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

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 specificbouncing 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.

• The paper demonstrates that bouncing geodesics encode black hole singularity signatures in asymptotically flat and de Sitter spacetimes.
• It employs rigorous geometric and microlocal analytic methods to relate Green’s function singularities with the spectrum of quasinormal modes.
• The derived cavity thermal product formula offers a universal protocol to infer interior properties of black holes from measurable external perturbations.

Bouncing Geodesics and the Cavity Thermal Product Formula for Asymptotically Flat and de Sitter Black Holes

Introduction and Motivation

The paper "Bouncing Geodesics, Singularities, and the Cavity Thermal Product Formula in Asymptotically Flat and de Sitter Black Holes" (2606.11297) investigates the manifestation of curvature singularities inside black holes via analytic signatures in external observables, using the structure of classical null and spacelike geodesics. Classical singularities in the black hole interior represent a longstanding theoretical challenge, especially regarding the detection or indirect probing of such features from regions accessible to external observers. Prior work exploited the AdS/CFT duality to explore related analytic structures for asymptotically AdS black holes, but analogous statements for physically realistic (asymptotically flat or de Sitter) black holes remained underdeveloped in the absence of holography.

This work extends the study of so-called "bouncing geodesics"—null geodesics that approach the singularity and return, generating "bouncing singularities" in correlation functions—beyond AdS, establishing their existence in asymptotically flat and de Sitter backgrounds. Crucially, the authors derive a "cavity thermal product formula" connecting the imprints of these geodesics to the asymptotic spectrum of cavity quasinormal modes (QNMs), thus demonstrating a robust, duality-independent mechanism by which black hole interiors encode signatures in accessible spectral data.

Bouncing Geodesics and Propagation of Singularities

The central geometric objects are "bouncing geodesics," defined as null limits of spacelike geodesics that enter the black hole interior, approach arbitrarily close to the singularity, and return to the exterior (or alternative boundary/horizon), traversing two-sided maximally extended spacetimes. The analysis begins by rigorously constructing these orbits in Schwarzschild and Schwarzschild–de Sitter geometries.

The propagation of Green's function singularities is formalized using the Hadamard expansion and microlocal analysis. Locally, retarded Green's functions in Lorentzian spacetimes are singular on the null cone; globally, the propagation of singularities theorem ensures these singularities track null geodesics, including those that "bounce" off the curvature singularity. The authors prove that bouncing geodesics thus enforce specific nontrivial singularities—“bouncing singularities”—in the analytic structure of retarded correlators, entirely independent of holography.

Existence and Structure of Bouncing Geodesics in Different Asymptotics

Asymptotically Flat Schwarzschild Spacetimes

The study explicitly constructs the families of symmetric spacelike geodesics in Schwarzschild geometry, with null limits becoming the bouncing geodesics of interest. The explicit calculation of "bouncing times"—the elapsed coordinate time along such geodesics—is carried out for various spacetime dimensions and anchoring points (timelike surfaces, null infinity). The analytic behavior of the Green's function and correlator singularities is rigorously mapped to these geometric objects, with the imaginary part of the bouncing time accounting for crossings of horizons.

Schwarzschild–de Sitter Spacetimes

For Schwarzschild–de Sitter geometries, the analysis is further complicated by the presence of both black hole and cosmological horizons. The authors focus on five-dimensional cases for analytic tractability and characterize geodesics anchored at generic points, the static sphere observer (where gravitational attraction and cosmological repulsion balance), and the cosmological horizon. All relevant bouncing times and singularity structures are obtained, and the analysis explicitly identifies the geometric dependence of these times on the black hole and cosmological horizon configuration. Notably, geodesics bouncing off spatial infinity—but not singularities—do not lead to singularities in the correlators, reinforcing the selective diagnostic power of the method.

The Cavity Thermal Product Formula and QNM Spectra

A core technical advance is the derivation of a generalized thermal product formula for black holes enclosed by a reflecting cavity, which plays the role of an artificial "boundary." The cavity setup allows robust anchoring for the analysis and sidesteps the absence of a natural conformal boundary in non-AdS geometries.

The causal structure ensures that the analytic properties of frequency-space two-point functions, particularly their pole structure (the QNM spectrum), are determined by the boundary conditions at the cavity wall and the black hole horizon. By leveraging Hadamard’s factorization theorem and the analyticity of the Green's function, the authors demonstrate that the two-sided (thermal) boundary correlator factorizes into a product over the zeros (QNMs) of the relevant boundary condition function, up to an overall exponential factor:

$$G_{12}(\omega) \propto \frac{\omega}{\sinh(\beta\omega/2)} \prod_n \left(1 - \frac{\omega^2}{\omega_n^2}\right)^{-1}$$

where $\omega_n$ are the cavity QNMs and $\beta$ is the inverse Hawking temperature.

Universal Relation Between Bouncing Time and QNM Spacing

A central result is the asymptotic relation between the "bouncing time" $t_*$ (computed from the geometric data of the black hole interior) and the high-overtone spacing of the cavity QNM frequencies:

$t_* \sim \frac{2\pi n}{\omega_n},\quad n\to\infty$

This provides a practical protocol: measuring the asymptotic QNM spectrum of a black hole in a reflective cavity encodes geometric and causal information about its interior, specifically, about the region adjacent to the singularity.

Explicit Numerical Verification for Scalar, Electromagnetic, and Gravitational Perturbations

The paper presents detailed numerical studies for scalar, electromagnetic, and gravitational cavity quasinormal mode spectra in both asymptotically flat and de Sitter spacetimes, considering all relevant physically-motivated boundary conditions (Dirichlet, Neumann, Robin, perfect conductor). In all cases, the predicted asymptotic linear QNM spacing, dictated by the geometric bouncing time (as above), is quickly approached at accessible overtone numbers. The analysis is robust with respect to boundary condition choice and perturbation sector, illustrating the universal character of the main result.

Theoretical and Practical Implications

These results have major implications for the theoretical understanding of both black hole interiors and the external observables that encode such data:

• Holography-Independent Singular Signature: The analytic structure of Green’s functions and the high-overtone QNM spectrum retain sensitivity to black hole singularities even in the absence of a dual field theory description (as is the case in flat or de Sitter asymptotics).
• Universal QNM–Geodesic Correspondence: The cavity thermal product formula applies generically to black holes with arbitrary cosmological constant and for a wide class of boundary conditions and perturbations.
• Observational Prospects: If a reflecting surface could (even artificially) be implemented around astrophysical black holes, analysis of measured QNM spectra might, in principle, extract information about the interior—especially signatures of classical or quantum-resolved singularities.
• Theoretical Extensions: The work opens several directions, including the extension to rotating (Kerr) geometries, analysis under higher-derivative (quantum gravity) corrections, and possible generalizations to coupled perturbation systems or alternative boundary constructions.

Conclusion

This paper establishes a robust geometric mechanism for encoding black hole interior information—including the presence and structure of curvature singularities—within the asymptotic spectrum of external perturbations, bridging the gap between classical general relativistic intuition and analytic properties of observables in non-AdS black hole spacetimes. The cavity thermal product formula provides both a practical computational tool and a conceptual framework for relating external spectral data to the elusive physics of black hole interiors. This development is expected to stimulate both further mathematical investigations of QNM spectra and their analytic structures, as well as speculative considerations regarding the empirical accessibility of black hole interiors in future gravitational wave observations.