Black Hole Entropy in f(Q) Gravity from the RVB Residue Method

Abstract: We extend the residue-based Robson-Villari-Biancalana (RVB) method from the calculation of Hawking temperature to the determination of black hole entropy within f(Q) gravity. Starting from the residue-corrected temperature prescription developed in recent RVB analyses of f(Q) black holes, we combine this approach with the first law of black hole thermodynamics to derive a general expression for the entropy of static, spherically symmetric configurations. By expressing the metric in a standard Schwarzschild-like decomposition with an additional correction term, we show that the entropy satisfies a universal integral relation. The integrand depends explicitly on horizon data as well as on a residue-induced temperature shift parameter. For the specific quadratic model, we obtain an explicit closed-form expression for the entropy at first order in the residue parameter. In the limit where the residue contribution vanishes, the standard Bekenstein-Hawking area law is recovered. However, once the complex contour contribution is retained, a correction beyond the area law naturally emerges. This framework should be interpreted as a residue-induced thermodynamic extension of the temperature-based method, rather than as a universal Noether charge formulation applicable to all f(Q) black hole solutions.

• The paper derives an explicit black hole entropy formula in f(Q) gravity using the RVB residue approach, extending the Bekenstein-Hawking area law.
• It utilizes a complex contour integral method to modify the Hawking temperature, with the residue term integrating via the first law of thermodynamics.
• Findings show that positive residue and nonmetricity corrections systematically reduce entropy, impacting black hole thermodynamic stability.

Black Hole Entropy in $f(Q)$ Gravity via the RVB Residue Method

Introduction

This work addresses the thermodynamics of black holes in metric-affine gravity, focusing specifically on the nonmetricity-based $f(Q)$ sector. It extends the residue-based Robson-Villari-Biancalana (RVB) contour method, which supplies an analytic/topological correction to Hawking temperature in modified gravity, to the black hole entropy sector. The approach adopts the RVB-modified temperature as an input and employs the first law of black hole thermodynamics to extract an explicit entropy formula for static, spherically symmetric black hole solutions in $f(Q)$ gravity. A detailed analytic treatment is presented for the quadratic model $f(Q) = Q + \alpha Q^2$, with the correction to the Bekenstein-Hawking area law exhibited in closed form to leading order in the residue parameter.

RVB Prescription and Residue-Corrected Hawking Temperature

Within $f(Q)$ gravity the action is parameterized by a general function of the nonmetricity scalar, with deviations from general relativity encoded in both the equations of motion and in horizon thermodynamics. The RVB approach leverages complex analysis and the residue theorem: the Hawking temperature receives a nonlocal contribution computed as a complex contour integral of $F'(z)/F(z)$, where $F(z)$ is determined by the metric or the nonmetricity.

As a result, the modified Hawking temperature is

$T_H(r_+) = \frac{g'(r_+)}{4\pi} + C_{\rm res}$

where $C_{\rm res}$ is the RVB residue shift, interpreted as a (potentially topological) analytic correction. This extends the usual surface gravity relation and automatically reduces to the GR value in the absence of the residue.

Entropy Formula: General Construction

The entropy is not derived via a Noether charge approach but rather obtained as the function compatible with the temperature in the first law,

$dM = T_H\, dS \,.$

For any static, spherically symmetric solution of the form

$f(Q)$0

with horizon at $f(Q)$1, the entropy is given by the integral

$f(Q)$2

where $f(Q)$3. The area law is instantly recovered by setting $f(Q)$4, yielding $f(Q)$5.

For perturbatively small $f(Q)$6, the correction to the area law is explicit, suppressing (or enhancing, depending on sign) the entropy through an additional horizon-integrated term controlled by the local metric deformation and the residue.

Entropy in the Quadratic Model $f(Q)$7

Specialization to the quadratic case,

$f(Q)$8

leads to $f(Q)$9 and $f(Q)$0.

The explicit entropy, valid to leading order in $f(Q)$1, is

$f(Q)$2

where $f(Q)$3 fixes the entropy's reference point. The standard area law is manifest when $f(Q)$4.

Notably, for fixed positive $f(Q)$5 and positive $f(Q)$6, the entropy is reduced relative to the area law at all horizon radii, with the difference governed by both the nonmetricity parameter and the residue, as made clear by the analytic form of the correction.

A nontrivial check is available by setting $f(Q)$7: the derived entropy reduces to the Schwarzschild case, confirming the formal limit and ensuring continuity with GR thermodynamics.

Theoretical Implications and Future Directions

The construction here is not a Noether charge approach but a thermodynamic extension dictated by the analytic/topological corrections found in the RVB temperature prescription. This marks a deviation from the traditional Wald formalism but remains consistent with the first law provided the temperature sector is residue-shifted.

Further investigation is needed to clarify the interpretational status of the residue parameter: whether it encodes intrinsic microphysical degrees of freedom, quantum anomalies, or analytic artifacts of complex analysis in gravity. Comparing this phenomenological entropy to a direct Noether charge computation in quadratic $f(Q)$8 gravity could shed light on whether the correction is a genuine modification of gravitational entropy or merely a model-specific prescription tied to the RVB analytic framework.

For larger values of the residue, pathologies or non-extensive behavior may emerge, suggesting the correction is physically meaningful primarily as a perturbation. The impact of the residue on thermodynamic stability, evaporation rates, and end states (such as black hole remnants) can be significant and may serve as distinguishing features in the landscape of modified gravity theories.

Conclusion

This study generalizes the residue-improved contour method of the RVB approach for Hawking temperature to black hole entropy in $f(Q)$9 gravity. For static spherically symmetric horizons, the entropy is given by a universal horizon integral dependent on the deformation function $f(Q) = Q + \alpha Q^2$0 and the RVB residue parameter. In the quadratic model, a closed first-order formula reveals a systematic reduction of entropy relative to the area law, proportional to the residue and the nonmetricity coupling. The construction offers a thermodynamically consistent route for encoding analytic/topological corrections into black hole entropy, motivating further comparisons with Noetherian derivations in modified geometry contexts.

Reference: "Black Hole Entropy in f(Q) Gravity from the RVB Residue Method" (2604.05240)