Dirac-Field Black Hole Entropy in \(f(Q)\) Gravity from the RVB Residue Method

Abstract: We compute the entropy of a Dirac quantum field near a static, spherically symmetric black hole in (f(Q)) gravity by combining the residue-based Robson--Villari--Biancalana method with the thin-film state-counting approach. The RVB prescription introduces a residue correction to the Hawking temperature, while the Dirac field entropy is obtained from the near-horizon WKB mode density and fermionic free energy. For an (f(Q))-deformed metric, we derive the Hamilton--Jacobi equation, radial momentum, mode number, and entropy at the residue-corrected temperature. The result shows that the Dirac-field entropy remains proportional to the horizon area after regularization, but its coefficient is modified by a cubic RVB temperature factor. An explicit expression is obtained for the quadratic model (f(Q)=Q+αQ^{2}).

• The paper computes Dirac-field entropy in static black holes within f(Q) gravity using the RVB residue and thin-film methods.
• It derives explicit formulas showing cubic temperature corrections and α-dependent modifications in the quadratic f(Q)=Q+αQ² model.
• The study confirms that the entropy remains area-proportional with renormalized prefactors, offering insights into quantum corrections in modified gravity.

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

Overview

This paper presents the computation of the entropy associated with a Dirac quantum field in the vicinity of a static, spherically symmetric black hole within the framework of $f(Q)$ gravity. The calculation leverages two core ingredients: the Robson--Villari--Biancalana (RVB) residue method, which modifies the horizon temperature via complex-analytic techniques, and the thin-film (brick-wall) state-counting method applied to the Dirac field near the horizon.

By carefully analyzing the mode density and integrating the fermionic free energy at the RVB-corrected temperature, explicit formulae for the Dirac-field black hole entropy in $f(Q)$ gravity are derived. Special attention is given to the quadratic $f(Q)$ model, $f(Q)=Q+\alpha Q^2$, for which closed-form results are presented.

$f(Q)$ Gravity Black Hole and RVB Temperature

$f(Q)$ gravity extends symmetric teleparallel gravity by making the gravitational action a general function of the nonmetricity scalar $Q$. Static, spherically symmetric solutions are parameterized with metric $ds^2 = -g(r) dt^2 + dr^2/g(r) + r^2 d\Omega^2$, where $g(r)$ incorporates $f(Q)$0-induced deformations.

The RVB method introduces a residue-based correction to the Hawking temperature. Specifically, the (inverse) temperature is determined as

$f(Q)$1

and the RVB temperature is

$f(Q)$2

where $f(Q)$3 arises from the residue of an appropriately chosen analytic function. In $f(Q)$4 gravity, both $f(Q)$5 and $f(Q)$6 acquire explicit corrections tied to the nonmetricity sector.

Dirac-Field Mode Counting and Entropy Calculation

The entropy of a quantum field in a black hole background can be computed via the thin-film (brick-wall) method by counting the modes of the field near the horizon, where the mode density becomes large and a proper ultraviolet cutoff must be introduced. For the Dirac field, the leading WKB radial equation reduces to

$f(Q)$7

where the near-horizon pole structure is identical to the scalar case except for spin and statistical prefactors.

By integrating out the angular momentum and energy, the total number of Dirac modes below energy $f(Q)$8 is given (leading order, neglecting field mass near the horizon) by

$f(Q)$9

with $f(Q)$0. The fermionic free energy is then

$f(Q)$1

The entropy follows directly via thermodynamic relations. The key result is:

$f(Q)$2

The entropy remains proportional to the horizon area after introducing proper distance cutoffs, but the numerical prefactor is modified by a cubic power of the RVB-corrected temperature.

Explicit Expressions and Quadratic $f(Q)$3 Model

For the quadratic model $f(Q)$4, analytic expressions for all relevant metric functions and their derivatives are given. The temperature and entropy formulae acquire explicit $f(Q)$5-dependent terms. The coordinate cutoff is mapped to a proper distance cutoff ($f(Q)$6) from the horizon, leading to an area-proportional entropy with a cutoff dependence:

$f(Q)$7

where $f(Q)$8 is the horizon area. For small residue corrections, a first-order expansion in the residue parameter is given, demonstrating how $f(Q)$9-induced $f(Q)$0 modifications propagate into the entropy.

Theoretical and Practical Implications

The main theoretical implication is that Dirac-field entropy in $f(Q)$1 gravity, calculated via the thin-film method, remains area-proportional after proper regularization. However, the prefactor is renormalized by the RVB residue, manifesting as a cubic temperature correction unique to the field statistics and the analytic structure of the gravitational background. This approach clarifies the interplay between field-theoretic entropy calculations and the geometric entropy prescriptions (first-law or Wald-type), particularly in nonmetricity-based modified gravity.

In practical terms, these results provide a template for evaluating quantum corrections to black hole entropy in generalized gravity theories. The explicit dependence on the cutoff remains, confirming the standard interpretation of thin-film entropy as entanglement or thermal-atmosphere entropy, which may be absorbed via renormalization of the gravitational coupling.

Prospective Developments and Future Directions

This framework can be extended to other field content (including vector and higher-spin fields), rotating or charged black holes in $f(Q)$2 gravity, and more general analytic gravity models. Investigating the universality, or lack thereof, of cubic temperature corrections and their compatibility with semiclassical and loop-level quantum gravity predictions will be important for a deeper understanding of gravitational thermodynamics in modified gravity.

Bridging the gap between the residue-based entropy branch and the Noether charge or Wald entropy prescriptions may yield further insights into the microscopic interpretation of black hole entropy beyond general relativity.

Conclusion

The paper rigorously establishes the calculation of Dirac-field black hole entropy in $f(Q)$3 gravity using the RVB residue-corrected temperature and thin-film method. The entropy retains its area dependence but acquires a field-statistics-specific cubic correction determined by the analytic structure of the horizon. These results elucidate the detailed impact of nonmetricity-based modifications of gravity on black hole thermodynamics and provide a solid foundation for further research in quantum gravitational entropy in extended theories.