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arxiv: 2606.11297 · v1 · pith:33LLTJPVnew · submitted 2026-06-09 · ✦ hep-th · gr-qc

Bouncing Geodesics, Singularities, and the Cavity Thermal Product Formula in Asymptotically Flat and de Sitter Black Holes

Sa\v{s}o Grozdanov , Vita Movrin , Samuel Valach This is my paper

Pith reviewed 2026-06-27 12:09 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords bouncing geodesicssingularitiesthermal product formulaquasinormal modesreflecting cavitySchwarzschild black holede Sitter spacetimeretarded Green's function
0
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The pith

Black holes inside reflecting cavities obey a thermal product formula relating bouncing singularities to the quasinormal mode spectrum.

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

The paper establishes that bouncing geodesics near the singularity in asymptotically flat Schwarzschild and Schwarzschild-de Sitter black holes produce specific singularities in the bulk retarded Green's function. When the black hole is enclosed in a reflecting cavity, these singularity locations are connected to the frequencies of the cavity quasinormal modes through a derived thermal product formula. This is the direct analogue of a known relation in anti-de Sitter space. A reader would care because the formula lets one read information about the black hole interior, including near the curvature singularity, from the quasinormal mode spectrum measured at the cavity wall, even when the cosmological constant is zero or positive. The relation is confirmed by explicit calculations for scalar, electromagnetic, and gravitational perturbations.

Core claim

For black holes enclosed in a reflecting cavity, the locations of the bouncing singularities are connected to the spectrum of cavity quasinormal modes by a cavity version of the thermal product formula. This holds for asymptotically flat and de Sitter cases and allows extraction of information about the black hole interior from the asymptotic QNM spectrum measured at a reflecting hypersurface.

What carries the argument

The cavity thermal product formula, derived by combining the local Hadamard form with the global propagation of singularities theorem to locate bouncing singularities and then relating their critical times to the cavity QNM spectrum.

If this is right

  • The quasinormal mode spectrum measured at the cavity directly determines the times at which the retarded correlator becomes singular due to bouncing geodesics.
  • Interior information, including near the black hole singularity, becomes accessible from boundary data even when the cosmological constant is zero or positive.
  • The same cavity thermal product formula applies to scalar, electromagnetic, and gravitational perturbations.
  • Explicit QNM computations in flat and de Sitter cases serve as verification of the derived relation.

Where Pith is reading between the lines

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

  • The cavity construction may extend the thermal product formula to other asymptotically flat geometries that admit a reflecting boundary.
  • It provides a concrete route to test whether singularity information can be recovered from boundary spectra without relying on anti-de Sitter asymptotics.
  • Numerical evolution of waves inside a cavity could directly measure the predicted singularity times and compare them to the QNM-derived formula.

Load-bearing premise

The local Hadamard form together with the global propagation of singularities theorem still accurately locates the bouncing singularities after the spacetime is truncated by a reflecting cavity boundary.

What would settle it

Explicitly compute the cavity QNM frequencies for scalar or gravitational perturbations on a Schwarzschild black hole inside a reflecting cavity at a chosen radius and check whether the resulting product formula reproduces the critical times of the bouncing singularities obtained from null geodesic analysis.

Figures

Figures reproduced from arXiv: 2606.11297 by Samuel Valach, Sa\v{s}o Grozdanov, Vita Movrin.

Figure 1
Figure 1. Figure 1: Penrose diagram for the maximally extended AdS-black brane in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Penrose diagram for the asymptotically flat Schwarzschild black hole with the bouncing geodesic [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The behaviour of t Λ=0 ∗ as a function of ri in D = 5 for µ = 1. For ri → 0, the bouncing time goes to 0, since the integration interval vanishes. The nontrivial radius for which the real part of t Λ=0 ∗ (ri) vanishes corresponds to the case when the Penrose diagram for the cavity forms a perfect square. The function diverges at the location of the black hole horizon ri = 1, since at this point, the anchor… view at source ↗
Figure 4
Figure 4. Figure 4: Penrose diagram for the Schwarzschild-de Sitter spacetime. Cosmological and black hole [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The behaviour of t Λ>0 ∗ as a function of ri in D = 5 for µ = 1/8. For ri → 0, the bouncing time goes to 0, since the integration interval vanishes. The nontrivial radius for which the real part of t Λ>0 ∗ vanishes corresponds to the case ri = rO. For the cases when the hypersurface becomes null-like the bouncing time diverges. The imaginary part is constant and counts how many times the black hole horizon… view at source ↗
Figure 6
Figure 6. Figure 6: Real and imaginary part of the bouncing time [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Penrose diagram for the Schwarzschild-de Sitter spacetime depicting a geodesic “bouncing off [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: Numerically computed scalar quasinormal frequencies [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Left: Numerically computed scalar quasinormal frequencies [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Left: Numerically computed electromagnetic [PITH_FULL_IMAGE:figures/full_fig_p036_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left: Numerically computed gravitational [PITH_FULL_IMAGE:figures/full_fig_p040_11.png] view at source ↗
read the original abstract

We investigate the existence and implications of ``bouncing geodesics'' in asymptotically flat Schwarzschild and Schwarzschild--de Sitter black holes. These trajectories, which probe the high-curvature regions near the black hole singularity, correspond to specific ``bouncing singularities'' in the bulk retarded Green's function. We provide a precise description of these singularities by combining the local Hadamard form with the global propagation of singularities theorem. We then derive the critical times at which the bulk retarded correlator becomes singular, considering all possible anchorings of the bouncing geodesics, including null infinity and the cosmological horizon. Finally, for black holes enclosed in a reflecting cavity, we establish a universal connection between the locations of the bouncing singularities and the spectrum of cavity quasinormal modes (QNMs) by deriving a cavity version of the thermal product formula, analogous to the one known for anti-de Sitter black holes. This relation allows one to extract information about the black hole interior from the asymptotic QNM spectrum measured at a reflecting hypersurface, even when the cosmological constant is zero or positive. We confirm this prediction through explicit examples by computing the cavity QNMs of scalar and electromagnetic fields, as well as gravitational waves, in spacetimes with asymptotically flat and de Sitter black holes.

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

1 major / 0 minor

Summary. The paper investigates bouncing geodesics in asymptotically flat Schwarzschild and Schwarzschild-de Sitter black holes. These correspond to singularities in the bulk retarded Green's function, located via the local Hadamard form combined with the global propagation of singularities theorem. Critical times are derived for various anchorings (including null infinity and cosmological horizon). For black holes inside a reflecting cavity, a cavity version of the thermal product formula is derived, relating bouncing singularity locations to the cavity QNM spectrum; this is verified by explicit computations of cavity QNMs for scalar, electromagnetic, and gravitational perturbations.

Significance. If the derivation is valid, the result extends the thermal product formula (previously known in AdS) to asymptotically flat and de Sitter cases with reflecting cavities. This would allow extraction of black hole interior information from boundary QNM spectra without AdS asymptotics, providing a concrete link between geodesic singularities and observable QNM poles.

major comments (1)
  1. [Derivation of cavity formula] The central derivation of the cavity thermal product formula relies on applying the propagation of singularities theorem to locate bouncing singularities after imposing reflecting boundary conditions at finite radius. The cavity truncates the manifold and introduces new reflected null geodesics, altering the global causal structure; the manuscript must explicitly show that the theorem continues to isolate the original bouncing singularities without modification or additional poles induced by the boundary (see the paragraph on derivation of critical times and cavity formula).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit justification in the cavity setting. We address the single major comment below and have revised the manuscript to strengthen the presentation of the derivation.

read point-by-point responses

  1. Referee: The central derivation of the cavity thermal product formula relies on applying the propagation of singularities theorem to locate bouncing singularities after imposing reflecting boundary conditions at finite radius. The cavity truncates the manifold and introduces new reflected null geodesics, altering the global causal structure; the manuscript must explicitly show that the theorem continues to isolate the original bouncing singularities without modification or additional poles induced by the boundary (see the paragraph on derivation of critical times and cavity formula).

    Authors: We agree that the interaction between the propagation of singularities theorem and the reflecting cavity boundary requires explicit clarification to avoid any ambiguity regarding the global causal structure. The theorem is applied to the bulk spacetime using the local Hadamard form to identify the singular support along null geodesics; the reflecting boundary conditions at finite radius are imposed subsequently and generate additional reflected geodesics that contribute to the cavity QNM spectrum. However, the specific bouncing singularities associated with geodesics that probe the black hole interior remain isolated by the theorem, as their locations are determined solely by the interior geodesic segments and are unaffected by the outer boundary. In the revised manuscript we have expanded the relevant paragraph (now a dedicated subsection) to demonstrate this explicitly: we show that the critical times for the original bouncing singularities are unchanged, that the boundary-induced geodesics produce distinct contributions to the QNM poles, and that no additional poles appear at the interior bouncing times. This separation follows from the fact that the wave operator characteristics propagate independently of the boundary until reflection occurs. We believe this addition resolves the concern while preserving the validity of the cavity thermal product formula. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation from Hadamard form + propagation theorem plus independent QNM computations

full rationale

The paper's central derivation identifies bouncing singularities via the local Hadamard form combined with the global propagation of singularities theorem, then derives critical times and a cavity thermal product formula relating those locations to cavity QNM poles. This is followed by explicit, independent computations of cavity QNMs for scalar, electromagnetic, and gravitational perturbations in asymptotically flat and de Sitter spacetimes to confirm the relation. No quoted step reduces a prediction to a fitted parameter, self-definition, or self-citation chain; the verification step is external to the analytic derivation and uses direct solution of the wave equation on the truncated manifold.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the local Hadamard form, the global propagation of singularities theorem, and the assumption that a reflecting cavity boundary preserves the correspondence between geodesic bounces and Green's function singularities; no free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)
  • standard math Local Hadamard form combined with global propagation of singularities theorem accurately locates bouncing singularities in the bulk retarded Green's function
    Invoked to provide precise description of singularities corresponding to bouncing geodesics (abstract).
  • domain assumption Reflecting cavity boundary allows a universal thermal product formula analogous to the AdS case
    Required for the cavity version of the formula to connect bouncing singularities to QNM spectrum.

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Reference graph

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