Pith. sign in

REVIEW 4 major objections 5 minor 22 references

Thermodynamics in space-times without horizons

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Horizonless spacetime junctions can still carry area-proportional entropy.

desk verdict The zero-energy junction result is real and new; the thermodynamic claims are not independently confirmed by the path integral and the dS/AdS examples do not satisfy the matching condition. read the letter →

arxiv 2507.03469 v1 pith:MXMA62JJ submitted 2025-07-04 gr-qc

Raymond Isichei , João Magueijo This is my paper
classification gr-qc
keywords zero-energysurfacetimelikejunctionIsraelconditionsareaentropyTolman-Ehrenfesttemperaturegravastarnegativehorizonlessspacetime
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 establish that horizonless spacetimes, when they contain a preferred timelike surface joining two static spherically symmetric geometries, can still have a well-defined thermodynamics. The joining surface has zero energy density as a consequence of the first Israel junction condition, and it carries only a transverse pressure set by the jump in extrinsic curvature. From this zero-energy condition the authors derive an area-proportional entropy $S=\alpha A$ and a temperature proportional to the surface pressure, $p_T=\alpha T$, and they show by Euclidean path integral that the total free energy vanishes once gravity is included. What makes this worth caring about is that entropy and temperature are no longer tied to an event horizon, and that some perfectly regular constructions are forced to negative temperature.

What carries the argument

The load-bearing object is the two-sided junction surface $\Sigma$ at fixed radius $r_0$, whose induced metric $ds_3^2=-f(r_0)dt^2+r_0^2d\Omega^2$ is common to both sides. Its partition function is built from the Euclidean action $I^E_\Sigma=\frac{1}{8\pi}\int_\Sigma d^3x\sqrt{\gamma}\,[k]-\int_\Sigma d^3x\sqrt{\gamma}\,p_T$, with the inverse temperature taken to be the Tolman-Ehrenfest period $\beta_{TE}=\int iN\,dt$, the proper time that elapses between the two boundary spheres. The zero-energy result $S^t{}_t=0$ follows from requiring the induced metrics to agree, and it is this condition that turns the Euler relation into $p_T=Ts$; the Maxwell relation then fixes $p_T=\alpha T$ and $s=\alpha$, yielding the area law for the entropy.

What would settle it

For a de Sitter interior glued to empty space at $r_0$, compute the on-shell Euclidean action (16) using a different temperature rule, such as the period $\beta=2\pi/\kappa$ of the would-be de Sitter horizon, and compare the resulting entropy with $S=\alpha A$; if the entropy becomes temperature-dependent or the total free energy no longer vanishes, the area law is an artifact of the Tolman-Ehrenfest choice.

Watch

Extended reading notes

Core claim

As the authors present it, the central discovery is that a timelike junction $\Sigma$ between two static spherically symmetric spacetimes obeying the matching condition (J1) necessarily has $S^t{}_t=0$: the surface energy density vanishes because the difference in Misner-Sharp energy across $\Sigma$ is proportional to the jump in $\gamma_{tt}$, which (J1) sets to zero. The only surface matter property left is a transverse pressure $p_T=[k]/(8\pi)$, determined by the jump in the trace of the extrinsic curvature. Feeding this zero-energy condition into general thermodynamics gives the Euler relation $p_T=Ts$, and the Maxwell relation upgrades it to the equations of state $p_T=\alpha T$ and $s=\alpha$, so the entropy is simply the area in units of a fundamental area, $S=\alpha A$. The Euclidean saddle point reproduces these relations only when the inverse temperature is the Tolman-Ehrenfest period; with that choice the matter free energy is $F_m=-p_T A$, and the gravitational junction action cancels it on shell, making the total free energy of the surface zero.

Load-bearing premise

The construction rests on assuming that the surface matter is described by the Euclidean action $\int_\Sigma d^3x\sqrt{\gamma}\,p_T$ and that the temperature is the Tolman-Ehrenfest period; the paper itself notes the perfect-fluid action is ambiguous, and without a horizon the temperature definition is not unique, so if either choice is changed the relations $p_T=\alpha T$ and $S=\alpha A$ need not follow.

Editorial extensions

If this is right

  • Gravastar-like regular black holes whose interiors are glued through a thin timelike shell would acquire an entropy equal to the shell area in fundamental units, despite having no horizon.
  • The junction temperature is set by the jump in surface gravity, replacing the horizon-based surface gravity with a boundary-based quantity that reduces to the Newtonian gradient in the weak-field limit.
  • Because the total free energy vanishes once the gravitational action is added, the thin shell contributes no net free energy, leaving the energetics of the glued spacetime unencumbered.
  • In configurations with negative transverse pressure, such as an anti-de Sitter interior matched to empty space, the temperature is negative, which undercuts the usual third-law intuition for gravitational boundaries.
  • The matter partition function is approximately $Z\approx e^S$ and does not depend on temperature, matching the zero-energy requirement.

Reading between the lines

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

  • This is an editorial inference: if $S=\alpha A$ is temperature-independent, the entropy is a purely geometric area law, pointing toward counting boundary degrees of freedom on the timelike surface rather than on a horizon.
  • As an extension of the paper's method, allowing the junction to rotate or carry charge breaks the static spherical setup, so (J1) will no longer force $S^t{}_t=0$; computing the modified surface stress tensor would delimit how universal the area law is.
  • The paper leaves the fundamental scale $\alpha$ unfixed; an obvious next step, inferred here, is that if $\alpha$ is the Planck scale, the construction predicts a definite numerical entropy for horizonless compact objects, a number that could be compared with numerical simulations of gravastar-like collapse.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies static, spherically symmetric spacetimes with a preferred timelike junction surface, motivated by gravastar-like constructions. Using Israel's junction conditions for the restricted matching ansatz of Eq. (3), the authors show that the surface stress-energy has Stt=0 and only a transverse pressure pT=[k]/(8π), so the junction carries no Misner-Sharp energy. They then propose a thermodynamics for this zero-energy surface, deriving the Euler relation pT=Ts, the equations of state pT=αT and s=α, and area-proportional entropy S=αA. A Euclidean path integral with a Chamblin-Reall gravitational action and a matter action Im=∫√γ pT is used to claim confirmation, with temperature identified with the Tolman-Ehrenfest period (15), and to derive zero total free energy when gravity is included. The paper concludes with remarks on negative temperature for AdS interiors and analogies with black-hole thermodynamics.

Significance. Section II's zero-energy result is a clean and correct observation within the stated ansatz, and the paper is clearly written and appropriately referenced. If the thermodynamic relations were rigorously established, they would constitute an interesting horizonless analogue of black-hole thermodynamics with area entropy, relevant to gravastar models and to the question of whether area entropy requires horizons. However, the load-bearing steps are not yet established: the Euclidean path integral confirmation is circular, the matter action is admitted to be ambiguous, and the total-action cancellation leaves the full system with zero entropy. The area law S=αA also contains an undetermined constant α (a fundamental inverse area), so the paper does not make a parameter-free prediction. The work is therefore best viewed as a proposal requiring further justification rather than a derivation.

major comments (4)
  1. [Section IV, Eqs. (16)-(18) and Table I] The Euclidean path integral does not confirm the matter thermodynamics; it presupposes it. From (16) one has Z_m = exp(β_TE A p_T), so the standard canonical formulas (used elsewhere in the paper) give U_m = -A p_T and S_m = 0, since p_T=[k]/(8π) is fixed by the junction condition and does not depend on β_TE. The paper instead sets S = ln Z_m and postulates that this is temperature-independent 'so that U=0' (Eq. (18)); but β_TE-independence of β_TE A p_T is precisely the desired equation of state p_T = α T_TE. Additionally, the Tolman-Ehrenfest period (15) contains an unspecified integration range in a horizonless spacetime, so the temperature is not uniquely fixed. As written, the path integral argument is therefore circular.
  2. [Section IV, closing paragraph and Conclusions] The total Euclidean action vanishes on shell because I_g^E and I_m^E cancel (Eq. (16) with p_T=[k]/(8π)), so the total partition function is unity, implying zero total free energy and zero total entropy. The text nonetheless states that the matter content has entropy S=αA and that the surface carries the thermodynamic entropy of the space. If the gravitational action is part of the system, its contribution to ln Z is -S, and the full system has S=0; the paper needs to state clearly whether the area law refers to the matter subsystem only, and if so, why that subsystem is physically meaningful when the combined action has no entropy.
  3. [Section III, Eq. (10)] The Maxwell relation is written as ∂p_T/∂T = ∂S/∂A, but from F = -A p_T(T,A) the general relation is T ∂p_T/∂T = p_T + A ∂p_T/∂A; the printed relation is valid only when ∂p_T/∂A=0. The junction pressure (5) generally depends on r0 and hence on A, so this condition is not automatic. The conclusion s=α can be obtained directly from Eq. (7) with ρ=0, which indeed forces ds=0, but the paper should present that derivation instead of the Maxwell-relation shortcut, and should state any additional assumption explicitly.
  4. [Section IV, footnote 3 and Eq. (14)] The matter action I_m = ∫√-γ p_T is admitted in footnote 3 to be ambiguous for perfect fluids. All subsequent thermodynamic statements (Eqs. (17)-(18), S=αA, p_T=αT) depend on this choice. A different perfect-fluid action would in general produce different free energy and entropy; the paper should justify the chosen action from a microscopic model or demonstrate that the main results are independent of this ambiguity.
minor comments (5)
  1. [Section II, Eq. (6)] The definition [E] ≡ E(+) − E(+) contains a typo; the second term should be E(−).
  2. [Section III, Eq. (9)] The symbol F is used both for the areal free energy density (F=-pT, Eq. (9)) and for the total free energy (F_mΣ=-pT A, Eq. (17)); please distinguish these with different notation.
  3. [Section III] The word 'Helmoltz' should be 'Helmholtz'.
  4. [Section VI] In the acknowledgments, 'ag tly supported' appears to be a typo for 'gratefully supported'.
  5. [Throughout] The hyphenation 'time-like' and 'timelike' is inconsistent; please choose one form and use it uniformly.

Circularity Check

1 steps flagged · score 6.0 of 10

Path integral confirmation is circular: β-independence of ln Z = βTE A pT is imposed, so pT = α TTE is an input, not a derived result.

  1. self definitional [Section IV.B, after Eq. (16), around Eq. (18) and the paragraph beginning 'Finally it is easy...']
    "In the saddle point approximation the matter partition function is approximated by the unusual: Z ≈ eS (18) and does not actually depend on the temperature. This must be true, so that here is no contradiction between the usual formula for U and the fact that U is zero on Σ. ... Finally it is easy to derive the equation of state (11) from the fact that ln Z = S = pT AβT Eand that this cannot depend on βT E, so that U = 0. Hence pT = αTT E."

    With the matter action (16), ln Z_m = β_TE A p_T, and p_T = [k]/(8π) is fixed by the junction condition with no β_TE dependence. The paper then defines S = ln Z and stipulates that ln Z 'cannot depend on β_TE' so that U = 0. But β_TE-independence of β_TE A p_T is exactly the relation p_T = α T_TE that is presented as the derived result; the equation of state is imposed as the condition, not obtained from the path integral. Under the standard canonical formulas used elsewhere, the same partition function would give U = -A p_T and S = 0, so the new entropy assignment in Table I is doing the work.

full rationale

The zero-energy junction result (Stt=0, pT=[k]/8π) is derived directly from Israel's junction conditions and the Misner-Sharp energy identity; that part is self-contained and not circular. The paper's Section III Euler relation pT=Ts follows from ρ=0, and S=αA is a consistent area-law form, though the constancy of α is an additional integration assumption rather than a forced consequence of the Maxwell relation. The circularity enters in Section IV.B, where the 'confirmation' of pT=αT is constructed: ln Z_m = βTE A pT depends on βTE unless pT ∝ 1/βTE, and the paper simply declares Z≈e^S temperature-independent because U=0, then reads off pT=αTTE. That is the desired equation of state inserted by hand, not a prediction. Because the central thermodynamic proportionality is thus not independently derived, but the independent junction energetics and on-shell action cancellation remain valid, the partial circularity score is 6. Footnote 3 also concedes the perfect-fluid action is ambiguous, so the path integral 'confirmation' is not unique; this is a limitation but not itself a circular step.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central thermodynamic results rest on one undetermined constant alpha, the choice of matter action (acknowledged ambiguous), and the choice of the Tolman-Ehrenfest temperature. The zero-energy junction condition itself is derived from standard Israel and Misner-Sharp inputs.

free parameters (1)
  • alpha (inverse fundamental area) = undetermined, positive constant
    Introduced in Eqs. (11)-(12) as the constant relating pT to T and s to alpha; sets S=alpha A and defines the fundamental area 1/alpha. Not fixed by the theory.
assumptions (6)
  • standard math Israel junction conditions (J1, J2) with continuity of induced metric and the standard thin-shell stress tensor S_ij = -(1/8pi)([k_ij]-[k] gamma_ij)
    Used in Section II to derive Stt=0 and pT=[k]/(8pi); cited to Israel and Poisson.
  • standard math Misner-Sharp energy E=(r/2)(1 - g^{mu nu} partial_mu r partial_nu r) is the correct energy measure for static spherically symmetric spacetimes
    Used to argue that a nonzero surface energy would contradict J1; cited to Misner-Sharp and Hayward.
  • domain assumption The first law for a 2+1 surface with no conserved particle number is dU = T dS - pT dA, with Euler relation U = TS - pT A
    Adopted in Section III as the thermodynamic framework for the zero-energy surface.
  • ad hoc to paper The matter action for the shell is I_m = integral sqrt(-gamma) pT
    Footnote 3 admits ambiguity for perfect fluids; this choice is needed to reproduce the junction condition and the derived thermodynamics.
  • ad hoc to paper The inverse temperature of the surface is the Tolman-Ehrenfest period betaTE = integral iN dt (Eq. 15)
    Adopted in Section IV A to make the Euclidean action factorize as betaTE A; the paper notes temperature is ambiguous without a horizon.
  • standard math The Euclidean path integral saddle point and identification Z approx exp(-I_E) with the partition function are valid
    Used in Section IV B to extract free energy, entropy, and pressure from the on-shell action.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Thermodynamics in space-times without horizons." pith.science (2026). https://pith.science/paper/MXMA62JJ

@misc{pith2026250703469,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics in space-times without horizons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXMA62JJ}},
  note         = {Machine review of arXiv:2507.03469}
}
read the original abstract

We consider the energetics and thermodynamics of spacetimes with no horizons, but endowed with a preferred timelike junction surface. They could arise as a limiting case of the gravastar and other constructions regularizing the interior of the horizon of a black hole, or from the conceptual cutting of a portion of a non-asymptotically flat space and gluing it with flat space. We find that such surfaces can be made to have zero energy, so that the energetics of such spaces is not encumbered by them. They do have a transverse pressure, fixed by the jump in surface gravity. A peculiar matter thermodynamics then follows, with well defined entropy, temperature and surface pressure, constrained by specific relations arising from the zero energy condition. This is confirmed by the Euclidean path integral, with the proviso that the Tolman-Ehrenfest temperature should be used. The entropy of the space is then the area of the timelike surface in units of a fundamental area, and the matter temperature is proportional to the transverse pressure and so the jump in surface gravity. However, when the gravitational action is added, the free energy of the surface is also zero. The fact that for some such spaces the temperature comes out negative raises interesting questions regarding the third law of thermodynamics.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

22 extracted references · 8 canonical work pages

  1. [1]

    J. D. Bekenstein, Lett. Nuovo Cim. 4 (1972) 737-740 doi:10.1007/BF02757029

    doi:10.1007/bf02757029 1972

  2. [2]

    J. D. Bekenstein, Phys. Rev .D 7 (1973) 2333-2346, doi:10.1103/PhysRevD.7.2333

    doi:10.1103/physrevd.7.2333 1973

  3. [3]

    J. D. Bekenstein, Phys. Rev .D 9 (1974) 3292-3300, doi:10.1103/PhysRevD.9.3292

    doi:10.1103/physrevd.9.3292 1974

  4. [4]

    J. M. Bardeen, B. Carter, S. W. Hawking, Commun. Math. Phys. 31 (1973) 161-170, doi:10.1007/BF01645742

    doi:10.1007/bf01645742 1973

  5. [5]

    G. W. Gibbons, S. W. Hawking, Phys. Rev. D 15 (1977) 2738-2751, doi:10.1103/PhysRevD.15.2738

    doi:10.1103/physrevd.15.2738 1977

  6. [6]

    P. O. Mazur, E. Mottola, Proc. Nat. Acad. Sci. 101 (2004) 9545-9550, doi:10.1073/pnas.0402717101, [arXiv:gr-qc/0407075 [gr-qc]]

    arXiv 2004

  7. [7]

    P. O. Mazur, E. Mottola, Universe 9(2) (2023), doi:10.3390/universe9020088, [arXiv:gr-qc/0109035 [gr- qc]]

    arXiv 2023

  8. [8]

    Mottola, Regular Black Holes

    E. Mottola, Regular Black Holes. Springer Series in Astrophysics and Cosmology: Gravitational Vac- uum Condensate Stars, Springer Singapore, 2023, doi: 10.1007/978-981-99-1596-5 8

    doi:10.1007/978-981-99-1596-5 2023

Show all 22 references
  1. [9]

    (or e.g. [10]). We consider an internal geometry M(−) with metric gµν(−) which induces metric γij(−) on a constant r = r0 time-like hypersurface Σ, acting as a junction surface to corresponding external struc- tures M(+), gµν(+) and γij(+). The difference of tensorial quantiti...

  2. [10]

    Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics: Hypersurfaces, Cambridge Uni- versity Press, 2004, doi:10.1017/CBO9780511606601

    E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics: Hypersurfaces, Cambridge Uni- versity Press, 2004, doi:10.1017/CBO9780511606601

    2004 doi

  3. [11]

    Israel, Nuovo Cim.B 44 (1966) 1, doi:10.1007/BF02710419; Nuovo Cim

    W. Israel, Nuovo Cim.B 44 (1966) 1, doi:10.1007/BF02710419; Nuovo Cim. B 48 (1967) 463 (erratum), doi:10.1007/BF02712210

    1966 doi

  4. [12]

    S. A. Hayward, Class. Quant. Grav. 15 (1998) 3147-3162, doi:10.1088/0264-9381/15/10/017, [arXiv: arXiv:gr- qc/9710089]

    1998

  5. [13]

    C. W. Misner, D. H. Sharp, Phys. Rev. 136 (1964) B571- B576,

    1964

  6. [14]

    R. C. Tolman, Phys. Rev. 35 (1930) 904–924; R. C. Tolman and P. Ehrenfest, Phys. Rev. 36 (1930) no. 12, 1791–1798. 6

    1930

  7. [15]

    H. A. Chamblin and H. S. Reall, Nucl. Phys. B 562, 133-157 (1999) doi:10.1016/S0550-3213(99)00520-9 [arXiv:hep-th/9903225 [hep-th]]

    1999 arXiv

  8. [16]

    Rovelli and M

    C. Rovelli and M. Smerlak, Class. Quant. Grav. 28, 075007 (2011) doi:10.1088/0264-9381/28/7/075007 [arXiv:1005.2985 [gr-qc]]

    2011 arXiv

  9. [17]

    J. D. Brown and J. W. York, Jr., ‘The Path in- tegral formulation of gravitational thermodynam- ics,” [arXiv:gr-qc/9405024 [gr-qc]]; J. D. Brown and J. W. York, Jr., Phys. Rev. D 47 (1993), 1407-1419 doi:10.1103/PhysRevD.47.1407 [arXiv:gr-qc/9209012 [gr-qc]]

    1993 arXiv

  10. [18]

    J. D. Brown, Class. Quant. Grav. 10, 1579-1606 (1993) doi:10.1088/0264-9381/10/8/017 [arXiv:gr-qc/9304026 [gr-qc]]

    1993 arXiv

  11. [19]

    on-shell

    in which ρ = ρ(s), so that dρ = ∂ρ ∂s A ds = T ds. Then, the right hand side of (7) vanishes, and we ob- tain the equivalent of the Euler relation for the surface ρ + pT = T s. Imposing now ρ = 0 upon this general ex- pression we finally get the equivalent of the Euler relatio...

  12. [20]

    Misner, K

    C. Misner, K. Thorne and J.A.Wheeler, Gravitation, W. H. Freeman, 1973

    1973

  13. [21]

    Ballesteros, D

    G. Ballesteros, D. Comelli and L. Pilo, Phys. Rev. D 94 (2016) no.2, 025034 doi:10.1103/PhysRevD.94.025034 [arXiv:1605.05304 [hep-th]]

    2016 arXiv

  14. [22]

    R. R. Caldwell, A. Chamblin and G. W. Gib- bons, Phys. Rev. D 53 (1996), 7103-7114 doi:10.1103/PhysRevD.53.7103 [arXiv:hep-th/9602126 [hep-th]]

    1996 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.