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arxiv: 2605.03311 · v1 · submitted 2026-05-05 · 🌀 gr-qc · hep-th

Recognition: unknown

Noether charges and the first law of thermodynamics for multifractional Schwarzschild black hole in the q-derivative theory

Reggie C. Pantig

Pith reviewed 2026-05-07 14:32 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords multi-fractional gravityblack hole thermodynamicsNoether chargesextended first lawq-derivative theorySchwarzschild solutionintegrable entropyprofile parameters
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0 comments X

The pith

Enlarging the thermodynamic state space with multi-fractional profile parameters restores integrability to the first law.

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

The paper examines thermodynamics of static spherically symmetric vacuum solutions in multi-fractional q-derivative gravity. In the geometric frame the metric reduces exactly to Schwarzschild in the areal radius q, so that Noether charges give a profile-independent mass and the Iyer-Wald entropy obeys the standard area law. When temperature is instead computed in the fractional radial coordinate, it acquires an explicit dependence on the profile through the derivative factor at the horizon, which generically prevents the Clausius relation from being integrable in the usual variables. The obstruction is removed by treating the profile parameters as additional thermodynamic coordinates and defining an entropy via a radial integral over the geometric radius. The resulting extended first law contains extra work terms conjugate to those parameters, and the construction is shown to be consistent for both binomial and log-oscillating profiles under conditions that guarantee a single exterior horizon.

Core claim

In the multi-fractional theory with q-derivatives the vacuum solution in the spherical-coordinate approximation is exactly Schwarzschild when expressed in the geometric areal radius q. Canonical Noether charges therefore yield a conserved mass that depends only on the Schwarzschild integration constant, while the Iyer-Wald entropy satisfies the area law evaluated at the geometric horizon radius. Hawking temperature defined in the fractional coordinate r nevertheless depends on the multi-fractional profile through the local factor q'(r_h). Variations of the non-dynamical profile parameters then obstruct integrability of the naive Clausius relation. This is resolved by enlarging the state空间 to

What carries the argument

Enlargement of the thermodynamic state space to include the multi-fractional profile parameters, together with the integrable entropy functional obtained from a radial integral of the geometric radius.

If this is right

  • The conserved mass extracted from Noether charges is insensitive to the choice of multi-fractional profile.
  • The entropy obeys the geometric area law but requires the radial integral construction to remain integrable when profiles are varied.
  • The extended first law acquires additional work terms whose conjugates are the multi-fractional coupling constants.
  • Both binomial and log-oscillating profiles admit consistent single-horizon exterior solutions once the stated conditions on the profile functions are met.
  • Presentation dependence appears in explicit expressions but does not affect the existence of the integrable structure.

Where Pith is reading between the lines

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

  • The same state-space enlargement may be needed in other scale-dependent or background-profile theories where standard thermodynamic relations cease to be integrable.
  • The separation between profile-insensitive Noether charges and profile-sensitive thermal quantities indicates that black-hole thermodynamics in these models is inherently frame-dependent.
  • Consistency conditions derived for the exterior branch could be used to restrict admissible profile functions in phenomenological applications.
  • Numerical evaluation of the radial entropy integral for concrete profile choices would provide a concrete check of the extended law.

Load-bearing premise

The vacuum solution remains exactly Schwarzschild in the geometric areal radius q after the spherical-coordinate approximation, and the profile parameters can be varied independently as non-dynamical backgrounds.

What would settle it

A direct variation of the profile parameters while holding the geometric metric fixed shows that the proposed entropy functional fails to satisfy the extended first law, or an explicit computation of surface gravity in the fractional coordinate lacks the predicted q'(r_h) factor.

Figures

Figures reproduced from arXiv: 2605.03311 by Reggie C. Pantig.

Figure 1
Figure 1. Figure 1: FIG. 1. Temperature ratio view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Integrable thermodynamic entropy relative to the Noether/area entropy, view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dimensionless multifractional work potential view at source ↗
read the original abstract

In this paper, we investigate black-hole thermodynamics in the multi-fractional theory with $q$-derivatives, focusing on static, spherically symmetric vacuum solutions in the spherical-coordinate approximation. In the geometric frame the solution is exactly Schwarzschild in the areal radius $q$, so that canonical charges can be defined using standard covariant methods. The conserved mass depends only on the Schwarzschild integration constant, and the Iyer--Wald entropy satisfies the usual area law in terms of the geometric horizon radius. When the Hawking temperature is defined in the fractional radial coordinate $r$, however, it acquires an explicit dependence on the multi-fractional profile through the local factor $q'(r_{\rm h})$ at the horizon. As a result, variations of the non-dynamical profile parameters generically obstruct integrability of a naive Clausius relation expressed solely in terms of mass and entropy. We show that this obstruction is resolved by enlarging the thermodynamic state space to include the profile parameters and by constructing an integrable entropy functional obtained from a radial integral of the geometric radius. The corresponding extended first law contains additional work terms conjugate to the multi-fractional couplings. We analyze both binomial and log-oscillating profiles, clarify the role of presentation dependence, and delineate the consistency conditions required for a well-defined exterior branch with a single horizon. Our results make explicit the separation between profile-insensitive canonical charges and profile-sensitive thermal quantities in multi-fractional black-hole thermodynamics.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates black-hole thermodynamics in multi-fractional q-derivative gravity for static spherically symmetric vacuum solutions under the spherical-coordinate approximation. In the geometric frame the metric is claimed to be exactly Schwarzschild in the areal coordinate q, permitting standard Iyer-Wald Noether charges (mass independent of the profile) and area-law entropy at the geometric horizon. Temperature defined in the fractional radial coordinate r acquires explicit dependence on the profile through q'(r_h), obstructing integrability of the naive first law. The obstruction is resolved by enlarging the thermodynamic state space to include the non-dynamical profile parameters and constructing an integrable entropy via radial integration of the geometric radius; the resulting extended first law includes additional work terms conjugate to the multi-fractional couplings. Explicit results are given for binomial and log-oscillating profiles together with consistency conditions for a single-horizon exterior branch.

Significance. If the central claims hold, the work supplies a concrete procedure for restoring thermodynamic integrability in theories whose action contains non-dynamical background profiles by systematically enlarging the state space while retaining covariant Noether-charge methods. The explicit separation between profile-insensitive canonical charges and profile-sensitive thermal quantities, together with the analysis of two representative profiles, offers a useful template for similar constructions in other modified-gravity settings with coordinate-dependent measures.

major comments (2)
  1. [Abstract / vacuum-solution section] Abstract and the derivation of the vacuum solution: the claim that the solution remains exactly Schwarzschild in the geometric areal radius q under the spherical-coordinate approximation is load-bearing for the profile-independence of the Noether mass and the Iyer-Wald entropy. The q-derivative modifications to the connection or measure could generate residual profile-dependent contributions to the Einstein tensor or to the surface integrals after the coordinate redefinition r → q(r); an explicit verification that these terms vanish at the horizon and at infinity is required.
  2. [Entropy-construction paragraph] Construction of the integrable entropy (abstract): the radial integral that defines the extended entropy functional must be shown to yield a closed one-form on the enlarged state space that includes the profile parameters. The explicit variation of this functional should be displayed and demonstrated to cancel the non-integrable term proportional to q'(r_h) δr_h that appears in the naive Clausius relation.
minor comments (2)
  1. [Temperature definition] The notation distinguishing the geometric radius q(r) from its derivative q'(r) should be introduced once and used consistently when the Hawking temperature is expressed in the fractional coordinate.
  2. [Results section] A brief table comparing the thermodynamic quantities (mass, entropy, temperature, and work terms) for the binomial versus log-oscillating profiles would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work's significance, and constructive major comments. We address each point below and have revised the manuscript to incorporate the requested explicit verifications and derivations.

read point-by-point responses

  1. Referee: [Abstract / vacuum-solution section] Abstract and the derivation of the vacuum solution: the claim that the solution remains exactly Schwarzschild in the geometric areal radius q under the spherical-coordinate approximation is load-bearing for the profile-independence of the Noether mass and the Iyer-Wald entropy. The q-derivative modifications to the connection or measure could generate residual profile-dependent contributions to the Einstein tensor or to the surface integrals after the coordinate redefinition r → q(r); an explicit verification that these terms vanish at the horizon and at infinity is required.

    Authors: We agree that an explicit verification strengthens the central claim. In the revised manuscript we have added a dedicated subsection (now Section 3.2) that computes the modified connection and measure contributions to the Einstein tensor after the redefinition r → q(r). We show term-by-term that all profile-dependent pieces cancel identically for the vacuum static spherically symmetric ansatz, both in the bulk and in the surface integrals evaluated at the geometric horizon and at spatial infinity. The resulting Noether mass and Iyer-Wald entropy therefore remain exactly those of the Schwarzschild solution in the q-coordinate, independent of the specific q-profile. revision: yes

  2. Referee: [Entropy-construction paragraph] Construction of the integrable entropy (abstract): the radial integral that defines the extended entropy functional must be shown to yield a closed one-form on the enlarged state space that includes the profile parameters. The explicit variation of this functional should be displayed and demonstrated to cancel the non-integrable term proportional to q'(r_h) δr_h that appears in the naive Clausius relation.

    Authors: We thank the referee for this observation. The original manuscript contained the integrated form but omitted the intermediate variation. In the revision we now display the explicit one-form variation of the extended entropy functional S_ext = ∫ 4π q(r) dr (integrated from a reference radius to r_h). Its exterior derivative on the enlarged state space (M, r_h, {profile parameters}) is shown to be closed; the term arising from δq'(r_h) precisely cancels the non-integrable contribution q'(r_h) δr_h that appears in the naive first law, leaving only the integrable combination δM − T δS_ext − ∑ W_i δλ_i = 0, where λ_i are the profile couplings and W_i their conjugate work terms. This is now written out in full in the new Section 4.3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard Noether methods to assumed metric and constructs extended entropy explicitly

full rationale

The paper takes as input the vacuum solution being exactly Schwarzschild in the geometric areal radius q under the spherical-coordinate approximation. It then applies standard Iyer-Wald Noether charges to obtain a profile-independent mass and area-law entropy in q. The non-integrability of the naive first law under profile variations is identified, after which an integrable entropy is constructed via an explicit radial integral over the geometric radius. The extended first law with additional work terms follows directly from this construction and the enlarged state space. No step reduces a claimed result to its own inputs by definition, fitting, or self-citation chain; the construction is presented as the resolution rather than a prediction forced by prior assumptions. The metric form and approximation are external inputs from the multi-fractional framework, not derived circularly within the paper.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the multi-fractional q-derivative framework (taken as given) together with standard general-relativity tools; no new particles or forces are introduced.

free parameters (1)
  • multi-fractional profile parameters
    Binomial and log-oscillating profile parameters are varied when checking integrability and appear as conjugate variables in the extended first law.
axioms (2)
  • domain assumption The static spherically symmetric vacuum solution in the geometric frame is exactly the Schwarzschild metric expressed in the areal radius q
    This assumption, stated in the spherical-coordinate approximation, permits the direct use of standard covariant Noether-charge methods.
  • standard math The Iyer-Wald formalism yields an entropy that obeys the area law in the geometric horizon radius
    Invoked to obtain the usual area-law expression before extending the thermodynamic space.

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