Thermodynamic-Geometric Phase Transition and Gravitational-Wave Quasinormal Modes of Schwarzschild Black Holes in $f(Q)$ Gravity: An RVB-Residue Approach

Abstract: We construct a residue-based framework connecting the thermodynamic geometry of a Schwarzschild-type black hole in $f(Q)$ gravity with its gravitational-wave quasinormal-mode spectrum. The analysis is based on the symmetric teleparallel formulation of gravity, in which the gravitational field is encoded by the nonmetricity scalar $Q$ rather than by curvature or torsion. For the Schwarzschild branch, the Robson--Villari--Biancalana (RVB) method gives the Hawking temperature through the simple-pole residue of the inverse blackening function. We show explicitly that the same residue also controls the logarithmic monodromy of the tortoise coordinate near the event horizon, and therefore enters the ingoing quasinormal-mode boundary condition. In the strict general-relativistic Schwarzschild limit the heat capacity is negative and finite, the one-dimensional Ruppeiner geometry contains no intrinsic curvature singularity, and no genuine thermodynamic phase transition occurs. In the extended $f(Q)$ state space, however, the modified horizon function and the effective Wald entropy generate a non-trivial thermodynamic Hessian. Its degeneracy condition coincides with singular behavior of the thermodynamic curvature and is reflected in the quasinormal-mode spectrum through shifts of the photon-sphere frequency, Lyapunov exponent, damping time, and near-horizon monodromy. This gives a precise statement of the internal relation between thermodynamic-geometric phase structure and gravitational-wave ringdown: both are different projections of the same analytic structure of the corrected black-hole metric.

• The paper introduces a novel RVB-residue approach that expresses the Hawking temperature via the near-horizon residue of the blackening function in f(Q) gravity.
• It demonstrates that thermodynamic phase transitions and quasinormal mode spectra share the same analytic origin tied to nonmetricity corrections.
• The study employs rigorous expansions and WKB analysis to quantify shifts in QNM frequencies, linking ringdown signals to modified gravity effects.

Thermodynamic-Geometric Phase Structure and Gravitational-Wave Quasinormal Modes in $f(Q)$ Gravity: An Expert Review

Overview and Scientific Context

This work presents a rigorous analytic framework relating the thermodynamic geometry of Schwarzschild-type black holes in the context of $f(Q)$ gravity to the gravitational-wave quasinormal mode (QNM) spectrum, using an RVB-residue approach. The analysis builds on the symmetric teleparallel formulation, where gravity is encoded in the nonmetricity scalar $Q$ rather than in curvature or torsion. The central thesis is that both thermodynamic-geometric features (including possible phase transitions) and the spectrum of QNMs are governed by the same analytic residue structure associated with the near-horizon expansion of the blackening function.

Distinctively, the study demonstrates that the main thermodynamic quantity, the Hawking temperature, acquires a residue representation (the Robson–Villari–Biancalana, RVB, formula), and that this same residue underpins both the tortoise coordinate monodromy used in QNM boundary conditions and the singular structure of the thermodynamic state space. The work rigorously elucidates the internal connection between equilibrium (thermodynamic geometry) and nonequilibrium (QNMs and ringdown) phenomena for black holes in modified gravity scenarios driven by nonmetricity corrections.

Technical Contributions

Formalism in $f(Q)$ Gravity

Static, spherically symmetric solutions in $f(Q)$ gravity are analyzed via a metric of the familiar form with a generalized blackening function $F(r;\lambda)$, where $\lambda$ parameterizes deviations from GR due to nonmetricity. The horizon position, surface gravity, and corrections to the entropy are computed systematically as a power series in $\lambda$.

The thermodynamic quantities are derived using Wald’s entropy formula, adapted to $f(Q)$ gravity, resulting in

$S_h = \frac{A_h}{4G}f_Q(Q_h)$

with explicit first-order corrections in the parameter $f(Q)$0.

RVB-Residue Method and Temperature

The RVB prescription encodes the Hawking temperature as

$f(Q)$1

where $f(Q)$2 is the simple pole residue of the inverse blackening function at the (corrected) horizon. The paper provides a detailed expansion of both $f(Q)$3 and its derivatives to obtain explicit corrections induced by $f(Q)$4 modifications, with all steps meticulously justified.

Thermodynamic Geometry and Phase Structure

Thermodynamic geometry is formulated using the Ruppeiner metric, taking the entropy Hessian in the extended state space $f(Q)$5. The determinant of the Hessian governs the scalar curvature, such that its vanishing signals a geometric singularity and the divergence of the heat capacity $f(Q)$6. Thus, thermodynamic phase transitions are only accessible in an extended state space, not in classical Schwarzschild GR.

The paper derives the criticality condition explicitly:

$f(Q)$7

and demonstrates that this is nontrivial only with nonzero $f(Q)$8. The response function divergence and curvature singularity are intrinsically linked.

Gravitational-Wave Quasinormal Modes

Black hole perturbations in this context yield a master equation for the perturbation field, with the effective Regge–Wheeler potential corrected by the nonmetricity-induced terms. The boundary conditions for QNMs near the horizon crucially depend on the same residue $f(Q)$9, as analytic continuation around the pole leads to the QNM monodromy directly proportional to $Q$0.

The WKB analysis yields shifted real (oscillation) and imaginary (damping) QNM frequencies, with corrections traced back to variations in $Q$1 and its derivatives at the photon sphere:

$Q$2

where both the orbital frequency $Q$3 and the Lyapunov exponent $Q$4 experience systematic shifts due to $Q$5.

Unified Analytic Structure

A key insight is the common analytic origin of all critical quantities: the entropy, temperature, response functions, thermodynamic curvature, photon sphere, and QNM spectrum are projections of the same corrected blackening function $Q$6 and its pole structure. This analytic chain determines both equilibrium (thermodynamic) and relaxation (spectral) phenomena.

Numerical and Conceptual Claims

• No true thermodynamic phase transition occurs for the Schwarzschild black hole in GR; the introduction of the $Q$7 coupling induces a thermodynamic Hessian degeneracy, which is reflected both as a heat capacity divergence and a curvature singularity in the Ruppeiner geometry.
• The same residue controls the Hawking temperature and the QNM monodromy, showing an exact analytic relation between the phase structure and the QNM spectral properties.
• Strong dependence of the QNM frequencies and damping times on the $Q$8 coupling is predicted near the geometric singularity in the thermodynamic state space.

Theoretical and Practical Implications

This work provides a precise theoretical identification of the analytic structures connecting black hole thermodynamics and perturbation spectra in generalized teleparallel scenarios. Practically, it suggests that ringdown observations—specifically, anomalous shifts or anomalous parameter dependences in QNM frequencies—could provide empirical diagnostics of modified gravity and nonmetricity effects. This connection creates avenues for gravitational-wave data to probe the nature of gravity beyond GR, especially in the context of phase transitions that have no analog in the classical Schwarzschild regime.

On the theoretical side, the residue-based approach offers a coordinate-independent, gauge-invariant methodology for extracting physically meaningful quantities directly from the near-horizon metric structure, amenable to analytic and numeric computation in a wide range of modified gravity frameworks.

Future Directions

Potential directions include generalization to rotating black holes in $Q$9 gravity, extension to higher-derivative models or broader classes of nonmetricity-corrected solutions, and the systematic computation of the explicit $f(Q)$0 correction functions for phenomenologically relevant actions. Furthermore, the application of this framework to dynamical or non-asymptotically flat spacetimes may reveal additional analytic structures or observational signatures.

Conclusion

This analysis establishes a precise analytic connection between thermodynamic phase structure and gravitational-wave relaxation for Schwarzschild-type black holes in $f(Q)$1 gravity. The residue formulation reveals that both Bekenstein–Hawking thermodynamic quantities and QNM frequencies are encoded in the same corrected near-horizon pole structure. This reinforces the interpretation of black hole phase transitions and gravitational-wave ringdown as distinct manifestations of a common geometric and analytic substrate in modified gravity, with direct implications for both theory and observation.