⚛️ general relativity

Redshift Spectroscopy as a Probe of Regular Black Holes, Black Bounces, and Scalar-Hair Compact Objects

This paper develops a unified, model-independent framework for analyzing photon frequency shifts in generic static, spherically symmetric spacetimes, extending the formalism to include peculiar motion and plasma effects while applying it to various regular black holes, black bounces, and scalar-hair compact objects to demonstrate a common spectroscopic language for distinguishing these horizonless and deformed geometries.

Original authors: Ali Övgün, Reggie C. Pantig, Joel Saavedra

Published 2026-09-03

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CC BY 4.0

Original authors: Ali Övgün, Reggie C. Pantig, Joel Saavedra

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

Deep in the heart of our galaxy, and in the dark spaces between stars, lie objects so dense that their gravity bends the very fabric of space and time. For decades, scientists have watched these cosmic giants, known as black holes, primarily by listening to the ripples they create in spacetime or by capturing the shadow they cast against the bright gas swirling around them. But there is another way to study them, one that relies on the light itself. When matter orbits these massive objects, it emits light that gets stretched or squeezed by the intense gravity and the speed of the orbit. This stretching, known as redshift, and the squeezing, known as blueshift, act like a cosmic fingerprint. By measuring how much the color of this light changes, astronomers can deduce the shape of the space around the object, revealing whether it is a standard black hole or something more exotic.

A team of researchers has now built a powerful new mathematical toolkit to decode these color shifts. They created a single, unified framework that works for a wide variety of strange cosmic objects, not just the classic black holes predicted by Albert Einstein. Their work allows scientists to take the light from an orbiting star or gas cloud and calculate exactly how its frequency changes, regardless of whether the central object is a regular black hole, a "bouncing" object that avoids a singularity, or a wormhole-like structure. By applying this method to several specific theoretical models, the team found that the same geometric rules that govern the orbit of a star also determine the size of the black hole's shadow, linking two different ways of observing the universe into one coherent picture.

The researchers began by imagining a generic, static sphere of mass, described by three flexible mathematical functions that define how space and time behave at different distances. Instead of assuming a specific type of object, they derived exact formulas for how a particle moves in a circle around such a mass and how a photon of light travels from that particle to a distant observer. They calculated the precise conditions required for a stable circular orbit and determined how the angle at which light is emitted affects the signal received far away. Crucially, they also accounted for the fact that light often travels through a thin, cold gas called plasma, which can slow down light waves and alter their path. They showed how this gas modifies the signal, specifically changing the light that travels sideways relative to the orbit, while leaving the light traveling directly toward or away from the observer unchanged.

To make these complex formulas useful for real-world astronomy, the team developed a way to treat these exotic objects as small deviations from the familiar black hole described by Einstein. They expanded their equations to show how tiny changes in the object's structure would ripple through the observed light shifts. This approach allows scientists to test specific theories against observations without needing to solve a completely new set of equations for every single model. They then applied this framework to three distinct types of theoretical objects. First, they looked at regular black holes created by a specific type of electromagnetic theory that removes the infinite density point at the center. Second, they examined a "black bounce" model, which describes an object that looks like a black hole from the outside but has a smooth, non-singular core that could theoretically connect to another region of space. Third, they studied a naked singularity, an object with a scalar field that lacks an event horizon entirely.

The results revealed a striking consistency across these very different models. In every case, the same underlying geometry that dictates how a star orbits the object also dictates the size of the shadow the object casts on the sky, provided a critical orbit for light exists. For the regular black holes based on nonlinear electrodynamics, the researchers found that the relationship between the orbital radius and the maximum redshift is smooth and predictable. As the object gets closer to the limit of being an extreme black hole, the redshift increases steadily, allowing astronomers to map the orbit directly to the observed color shift without confusion. This monotonic behavior means that if we measure a specific redshift, we can uniquely determine the distance of the orbiting matter, a vital step for reconstructing the shape of spacetime.

For the black-bounce and wormhole models, the study highlighted how the transition from a black hole to a traversable wormhole changes the optical landscape. As the central object becomes more wormhole-like, the inner structure changes, but the outer shadow remains surprisingly similar to that of a standard black hole until a specific threshold is crossed. The team also discovered that for the scalar-field objects, there is a sharp dividing line. If the object is deformed enough, the standard unstable orbit for light disappears, meaning no shadow forms in the traditional sense. Yet, even in this case, the light from orbiting matter still carries a clear, well-defined signal. This finding suggests that spectroscopy, or the study of light frequencies, can probe these exotic objects even when traditional imaging techniques like shadow detection might fail.

The paper concludes that this unified language of light shifts can be applied across the entire spectrum of compact objects, from regular black holes to horizonless singularities. By providing exact formulas that include the effects of plasma and peculiar motion, the researchers have given astronomers a robust method to interpret future observations from telescopes like the Event Horizon Telescope. The work does not claim to have discovered a new object, but rather provides the essential map for reading the light from any such object we might find. It confirms that the physics of orbiting matter and the physics of light bending are deeply intertwined, offering a single, coherent way to test the limits of gravity and explore the true nature of the universe's most extreme environments.

Technical Summary: Redshift Spectroscopy as a Probe of Regular Black Holes, Black Bounces, and Scalar-Hair Compact Objects

Problem Statement
Recent observational advances, including gravitational-wave detections, Event Horizon Telescope (EHT) imaging, and precision stellar motion tracking (e.g., the S2 star), have necessitated a unified theoretical framework to interpret strong-gravity phenomena. While redshift spectroscopy, orbital dynamics, photon-sphere physics, and black-hole imaging are often treated as distinct observational facets, there is a lack of a single, model-independent analytic structure capable of connecting these observables across diverse compact-object geometries. Specifically, existing literature often focuses on specific metrics (e.g., polymerized black holes) or neglects the interplay between local spectroscopic signals, critical null orbits (photon spheres), and the effects of dispersive media like cold plasma. The authors aim to formulate a unified spectroscopic framework for photon frequency shifts in generic static, spherically symmetric spacetimes that can accommodate regular black holes, black bounces, wormholes, and scalar-hair solutions without presupposing a specific matter source or causal structure.

Methodology
The authors develop a general formalism based on a line element characterized by three arbitrary radial metric functions, $\alpha(r, \delta)$ , $\beta(r, \delta)$ , and $\gamma(r, \delta)$ , where $\delta$ is a dimensionless deformation parameter. The methodology proceeds through several stages:

Exact Geodesic Derivation: Working in the equatorial plane, the authors derive exact expressions for the conserved energy ( $E$ ) and angular momentum ( $L$ ) for massive particles and photons. They establish the conditions for circular timelike geodesics and derive the emitter's four-velocity components ( $u^t, u^\phi$ ).
Spectroscopic Formalism: The local frequency shift is defined for a photon emitted by a circularly orbiting source and detected by a distant static observer. The authors derive the exact redshift/blueshift formula, $1+z$ , as a function of the local emission angle $\psi$ . They distinguish between radial emission ( $\psi=0$ ) and tangential emission ( $\psi=\pm \pi/2$ ), identifying the latter as the branches yielding extremal redshifts and blueshifts.
Plasma Extension: The formalism is extended to include propagation through a nonmagnetized cold plasma. By distinguishing between the wave covector and the ray tangent, the authors derive how the refractive index modifies the photon impact parameter and, consequently, the nonradial frequency-shift branches, while noting that the radial branch remains locally unchanged.
Perturbative Expansion: To facilitate application to deformed models, the authors construct a perturbative expansion of the metric functions and all derived observables (orbital energy, angular momentum, four-velocity, impact parameter, and frequency shifts) up to second order in the deformation parameter $\delta$ around the Schwarzschild background.
Model Applications: The framework is applied to three specific classes of spacetimes:
- Regular black holes from nonlinear electrodynamics (NED), specifically the Fan-Wang family.
- The Simpson-Visser black-bounce spacetime, which interpolates between regular black holes and traversable wormholes.
- The Fisher-Janis-Newman-Winicour-Wyman (FJNW) geometry, a scalar-hair naked singularity.

Key Contributions and Results

Unified Geometric Kernel: The analysis reveals that the combination $D(r) \equiv \alpha \beta' - \beta \alpha'$ acts as a universal kernel. This single geometric quantity governs both the existence of circular timelike emitters and the spectroscopic quantities derived from them. In vacuum, this same structure determines the photon sphere and the shadow impact parameter, establishing a direct link between orbital spectroscopy and imaging observables.
Local Spectroscopic Triplet: The authors introduce a compact triplet of observables ( $R, S, T$ ) representing the total frequency shifts for specific emission angles. These provide a model-independent way to encode spectroscopic information.
Plasma Effects: The study explicitly demonstrates that while plasma modifies the nonradial spectroscopic branches via the refractive index, the radial emission branch ( $\psi=0$ ) remains unaffected at the level of the local frequency shift. The authors clarify that full plasma-modified shadow calculations require solving the dispersive ray problem, which is beyond the scope of their local analytic treatment.
Fan-Wang Family Analysis: For the Fan-Wang regular black holes, the authors investigate the invertibility of the spectroscopic map (mapping maximal redshift to orbital radius). They derive an exact condition for the existence of local extrema in the redshift profile. Their analysis shows that for the representative physical black-hole branches considered, the local maximal-redshift branch remains globally monotonic and invertible in the exterior domain.
Simpson-Visser Geometry: The application to the Simpson-Visser metric highlights the transition from regular black holes to wormholes. The authors identify that while the outer photon sphere persists up to a specific bounce parameter ( $b_0 = 3M$ ), the horizon disappears at $b_0 = 2M$ . This creates a regime ( $2M < b_0 < 3M$ ) where the geometry is wormhole-like but still supports an outer photon sphere and standard shadow formation.
FJNW Spacetime: In the scalar-hair FJNW geometry, the authors identify a sharp geometric threshold at $\nu = 1/2$ . For $\nu > 1/2$ , an exterior photon sphere exists, linking spectroscopy and shadow formation. For $0 < \nu \le 1/2$ , no exterior photon sphere exists, yet the spectroscopic observables remain well-defined throughout the exterior region. This decouples the necessity of a null critical orbit for the validity of redshift spectroscopy.

Significance and Claims
The paper claims to provide a "common language" for analyzing a broad spectrum of compact objects, including regular black holes, black bounces, wormholes, and scalar-supported horizonless spacetimes. The primary significance lies in demonstrating that:

Orbital spectroscopy, photon-sphere physics, and shadow observables share a common geometric origin in static, spherically symmetric spacetimes.
The same local spectroscopic framework can be applied to horizonless and singular spacetimes where traditional black-hole imaging assumptions might fail.
The perturbative expansion offers a practical tool for parameter estimation and inverse problems, separating metric deformations that affect orbital dynamics from those affecting optical trajectories.

The authors remain modest regarding the scope of their work, explicitly stating that their formulas serve as "local spectroscopic building blocks" and analytic expressions for extremal directions. They note that they do not perform full ray-tracing or compute complete observed line profiles, which would require solving the boundary-value problem for null rays between the emitter and observer. The work is presented as a foundational analytic structure intended to complement, rather than replace, full numerical ray-tracing and shadow studies.

Technical Summary: Redshift Spectroscopy as a Probe of Regular Black Holes, Black Bounces, and Scalar-Hair Compact Objects

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