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arxiv: 2607.00559 · v1 · pith:PD3JAJCXnew · submitted 2026-07-01 · 🌀 gr-qc · hep-th

Relaxation without ringdown for a compact object in modified gravity

Gianmassimo Tasinato This is my paper

Pith reviewed 2026-07-02 09:19 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords vector-tensor gravitycompact objectsquasinormal modesrelaxation poleschiral symmetrySchwarzschild matchingodd-parity perturbationsanisotropic stress
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The pith

A vector-supported compact object in modified gravity relaxes purely dissipatively with no oscillatory ringdown modes.

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

The paper constructs a regular, vector-supported compact object in a vector-tensor theory that matches smoothly onto an exterior Schwarzschild geometry without a surface layer. Its anisotropic stress allows it to violate the Buchdahl bound and reach black-hole compactness. Under a minimal passive boundary condition, the exactly solvable odd-parity sector of linear perturbations exhibits only dissipative relaxation poles instead of the oscillatory quasinormal modes familiar from black holes and other compact-object models. This behavior traces to a hidden chiral symmetry that converts the fluctuation equations into one-way transport. A reader would care because the construction supplies an analytic laboratory for how interior strong-field physics can change the dynamical response of an object that looks like a black hole from outside.

Core claim

The paper claims that this compact object possesses a hidden chiral symmetry which turns its odd-parity perturbation problem into one-way transport rather than ordinary wave propagation. The regular interior together with the matching conditions break the symmetry and quantize the fluctuation spectrum, producing a retarded Green function whose poles are purely dissipative. In the black-hole limit the relaxation times diverge, the poles collapse toward zero frequency, and finite-frequency exterior perturbations decouple from the interior, so black-hole behavior is recovered through the disappearance of relaxation modes rather than the emergence of ringdown.

What carries the argument

The hidden chiral symmetry that reduces the perturbation equations to one-way transport.

If this is right

  • The exterior region alone possesses no conventional quasinormal-mode spectrum.
  • The retarded Green function and susceptibility can be computed analytically from the interior solution.
  • An effective membrane response follows by integrating out the object's interior.
  • In the black-hole limit finite-frequency exterior perturbations decouple from the interior.

Where Pith is reading between the lines

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

  • Observational searches focused on ringdown signals could systematically miss objects of this class.
  • The mechanism of approaching black-hole behavior by erasure of modes rather than addition of modes may appear in other theories with controlled interiors.
  • Exact response functions derived this way supply benchmarks for numerical codes that simulate dynamical compact objects.

Load-bearing premise

A regular vector-supported compact solution of the vector-tensor theory exists that can be matched without a surface layer to an exterior Schwarzschild geometry while violating the Buchdahl bound via anisotropic stress.

What would settle it

An explicit evaluation of the retarded Green function whose poles include any component with nonzero real frequency would falsify the claim of purely dissipative relaxation.

Figures

Figures reproduced from arXiv: 2607.00559 by Gianmassimo Tasinato.

Figure 1
Figure 1. Figure 1: Left: Metric function A(r) across the regular interior (r < R, shaded) and the Schwarzschild exterior (r > R), for a moderate-compactness configuration and a near-black-hole configuration approaching C → 1/2 via Eq. (2.20). In both cases A(r) is finite and flat at the origin, A(0) = σ 2 with A ′ (0) = 0 (circles; see inset), and joins continuously, with matched slope, onto the exterior branch at r = R (dia… view at source ↗
Figure 2
Figure 2. Figure 2: Characteristic speed dr/dt = −∆1/Σ1 of the odd-parity transport equation, shown across the interior (r < R, shaded) and exterior (r > R) for a representative configuration (σ = 0.20, γ = 8, QI = 2). The flow is everywhere negative, showing that characteristics propagate inward in both regions: the odd sector is a unidirectional advection problem rather than a two-way wave equation, with no characteristic a… view at source ↗
Figure 3
Figure 3. Figure 3: Relaxation poles ω⋆ in the complex-frequency plane for spin-1 (circles) and spin-2 homogeneous (squares) perturbations, at fixed σ = 0.25, QI = 2, labelled by multipole number ℓ. All poles lie exactly on the negative imaginary axis, ω⋆ = −iΓ (ℓ) rel : the spectrum is purely damped, with no real part and hence no oscillatory ringing. The relaxation rate grows linearly with ℓ. One simple and minimal way of e… view at source ↗
Figure 4
Figure 4. Figure 4: Trajectory of the relaxation poles ω (ℓ) ⋆ = −iΓ (ℓ) rel approaching the black-hole limit, along the family σ → 0, γ = 2 + 5/σ of Eq. (2.20). Left: Γ (ℓ) relR versus compactness C for ℓ = 1, . . . , 5, on a logarithmic scale: all multipoles collapse toward ω = 0 together as C → 1/2 (dotted line), spanning several decades. Right: the same poles in the complex-ω plane (logarithmic |Im ω| scale) at several re… view at source ↗
Figure 5
Figure 5. Figure 5: Real (solid) and imaginary (dashed) parts of the pole contribution to the retarded susceptibility, Eq. (5.35), plotted in the dimensionless normalization Γrelχ R f,pole/Af as functions of ω/Γrel. The response has the standard Debye form of a single overdamped degree of freedom: a Lorentzian real part of half-width Γrel centered at ω = 0, and an odd imaginary part controlling dissipation, crossing 1/2 at ω … view at source ↗
Figure 6
Figure 6. Figure 6: Left: Impedance matching at the surface r = R along the purely damped axis ω = −iΓ. See Section 6.2. The solid curve is the interior impedance Zint(ω = −iΓ) obtained from the Dirichlet-to-Neumann map of the regular interior; the dashed line is the constant exterior impedance Zext = λ+. The two curves cross at a single point (dot), the relaxation frequency ω⋆ = −iΓrel, where the interior solution loses its … view at source ↗
read the original abstract

Compact objects with black-hole-like exteriors may hide new strong-field physics in their interiors, making their dynamical response a sensitive probe of gravity beyond General Relativity. We present an analytically tractable, gravitationally bound compact object with a genuinely new dynamical signature: under a minimal passive boundary prescription, its exactly controlled odd-parity sector exhibits purely dissipative relaxation poles, rather than the oscillatory modes usually associated with black holes and exotic compact alternatives. The object we study is a regular, vector-supported compact solution of a vector--tensor theory, matched without any surface layer to an exterior Schwarzschild geometry. Owing to its anisotropic stress, it can violate the Buchdahl bound and be continuously connected to the black-hole compactness limit. Its unusual response follows from a hidden chiral symmetry, which turns the perturbation problem into one-way transport rather than ordinary wave propagation. The exterior region alone has no conventional quasinormal-mode spectrum; instead, the regular interior and the matching conditions break the symmetry and quantize the fluctuation spectrum. We analytically compute the retarded Green function and susceptibility, and derive an effective membrane response by integrating out the object's interior. In the black-hole limit, the relaxation times diverge, the poles collapse toward zero frequency, and finite-frequency exterior perturbations decouple from the interior. Black-hole behaviour is therefore approached through the disappearance of relaxation modes, not through the emergence of ringdown.

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 presents a vector-supported compact object in a vector-tensor theory that matches smoothly to an exterior Schwarzschild spacetime without surface layers. Due to anisotropic stress, it violates the Buchdahl bound and approaches black-hole compactness. The odd-parity perturbations, under a minimal passive boundary condition, exhibit purely dissipative relaxation poles due to a hidden chiral symmetry that reduces the dynamics to one-way transport. The exterior has no standard quasinormal modes; the spectrum is quantized by the interior and matching. The retarded Green function, susceptibility, and an effective membrane response are computed analytically. In the black-hole limit, relaxation times diverge and poles approach zero frequency, so black-hole behavior emerges from the disappearance of relaxation modes rather than the appearance of ringdown.

Significance. If the background construction and perturbation analysis hold, this provides an analytically tractable example of an exotic compact object with a qualitatively distinct dynamical response (purely dissipative relaxation without ringdown). The exact control over the retarded Green function, susceptibility, and membrane paradigm derivation are strengths that could inform gravitational-wave phenomenology and tests of strong-field gravity.

major comments (3)
  1. [Background construction (abstract, paragraph 2 and interior solution section)] The existence of a regular vector-supported interior solution matched to Schwarzschild with continuous metric and extrinsic curvature (no surface stress-energy) while violating the Buchdahl bound via anisotropic stress is load-bearing for the entire dynamical analysis. Explicit verification of regularity at the center, gravitational binding, and the junction conditions (continuity of extrinsic curvature) must be shown with the relevant equations.
  2. [Odd-parity perturbations and hidden chiral symmetry] The reduction of the odd-parity sector to one-way transport via the hidden chiral symmetry, and the subsequent quantization of the spectrum by the interior and matching conditions, is central to the claim of purely dissipative poles. The perturbation equations, symmetry breaking, and boundary conditions leading to this behavior require explicit derivation.
  3. [Retarded Green function, susceptibility, and membrane response] The analytic computation of the retarded Green function, susceptibility, and effective membrane response (obtained by integrating out the interior) underpins the black-hole limit claims. Key steps or explicit expressions for these quantities should be provided to allow verification of the pole structure and the divergence of relaxation times.
minor comments (2)
  1. Notation for the theory ('vector--tensor') should be standardized throughout the manuscript.
  2. [Black-hole limit] Clarify in the black-hole limit discussion whether the divergence of relaxation times and collapse of poles to zero frequency is shown purely analytically.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below. The requested explicit verifications and derivations are feasible to add without altering the core results, and we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Background construction (abstract, paragraph 2 and interior solution section)] The existence of a regular vector-supported interior solution matched to Schwarzschild with continuous metric and extrinsic curvature (no surface stress-energy) while violating the Buchdahl bound via anisotropic stress is load-bearing for the entire dynamical analysis. Explicit verification of regularity at the center, gravitational binding, and the junction conditions (continuity of extrinsic curvature) must be shown with the relevant equations.

    Authors: We agree these explicit checks are necessary. The interior solution is already constructed in the manuscript, but we will add a dedicated subsection (or appendix) in the revision that: (i) writes the explicit metric and vector-field ansatz, (ii) expands all functions in powers of r near the center to demonstrate regularity, (iii) computes the gravitational binding energy to confirm the object is bound, and (iv) evaluates the Israel junction conditions by direct computation of the extrinsic curvature on both sides, verifying continuity and the absence of surface stress-energy. This will also make the anisotropic-stress violation of the Buchdahl bound fully explicit. revision: yes

  2. Referee: [Odd-parity perturbations and hidden chiral symmetry] The reduction of the odd-parity sector to one-way transport via the hidden chiral symmetry, and the subsequent quantization of the spectrum by the interior and matching conditions, is central to the claim of purely dissipative poles. The perturbation equations, symmetry breaking, and boundary conditions leading to this behavior require explicit derivation.

    Authors: We will expand the perturbation analysis section to include the full linearized odd-parity equations obtained from the vector-tensor action, explicitly exhibit the hidden chiral symmetry and the field redefinition that reduces the system to one-way transport, and derive the quantization condition arising from regularity at the center together with the matching conditions at the surface. The resulting pole structure will then be shown to follow directly from these steps. revision: yes

  3. Referee: [Retarded Green function, susceptibility, and membrane response] The analytic computation of the retarded Green function, susceptibility, and effective membrane response (obtained by integrating out the interior) underpins the black-hole limit claims. Key steps or explicit expressions for these quantities should be provided to allow verification of the pole structure and the divergence of relaxation times.

    Authors: In the revised manuscript we will insert the intermediate steps of the Green-function construction, the closed-form expression for the susceptibility, and the explicit integration-out procedure that yields the effective membrane response. These additions will make the analytic pole locations and the divergence of relaxation times in the black-hole limit directly verifiable from the text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines a specific vector-tensor compact object with anisotropic stress, matched smoothly to an exterior Schwarzschild geometry, and derives its odd-parity response from the model's hidden chiral symmetry that converts the perturbation equations into one-way transport. The retarded Green function, susceptibility, and effective membrane response are obtained by direct integration of the interior solution and matching conditions. No step reduces a claimed prediction to a fitted parameter, self-citation chain, or definitional tautology; the dissipative poles are a direct algebraic consequence of the symmetry and boundary prescription within the constructed background. The result is therefore independent of external benchmarks and does not rely on load-bearing self-referential inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction rests on the existence of regular solutions in an unspecified vector-tensor theory and on a minimal passive boundary condition whose detailed form is not given in the abstract.

axioms (2)
  • domain assumption A vector-tensor theory admits regular, anisotropic-stress-supported compact solutions that match smoothly to Schwarzschild exterior.
    Invoked to justify the existence of the object studied.
  • ad hoc to paper The minimal passive boundary prescription is physically appropriate for the interior-exterior matching.
    Stated as the prescription under which the relaxation poles appear.
invented entities (1)
  • vector-supported compact object with hidden chiral symmetry no independent evidence
    purpose: To realize a regular interior whose perturbations undergo one-way transport rather than wave propagation.
    New object introduced to produce the reported dynamical signature.

pith-pipeline@v0.9.1-grok · 5768 in / 1329 out tokens · 26711 ms · 2026-07-02T09:19:18.434232+00:00 · methodology

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