⚛️ general relativity

Shadows and gravitational perturbations of black bounces

This paper investigates axial gravitational perturbations and shadow formation in symmetric and asymmetric black-bounce geometries supported by anisotropic fluids, revealing that while horizon-containing solutions closely mimic standard black holes, horizonless symmetric configurations exhibit distinct features like gravitational-wave echoes and multiple photon rings, whereas horizonless asymmetric solutions remain phenomenologically similar to the Reissner-Nordström case.

Original authors: Alana C. L. Santos, Leandro A. Lessa, Roberto V. Maluf, Gonzalo J. Olmo

Published 2026-09-10

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CC BY 4.0

Original authors: Alana C. L. Santos, Leandro A. Lessa, Roberto V. Maluf, Gonzalo J. Olmo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, our best understanding of gravity has been shaped by a theory that predicts the existence of objects so dense that not even light can escape them. These are black holes, regions of space where the fabric of the universe is stretched to its breaking point, collapsing into a single, infinitely dense point known as a singularity. While this prediction has been incredibly successful at explaining the motion of planets and the bending of starlight, many physicists suspect the theory breaks down at that very center. The idea of a point with infinite density is mathematically troublesome, suggesting that our current laws of physics are incomplete. To fix this, researchers have proposed alternative models where the center of a black hole does not collapse into a singularity but instead "bounces" back, creating a smooth, finite core. This concept, known as a black bounce, suggests that the extreme gravity might curve space so sharply that it opens a passage to another region of the universe, or simply creates a tiny, stable universe hidden inside the event horizon.

The question remains: if these exotic objects exist, how would we tell them apart from the standard black holes predicted by Einstein? The answer lies in two distinct ways we observe the cosmos: by listening to the ripples in space-time caused by violent collisions, and by looking at the dark silhouettes these objects cast against the glowing gas swirling around them. A new study by a team of researchers investigates exactly this, simulating how these "bouncing" black holes would behave when disturbed and how they would appear to a telescope. They focused on two specific types of these objects: one that is perfectly symmetrical, like a sphere, and another that is lopsided, resembling a distorted version of a charged black hole. By running detailed computer simulations, they mapped out how gravitational waves would ripple through these objects and how light would bend around them, looking for any subtle fingerprints that would reveal their true nature.

The researchers found that if a black bounce possesses an event horizon—the point of no return that defines a traditional black hole—it is incredibly difficult to distinguish from a standard black hole. Whether the object is symmetrical or lopsided, the gravitational waves it emits as it settles down after a disturbance look almost identical to those of a normal black hole. The signal is a clean, single pulse that fades away quickly, with no strange echoes or extra bumps. Similarly, when they calculated the shadow these objects would cast, the result was a single, dark circle surrounded by a bright ring of light, looking nearly indistinguishable from the famous images captured by the Event Horizon Telescope. In these cases, the event horizon acts as a curtain, hiding the exotic interior from the outside world and making the object behave exactly as our standard theories predict.

However, the story changes dramatically if the object has no event horizon. Without that point of no return, the exotic interior is exposed to the universe, and the physics becomes far more complex. For the symmetrical, horizonless objects, the simulations revealed a startling phenomenon: the gravitational waves did not just fade away in a single pulse. Instead, the waves bounced back and forth inside the object's core, creating a series of repeating signals known as echoes. These echoes appeared as faint, delayed repetitions of the main signal, with the time between them and their strength depending on the specific size and density of the object's core. It is as if the object were a hollow bell that rings once and then continues to chime softly for a long time, whereas a standard black hole is more like a solid stone that makes a single, sharp sound.

This complexity extended to the way light behaved around these horizonless objects as well. Instead of a single ring of light, the simulations showed that these objects could support multiple rings, creating a complex, nested structure of bright circles. The number of these rings and their brightness depended heavily on the specific parameters of the object. In some cases, light rays would be trapped in a chaotic dance, bouncing off an invisible wall near the center before escaping, creating a dense pattern of concentric rings that would look very different from the simple shadow of a standard black hole. The researchers also noted that for certain configurations, there were regions of space that light simply could not reach, creating a "forbidden zone" where no photons could travel, a feature that does not exist around ordinary black holes.

The lopsided, horizonless objects told a different story. Unlike their symmetrical counterparts, these irregular shapes did not produce the repeating echoes in the gravitational waves, nor did they generate the complex web of multiple light rings. Their behavior remained surprisingly close to that of a standard black hole, with a single potential barrier and a single ring of light. This suggests that the symmetry of the object is a crucial factor in determining whether these exotic signatures appear. If the object is perfectly round, the interior structure creates a complex environment that traps waves and light in unique ways. If it is distorted, the interior smooths out, and the object behaves much more like the familiar black holes we already know.

Ultimately, the study confirms that the presence of an event horizon is the great equalizer. As long as a black hole has a horizon, the strange physics happening inside is hidden, and the object looks and sounds exactly like a standard black hole. It is only when that horizon is removed that the true nature of the object is revealed, either through the haunting echoes of gravitational waves or the intricate, multi-layered shadows cast by light. While the researchers could not prove that these objects exist in our universe, their work provides a clear roadmap for what to look for. If future telescopes and gravitational wave detectors ever spot a signal with a repeating echo or a shadow with multiple rings, it would be a strong indication that the center of a black hole does not end in a singularity, but instead bounces back, offering a glimpse into a universe where the laws of physics remain whole.

Technical Summary: Shadows and Gravitational Perturbations of Black Bounces

Problem Statement
General Relativity (GR) remains incomplete due to the inevitable formation of singularities and tensions with cosmological observations. To address these issues, theoretical models such as "black bounces"—regular spacetimes that interpolate between black holes and wormholes—have been proposed. These geometries, characterized by a non-vanishing minimum in the areal function (the "bounce"), remove central singularities. While scalar perturbations of such geometries have been studied, a comprehensive analysis of axial gravitational perturbations and their corresponding shadow formation (optical appearance) is required to fully characterize their phenomenological imprints. Specifically, the relationship between quasinormal modes (QNMs) and photon sphere instability, well-established for standard black holes with horizons, requires re-evaluation in horizonless black-bounce configurations where effective potentials may exhibit complex structures.

Methodology
The authors analyze two distinct classes of regular black-bounce solutions supported by anisotropic fluids within GR, derived from a general spherically symmetric metric $ds^2 = -A(r)dt^2 + B(r)^{-1}dr^2 + \Sigma(r)^2 d\Omega^2$ .

Model I (Symmetric): A Simpson–Visser-like geometry where the areal function is $\Sigma_I(r) = \sqrt{a^2 + r^2}$ . This model features a symmetric bounce at $r=0$ .
Model II (Asymmetric): A deformation of the Reissner–Nordström metric where the areal function $\Sigma_{II}(r)$ is defined for $r \geq 0$ . Depending on the parameter $l_0$ , this model yields either a bounded interior universe ( $l_0 > 0$ ) or an unbounded wormhole-like extension ( $l_0 < 0$ ).

The study employs two primary analytical frameworks:

Axial Gravitational Perturbations: Linear perturbations are introduced around the background metric using the Regge–Wheeler gauge. The perturbation equations are reduced to a Schrödinger-like wave equation with an effective Regge–Wheeler potential $V_{RW}$ . The time-domain evolution is solved numerically using a finite difference method in null coordinates to extract QNM frequencies and identify gravitational-wave echoes.
Shadow Formation: Null geodesic equations are solved to determine the shadow potential $V_{Sh}(r)$ . The critical impact parameter and photon sphere radii are calculated. The optical appearance is modeled by integrating the intensity of an optically thin accretion disk (using a Johnson distribution profile) to generate transfer functions and synthetic images, analyzing the formation of direct, lensing, and higher-order photon rings.

Key Contributions and Results

Horizon Configurations (Symmetric and Asymmetric):
- Both symmetric and asymmetric solutions possessing event horizons exhibit an effective potential with a single barrier.
- The resulting QNM spectra are nearly indistinguishable from those of standard black holes (specifically the Reissner–Nordström case).
- Optically, these configurations possess a single relevant photon sphere. Their shadow profiles and transfer functions are virtually identical to their black-hole counterparts, as the horizon masks the internal regularization.
Horizonless Symmetric Configurations:
- The absence of a horizon allows the internal structure to influence the exterior. The effective potential can develop multiple barriers (a central peak flanked by secondary maxima).
- Gravitational Waves: The multi-barrier potential leads to the production of gravitational-wave echoes. The amplitude and time separation ( $\Delta t$ ) of these echoes depend sensitively on the bounce parameter $a$ and the density parameter $\rho_0$ . Smaller $a$ values generate additional peaks and longer-lived echoes, while higher $\rho_0$ compresses the barriers, reducing the echo delay.
- Optical Appearance: These geometries can sustain multiple photon rings. The number of rings decreases as the bounce parameter $a$ increases. For small $a$ , the central potential barrier acts as a "gravitational mirror," reflecting rays with small impact parameters and creating a dense structure of nested concentric sub-rings and a distinct central shadow. As $a$ increases, the central barrier flattens, and the structure transitions toward a single-ring profile.
- Geodesic Forbidden Regions: For small $a$ , a "forbidden" region exists in the $(r, b)$ plane where null geodesics cannot penetrate, leading to a "snail-like" or "cashew-nut" shaped inaccessible region in the spacetime geometry.
Horizonless Asymmetric Configurations:
- Unlike the symmetric case, these geometries exhibit regularized effective potentials that do not develop multiple barriers.
- Consequently, they do not produce gravitational-wave echoes. Their ringdown signals remain phenomenologically close to the Reissner–Nordström case.
- Optically, they may support either a single photon ring or none, depending on parameters, but do not generate the hierarchy of additional rings seen in the symmetric horizonless case.

Correspondence Between QNMs and Shadows
The paper verifies the standard eikonal correspondence between QNM frequencies and the instability of circular null geodesics (Lyapunov exponent) for configurations with horizons. However, for horizonless geometries, the emergence of multiple potential barriers prevents a straightforward extension of this analytic relation, as the standard single-barrier WKB approximation is no longer directly applicable. Despite this, the authors note a strong similarity between the wave and optical sectors, suggesting that a meaningful phenomenological correspondence may still persist even if the analytic derivation is not directly transferable.

Significance and Claims
The paper claims to clarify the phenomenological differences between horizon and horizonless black-bounce geometries. It establishes that while event horizons effectively hide the regularization of the interior (making the object observationally similar to a standard black hole in both gravitational wave and optical sectors), the absence of a horizon in symmetric configurations leads to distinct signatures: gravitational echoes and complex multi-ring shadow structures. Conversely, asymmetric horizonless geometries remain observationally subtle, lacking the echo signatures of their symmetric counterparts. The work underscores that distinguishing these geometries requires precise measurements of echo timing and shadow ring multiplicity, which are sensitive to the specific bounce parameters ( $a, \rho_0, l_0$ ).

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