arXiv 2024
[15]

Quasinormal modes of Einstein–scalar–Gauss–Bonnet black holes,

P. Paul, “Quasinormal modes of Einstein–scalar–Gauss–Bonnet black holes,” Eur. Phys. J. C84(2024) no.3, 218 [arXiv:2312.16479 [gr-qc]]

work page arXiv 2024
[16]

Instabilities of black holes in Einstein-scalar–Gauss–Bonnet theories,

J. L. Bl´ azquez-Salcedo, B. Kleihaus and J. Kunz, “Instabilities of black holes in Einstein-scalar–Gauss–Bonnet theories,” Gen. Rel. Grav.56(2024) no.8, 99 14

2024
[17]

Spin-induced black hole spontaneous scalarization,

A. Dima, E. Barausse, N. Franchini and T. P. Sotiriou, “Spin-induced black hole spon- taneous scalarization,” Phys. Rev. Lett.125(2020) no.23, 231101 [arXiv:2006.03095 [gr-qc]]

work page arXiv 2020
[18]

Berti, L

E. Berti, L. G. Collodel, B. Kleihaus and J. Kunz, “Spin-induced black-hole scalar- ization in Einstein-scalar-Gauss-Bonnet theory,” Phys. Rev. Lett.126(2021) no.1, 011104 [arXiv:2009.03905 [gr-qc]]

work page arXiv 2021
[19]

Spinning and excited black holes in Einstein-scalar-Gauss–Bonnet theory,

L. G. Collodel, B. Kleihaus, J. Kunz and E. Berti, “Spinning and excited black holes in Einstein-scalar-Gauss–Bonnet theory,” Class. Quant. Grav.37(2020) no.7, 075018 [arXiv:1912.05382 [gr-qc]]

work page arXiv 2020
[20]

Hod, Phys

S. Hod, “Onset of spontaneous scalarization in spinning Gauss-Bonnet black holes,” Phys. Rev. D102(2020) no.8, 084060 [arXiv:2006.09399 [gr-qc]]

work page arXiv 2020
[21]

Spin-induced scalarized black holes,

C. A. R. Herdeiro, E. Radu, H. O. Silva, T. P. Sotiriou and N. Yunes, “Spin-induced scalarized black holes,” Phys. Rev. Lett.126(2021) no.1, 011103 [arXiv:2009.03904 [gr-qc]]

work page arXiv 2021
[22]

Rotating scalarized black holes in scalar couplings to two topological terms,

D. C. Zou and Y. S. Myung, “Rotating scalarized black holes in scalar couplings to two topological terms,” Phys. Lett. B820(2021), 136545 [arXiv:2104.06583 [gr-qc]]

work page arXiv 2021
[23]

Beyond the spontaneous scalarization: New fully nonlinear mechanism for the formation of scalarized black holes and its dynamical development,

D. D. Doneva and S. S. Yazadjiev, “Beyond the spontaneous scalarization: New fully nonlinear mechanism for the formation of scalarized black holes and its dynamical development,” Phys. Rev. D105(2022) no.4, L041502 [arXiv:2107.01738 [gr-qc]]

work page arXiv 2022
[24]

Radial perturbations of scalar-Gauss-Bonnet black holes beyond spontaneous scalarization,

J. L. Bl´ azquez-Salcedo, D. D. Doneva, J. Kunz and S. S. Yazadjiev, “Radial pertur- bations of scalar-Gauss-Bonnet black holes beyond spontaneous scalarization,” Phys. Rev. D105(2022) no.12, 124005 [arXiv:2203.00709 [gr-qc]]

work page arXiv 2022
[25]

Effects of mass and self-interaction on nonlinear scalarization of scalar-Gauss-Bonnet black holes,

A. M. Pombo and D. D. Doneva, “Effects of mass and self-interaction on nonlinear scalarization of scalar-Gauss-Bonnet black holes,” Phys. Rev. D108(2023) no.12, 124068 [arXiv:2310.08638 [gr-qc]]

work page arXiv 2023
[26]

Nonlinear black hole scalarization in multi-scalar Gauss-Bonnet gravity,

K. V. Staykov and D. D. Doneva, “Nonlinear black hole scalarization in multi-scalar Gauss-Bonnet gravity,” J. Phys. Conf. Ser.2719(2024) no.1, 012007 15

2024
[27]

Multiscalar Gauss-Bonnet gravity: Scalarized black holes beyond spontaneous scalarization,

K. V. Staykov and D. D. Doneva, “Multiscalar Gauss-Bonnet gravity: Scalarized black holes beyond spontaneous scalarization,” Phys. Rev. D106(2022) no.10, 104064 [arXiv:2209.01038 [gr-qc]]

work page arXiv 2022
[28]

Nonlinear scalar- ization of Schwarzschild black holes in scalar-torsion teleparallel gravity,

P. A. Gonz´ alez, E. Papantonopoulos, J. Robledo and Y. V´ asquez, “Nonlinear scalar- ization of Schwarzschild black holes in scalar-torsion teleparallel gravity,” Phys. Rev. D111(2025) no.4, 044064 [arXiv:2407.13557 [gr-qc]]

work page arXiv 2025
[29]

Spontaneous nonlinear scalarization of Kerr black holes,

D. D. Doneva, L. G. Collodel and S. S. Yazadjiev, “Spontaneous nonlinear scalarization of Kerr black holes,” Phys. Rev. D106(2022) no.10, 104027 [arXiv:2208.02077 [gr-qc]]

work page arXiv 2022
[30]

Lai, D.-C

M. Y. Lai, D. C. Zou, R. H. Yue and Y. S. Myung, “Nonlinearly scalarized rotating black holes in Einstein-scalar-Gauss-Bonnet theory,” Phys. Rev. D108(2023) no.8, 084007 [arXiv:2304.08012 [gr-qc]]

work page arXiv 2023
[31]

Non-linearly scalarized supermassive black holes,

S. Liu, Y. Liu, Y. Peng and C. Y. Zhang, “Non-linearly scalarized supermassive black holes,” Eur. Phys. J. C85(2025) no.12, 1370 [arXiv:2509.17892 [gr-qc]]

work page arXiv 2025
[32]

The rotating solutions beyond the spontaneous scalarization in Einstein-Maxwell-scalar theory,

W. Xiong, C. Y. Zhang and P. C. Li, “The rotating solutions beyond the spontaneous scalarization in Einstein-Maxwell-scalar theory,” [arXiv:2312.11879 [gr-qc]]

work page arXiv
[33]

Existence of nonlinearly scalarized black holes in Einstein-scalar-Gauss-Bonnet theory with polynomial couplings

D. C. Zou, X. Yang, M. Y. Lai, H. Huang, B. Liu, J. Kunz, Y. S. Myung and R. H. Yue, “Existence of nonlinearly scalarized black holes in Einstein-scalar-Gauss-Bonnet theory with polynomial couplings,” [arXiv:2404.19521 [gr-qc]]

work page internal anchor Pith review arXiv
[34]

Herdeiro, H

C. Herdeiro, H. Huang, J. Kunz, M. Y. Lai, E. Radu and D. C. Zou, “Phase Structure of Scalarized Black Holes in Einstein-Scalar-Gauss-Bonnet Gravity,” [arXiv:2603.24164 [gr-qc]]

work page arXiv
[35]

Action Integrals and Partition Functions in Quantum Gravity,

G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D15(1977), 2752-2756 16

1977