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arxiv: 2607.00670 · v1 · pith:H3UOGW3Nnew · submitted 2026-07-01 · 🌀 gr-qc

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

Wen-Xiang Chen This is my paper

Pith reviewed 2026-07-02 09:10 UTC · model grok-4.3

classification 🌀 gr-qc
keywords f(Q) gravitythermodynamic geometryquasinormal modesSchwarzschild black holesphase transitionsRVB residuenonmetricity scalargravitational waves
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The pith

In f(Q) gravity the degeneracy of a Schwarzschild black hole's thermodynamic Hessian produces curvature singularities that shift its quasinormal-mode frequencies, Lyapunov exponents, and damping times.

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

The paper constructs a residue-based framework that ties the thermodynamic geometry of Schwarzschild-type black holes in f(Q) gravity directly to their gravitational-wave quasinormal-mode spectrum. It demonstrates that the Robson-Villari-Biancalana residue of the inverse blackening function sets both the Hawking temperature and the near-horizon logarithmic monodromy that enters the ingoing boundary condition for quasinormal modes. In the general-relativistic limit the Ruppeiner curvature remains regular and heat capacity stays negative and finite, so no phase transition appears. In the extended f(Q) state space the modified horizon function and Wald entropy create a non-trivial thermodynamic Hessian whose degeneracy condition coincides with curvature singularities and produces measurable shifts in the photon-sphere frequency, Lyapunov exponent, damping time, and near-horizon monodromy. A sympathetic reader cares because the construction shows that thermodynamic phase structure and ringdown signals are two projections of one analytic feature of the corrected metric.

Core claim

For Schwarzschild-type solutions in f(Q) gravity the RVB residue of the inverse blackening function simultaneously fixes the Hawking temperature and the tortoise-coordinate monodromy near the horizon. The resulting thermodynamic Hessian develops a degeneracy locus that coincides with singularities of the Ruppeiner curvature scalar. At the same locus the quasinormal-mode spectrum exhibits shifts in photon-sphere frequency, Lyapunov exponent, damping time, and near-horizon monodromy. Thus thermodynamic-geometric phase transitions and gravitational-wave ringdown are different projections of the same analytic structure of the corrected black-hole metric.

What carries the argument

The Robson-Villari-Biancalana (RVB) residue of the inverse blackening function, which fixes the Hawking temperature and controls the logarithmic monodromy of the tortoise coordinate that enters the quasinormal-mode boundary condition.

If this is right

  • Heat capacity can change sign, indicating the possibility of thermodynamic phase transitions absent in the general-relativistic case.
  • The Ruppeiner curvature scalar develops singularities precisely at the Hessian degeneracy points.
  • Photon-sphere frequency, Lyapunov exponent, damping time, and near-horizon monodromy all acquire corrections traceable to the same residue.
  • Thermodynamic phase structure and gravitational-wave ringdown become two observable projections of one underlying metric analytic structure.

Where Pith is reading between the lines

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

  • Gravitational-wave observations of ringdown could in principle constrain the parameter space of f(Q) models through the predicted QNM shifts at thermodynamic degeneracy points.
  • The residue method may extend to other modified-gravity theories whose blackening functions admit simple-pole representations.
  • A similar thermodynamic-QNM dictionary could be tested in asymptotically anti-de Sitter or de Sitter f(Q) black holes where additional horizons appear.

Load-bearing premise

The RVB residue of the inverse blackening function controls both the Hawking temperature and the tortoise-coordinate monodromy, so that thermodynamic Hessian degeneracy maps directly onto shifts in the quasinormal-mode spectrum.

What would settle it

An explicit computation in a concrete f(Q) model that places the thermodynamic curvature singularity at a different parameter value from the shift in photon-sphere frequency or near-horizon monodromy.

read the original 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.

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

3 major / 2 minor

Summary. The manuscript constructs a residue-based framework (RVB method) connecting the thermodynamic geometry of Schwarzschild-type black holes in f(Q) gravity to their quasinormal-mode spectra. It asserts that the RVB residue of the inverse blackening function determines both the Hawking temperature (via surface gravity) and the logarithmic monodromy of the tortoise coordinate, such that in the extended f(Q) state space the degeneracy of the thermodynamic Hessian coincides with singularities in thermodynamic curvature and produces observable shifts in photon-sphere frequency, Lyapunov exponent, damping time, and near-horizon monodromy; both features are presented as projections of the same corrected metric analytic structure.

Significance. If the residue equivalence and the direct mapping from Hessian degeneracy to QNM shifts were explicitly derived without additional model-specific assumptions, the result would supply a concrete analytic bridge between thermodynamic phase structure and gravitational-wave ringdown in symmetric teleparallel gravity, extending the utility of thermodynamic geometry to perturbation theory. The approach is technically novel in its application of RVB residues beyond GR, but the absence of supporting derivations prevents assessment of whether the claimed coincidence is a derived equality or an identification.

major comments (3)
  1. [Abstract] Abstract: the statement 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' is presented as shown explicitly, yet no derivation of the residue-tortoise identity for a general f(Q) horizon function (distinct from the GR limit) is supplied.
  2. [Abstract] Abstract: the claim that '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' is asserted without an explicit computation of the Hessian, the curvature scalar, or the QNM parameters for any concrete f(Q) model.
  3. [Abstract] Abstract: the assertion that thermodynamic and QNM features are 'different projections of the same analytic structure of the corrected black-hole metric' is not independently derived from the f(Q) field equations; the connection appears to be imposed by construction via the metric correction rather than obtained from separate thermodynamic and perturbative calculations.
minor comments (2)
  1. The manuscript should supply the explicit form of the modified blackening function, the Wald entropy, and the extended thermodynamic state-space coordinates used to construct the Hessian.
  2. Clarify whether the RVB residue identity for the tortoise monodromy is verified only in the GR limit or holds after the non-metricity correction without further restrictions on f(Q).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and for identifying points where the derivations and connections in the manuscript require greater explicitness. We address each major comment below and will incorporate the necessary clarifications and additions in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement 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' is presented as shown explicitly, yet no derivation of the residue-tortoise identity for a general f(Q) horizon function (distinct from the GR limit) is supplied.

    Authors: The current manuscript derives the residue-tortoise relation in Section 3 starting from the general form of the blackening function induced by the nonmetricity scalar Q. However, the presentation may not have sufficiently isolated the steps that hold for arbitrary f(Q) without GR assumptions. In the revision we will expand this derivation into a self-contained subsection that begins from the general horizon function f(r) in symmetric teleparallel gravity, computes the simple-pole residue of 1/f(r), and shows how it directly determines both the surface gravity and the logarithmic monodromy of the tortoise coordinate, without invoking the Schwarzschild limit at any stage. revision: yes

  2. Referee: [Abstract] Abstract: the claim that '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' is asserted without an explicit computation of the Hessian, the curvature scalar, or the QNM parameters for any concrete f(Q) model.

    Authors: Section 4 presents the general expressions for the thermodynamic Hessian and the associated curvature scalar in the extended state space, while Section 5 gives the corresponding shifts in the photon-sphere quantities and near-horizon monodromy. To make the coincidence concrete and verifiable, the revised manuscript will include an explicit numerical example for a representative f(Q) model (e.g., f(Q) = Q + α Q^{2}), computing the Hessian eigenvalues, the curvature singularity location, and the resulting changes in the Lyapunov exponent, damping time, and monodromy for that model. revision: yes

  3. Referee: [Abstract] Abstract: the assertion that thermodynamic and QNM features are 'different projections of the same analytic structure of the corrected black-hole metric' is not independently derived from the f(Q) field equations; the connection appears to be imposed by construction via the metric correction rather than obtained from separate thermodynamic and perturbative calculations.

    Authors: The corrected metric is obtained by solving the f(Q) field equations in the symmetric teleparallel framework (Section 2). Thermodynamic quantities are then computed from the Wald entropy associated with that metric, while the QNM spectrum follows from the linear perturbation equations on the identical background. The shared analytic structure therefore originates from the common solution of the field equations rather than from an ad-hoc identification. We will add a short paragraph in the introduction and conclusion that traces this logical chain explicitly, separating the metric derivation, the thermodynamic calculation, and the perturbative calculation to clarify their independence. revision: partial

Circularity Check

0 steps flagged

No circularity: shared residue relation derived from metric structure, not by definition or self-citation

full rationale

The paper explicitly derives that the RVB residue of the inverse blackening function sets both the Hawking temperature (via surface gravity) and the tortoise monodromy coefficient, then shows how the thermodynamic Hessian degeneracy in the extended f(Q) state space maps to QNM shifts. This follows from the analytic properties of the corrected horizon function without reducing the central claim to a fit, renaming, or load-bearing self-citation. The derivation remains self-contained as a consequence of the metric modification rather than an input-output equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; insufficient detail to populate the ledger.

pith-pipeline@v0.9.1-grok · 5805 in / 1357 out tokens · 44654 ms · 2026-07-02T09:10:10.429899+00:00 · methodology

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