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

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.

• The paper develops a multifractional framework with q-derivatives that reproduces the Schwarzschild solution in a geometric frame while highlighting profile-induced thermal modifications.
• It employs covariant phase-space methods to clearly distinguish between profile-insensitive canonical charges and profile-sensitive physical temperature.
• The study extends the first law by incorporating multifractional parameters as external couplings, leading to an integrable thermal entropy formulation.

Noether Charges and Thermodynamics of Multifractional Schwarzschild Black Holes in the $q$-Derivative Theory

Introduction and Context

This work critically advances the thermodynamic analysis of black holes in the multi-fractional spacetime framework with $q$-derivatives. In this approach, the structure of spacetime is encoded by nontrivial geometric coordinates $q^\mu(x^\mu)$ designed to capture scale-dependent dimensional flow, a phenomenon present in diverse quantum gravity models. This "multi-scale" structure modifies gravitational dynamics via an anomalous integration measure and replacement of derivatives by $q$-derivatives.

Within the $q$-derivative theory ($T_q$), static, spherically symmetric vacuum solutions admit a geometric frame where the metric is exactly Schwarzschild in terms of the areal coordinate $q$. Canonical Noether charges are computable in this frame using standard covariant phase-space/Hamiltonian methods. However, physical observables such as the Hawking temperature, when defined in the fractional (physical) radial coordinate $r$, acquire explicit dependence on the multiscale profile parameters through the derivative $q'(r)$ evaluated at the horizon. This distinct separation between profile-insensitive canonical charges and profile-sensitive thermal quantities underpins much of the paper’s technical development.

The Multifractional Schwarzschild Solution and Profile Parametrization

The multifractional Schwarzschild geometry is constructed by importing the Schwarzschild solution to the geometric coordinate $q$, with the relation $q$0 set by the chosen profile:

$q$1

for $q$2 and characteristic scale $q$3. With log-oscillating modifications,

$q$4

$q$5

where $q$6, $q$7, and $q$8 encode amplitude and frequency of discrete scale invariance, with $q$9 a further scale.

The metric in the physical frame is then

$q^\mu(x^\mu)$0

$q^\mu(x^\mu)$1

The horizon is located at $q^\mu(x^\mu)$2.

Physical Temperature and Integrability Obstruction

The physical Hawking temperature is given in terms of the fractional coordinate as

$q^\mu(x^\mu)$3

$q^\mu(x^\mu)$4

This temperature inherits explicit profile dependence via $q^\mu(x^\mu)$5. For binomial profiles, $q^\mu(x^\mu)$6, so $q^\mu(x^\mu)$7. For log-oscillating profiles, the temperature acquires a log-periodic modulation due to the oscillatory structure of $q^\mu(x^\mu)$8.

Figure 1: Temperature ratio $q^\mu(x^\mu)$9 as a function of the rescaled Schwarzschild parameter, illustrating the dependence on binomial and log-oscillating profiles and the distinct thermal behavior arising from multifractional corrections.

A critical observation is that, on an extended state space including the profile parameters $q$0 (e.g., $q$1, $q$2, $q$3, $q$4, $q$5), the differential form $q$6 fails to be integrable: $q$7, which is nonzero if $q$8 depends on $q$9. This obstructs the definition of a naive thermodynamic entropy.

Covariant Phase Space: Canonical Mass and Noether Charge Entropy

Using the covariant phase-space formalism (Lee–Wald/Iyer–Wald), the canonical mass is identified with the coefficient of the $q$0 term in $q$1 in the geometric frame, yielding

$q$2

This is independent of the profile parameters at fixed $q$3.

The Noether (Iyer–Wald) entropy is

$q$4

again depending solely on the geometric horizon radius $q$5, and similarly insensitive to the multifractional profile.

Extended First Law: Integrability Restoration and Thermal Entropy

To resolve the integrability problem, the thermodynamic state space is enlarged to include the profile parameters as external couplings. Along this extended space, the entropy compatible with the physical temperature is defined via

$q$6

leading to an integrable entropy

$q$7

Figure 2: Ratio of the integrable thermal entropy $q$8 to the Noether entropy $q$9 as a function of $T_q$0, quantifying the departure from the geometric frame area law due to multifractional corrections for different profiles.

Allowing variations of $T_q$1, the full differential reads

$T_q$2

The extended first law becomes

$T_q$3

$T_q$4

For the binomial profile, explicit expressions for $T_q$5 and $T_q$6 are provided, with the dimensionless multifractional work potential $T_q$7 quantifying the work associated with varying $T_q$8.

Figure 3: Dimensionless multifractional work potential $T_q$9 for binomial and oscillatory profiles, highlighting the impact of profile variation on the extended thermodynamic law.

Profile and Presentation Dependence

Thermal quantities such as $q$0, $q$1, and $q$2 are strongly profile-dependent, with both the binomial sign (presentation) and log-oscillations generating $q$3 fractional changes in $q$4 and $q$5 at small $q$6, though all such effects decay in the infrared (large black holes).

For oscillatory profiles, the entropy correction integrates to a closed analytic form exhibiting log-periodic modulations, with the overall amplitude suppressed for $q$7. The presence of additional structure in $q$8 can generate further horizon loci and complexify the exterior region, a fact carefully treated via branch and monotonicity restrictions.

The canonical charges $q$9, $r$0 remain unaffected by detailed profile or presentation choices, sharpening the distinction between the underlying geometric frame charges and the profile-sensitive thermal observables.

Implications and Prospects

The theoretical implications are significant. Adopting the physical frame for temperature definitions in multifractional gravity necessitates a redefinition of black-hole thermodynamics, yielding an integrable thermal entropy that is not simply the area in the geometric frame, and enforcing extra work terms in the first law conjugate to the multifractional couplings. This structure mirrors, at a technical level, extended black hole thermodynamics with variable couplings and enriches the thermodynamic landscape with new operational observables.

The explicit separation between profile-insensitive and profile-sensitive charges may have practical ramifications for phenomenological searches for quantum gravity signatures, especially in multimessenger astrophysics, where constraints on multifractional parameters continue to tighten.

The formalism established here provides a foundation for future analyses of rotating, charged, or otherwise structured black holes in multifractional frameworks. The phase-space methodology and the treatment of external parameter variations are extensible, and it is anticipated that the presentation-induced uncertainty bands and log-periodicity will persist as operational features in more general solutions.

Conclusion

This work delivers a comprehensive and technically rigorous analysis of static vacuum black hole thermodynamics in the multifractional $r$1-derivative gravity framework. The careful distinction between geometric-frame Noether charges and profile-dependent thermal observables leads to a generalization of the first law on an extended state space, with integrability enforced by inclusion of work terms conjugate to all multifractional parameters. The methodology and results lay a robust groundwork for further study of black hole mechanics in multi-scale gravitational models and their phenomenological implications.

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References: See (2605.03311).