⚛️ general relativity

Corrected thermodynamics and radiation predictions of modified black bounce compact objects

This paper investigates the thermodynamic properties, radiation characteristics, and linear stability of a Simpson-Visser regularized modified gravity black hole, revealing phase transitions, deriving effective temperatures for horizonless wormhole branches, and demonstrating stability against monopole perturbations.

Original authors: Shokhzod Jumaniyozov, Javlon Rayimbaev, Yassine Sekhmani, Satimbay Palvanov, Olmos Tursunboyev, Dilshod Karshiev

Published 2026-09-09

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CC BY 4.0

Original authors: Shokhzod Jumaniyozov, Javlon Rayimbaev, Yassine Sekhmani, Satimbay Palvanov, Olmos Tursunboyev, Dilshod Karshiev

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

In the deepest corners of our universe, gravity is so intense that it warps space and time into a shape we can barely imagine. For decades, the prevailing theory of gravity, known as general relativity, has predicted that when massive stars collapse, they form black holes: regions where gravity becomes so strong that nothing, not even light, can escape. However, at the very center of these objects, the theory predicts a singularity, a point where the curvature of space becomes infinite and the laws of physics as we know them break down. Most physicists believe this singularity is not a real physical object but a sign that our current theories are incomplete, hinting that a deeper understanding of gravity is needed to describe what happens at such extreme scales. To explore this, scientists often build theoretical models that smooth out these infinite points, creating "regular" black holes that have a finite, gentle core instead of a violent tear in the fabric of space.

A team of researchers has recently taken a significant step in this direction by combining two distinct ideas to create a new model of a compact object. They merged a theory called Modified Gravity, which suggests that gravity behaves differently than Einstein predicted without needing invisible dark matter, with a mathematical trick that removes the central singularity. The result is a theoretical object that can exist in three different forms depending on its internal structure: a standard black hole with an event horizon, a one-way passage that connects two regions of space, or a fully traversable wormhole that allows travel in both directions. By studying the heat and radiation of these objects, the team has uncovered how these exotic shapes would behave if they were real, offering new ways to distinguish them from ordinary black holes using future telescopes.

The researchers began by constructing the geometry of this new object, which they call an SV-MOG compact object. This model is defined by two key numbers. One number represents the strength of the modified gravity effects, while the other determines how much the center of the object is "bounced" or smoothed out to avoid a singularity. When the smoothing number is small, the object behaves like a regular black hole with a surface from which nothing can escape. As the smoothing number increases, the event horizon shrinks until it vanishes entirely, transforming the object into a wormhole. In this wormhole state, there is no point of no return; instead, there is a throat connecting two sides of the universe that a traveler could theoretically cross. The team analyzed how the transition between these states happens, finding that stronger modified gravity effects make the black hole region larger, requiring a greater degree of smoothing to turn it into a wormhole.

To understand the thermal nature of these objects, the team calculated their temperature. For the black hole version, they determined the temperature of the radiation emitted from the event horizon, a process known as Hawking radiation. They found that as the object gets smaller, its temperature rises to a peak and then falls again, a behavior that signals a change in its stability. Crucially, they identified a specific point where the object undergoes a phase transition, shifting from a thermally unstable state to a stable one. This is a feature not seen in standard black holes. For the wormhole version, which has no event horizon and therefore no traditional Hawking radiation, the team devised a different way to define temperature. They looked at the behavior of light orbiting the object in an unstable circle. The rate at which these light orbits break apart provided a measure of instability that they translated into an effective temperature. This allowed them to compare the thermal properties of the wormhole to the black hole, revealing that the wormhole does not undergo the same kind of stability transition.

The study also explored how quantum mechanics might alter the entropy, or the measure of disorder, of these objects. In standard physics, the entropy of a black hole is directly proportional to the area of its event horizon. However, theories of quantum gravity suggest that at very small scales, this relationship receives corrections. The researchers applied these corrections to their model and found that the smoothing parameter plays a vital role. Because the object's core never shrinks to zero size, the entropy remains finite and well-behaved even as the object becomes very small. This prevents the mathematical infinities that plague traditional black hole models. They showed that the strength of the modified gravity and the degree of smoothing both influence how much the quantum corrections change the object's entropy, with the effects becoming most significant when the object is small.

Finally, the team simulated how these objects would look if they were surrounded by a disk of hot, swirling gas, a common feature around real black holes. They calculated the light emitted by this disk, including its brightness, temperature, and the specific colors of light it produces. They discovered that the modified gravity parameter significantly changes the outcome. Increasing this parameter pushes the inner edge of the gas disk further out, which makes the disk emit more total energy but at a lower peak temperature, shifting the light toward redder frequencies. In contrast, increasing the smoothing parameter, which turns the object toward a wormhole, suppresses the peak brightness and softens the spectrum. Perhaps most strikingly, the wormhole version produces a distinct signature: because there is no event horizon to block the view, the gas disk can extend all the way to the central throat, creating a bright, central region of emission that would be absent in a standard black hole.

The findings suggest that if we can observe the X-ray light from compact objects with enough precision, we might be able to tell the difference between a standard black hole, a modified gravity black hole, and a wormhole. The specific patterns in the light, such as the temperature of the inner disk and the frequency of the emitted radiation, carry the fingerprints of the object's internal structure. While these results are based on theoretical models and simulations rather than direct observation, they provide a clear roadmap for what astronomers should look for. The study highlights that the universe might contain objects that are not just black holes, but smooth, traversable bridges in space, and that the light they emit holds the key to revealing their true nature.

Technical Summary: Corrected Thermodynamics and Radiation Predictions of Modified Black Bounce Compact Objects

Problem Statement
General Relativity (GR) predicts spacetime singularities where curvature diverges, signaling the theory's incompleteness. While Modified Gravity theories like Scalar-Tensor-Vector Gravity (STVG or MOG) offer alternatives to dark matter, their static, spherically symmetric vacuum solutions (Schwarzschild-MOG) still harbor curvature singularities at the origin. Conversely, regular black hole models, such as those constructed via the Simpson–Visser (SV) regularisation, remove singularities by replacing the center with a smooth, geodesically complete core but often lack the specific phenomenological corrections provided by modified gravity. There is a need to understand the thermodynamic stability and radiative signatures of compact objects that simultaneously incorporate MOG corrections and singularity regularisation, particularly distinguishing between black hole and horizonless wormhole phases.

Methodology
The authors construct the SV-MOG spacetime by applying the Simpson–Visser substitution ( $r^2 \to r^2 + l^2$ ) to the Schwarzschild-MOG metric. This introduces two parameters: the MOG coupling constant $\alpha$ and the black-bounce parameter $l$ . The study proceeds through four main analytical stages:

Geometric Classification: The horizon structure is analyzed by solving for the zeros of the lapse function. The geometry is classified into three regimes based on the ratio $l/l_{cr}(\alpha)$ : regular black holes ( $l < l_{cr}$ ), extremal one-way wormholes ( $l = l_{cr}$ ), and traversable wormholes ( $l > l_{cr}$ ).
Thermodynamic Analysis:
- Black Hole Branch ( $l \leq l_{cr}$ ): The Hawking temperature ( $T_H$ ) is derived from the surface gravity at the outer event horizon. The heat capacity ( $C_V$ ) is calculated to assess thermodynamic stability and identify phase transitions.
- Wormhole Branch ( $l > l_{cr}$ ): Recognizing the absence of a causal horizon, the authors reject a direct continuation of $T_H$ . Instead, they construct a physically well-defined effective temperature ( $T_{Lyap}$ ) and heat capacity ( $C_{Lyap}$ ) based on the Lyapunov exponent of the unstable photon sphere.
- Quantum Corrections: Logarithmic corrections to the Bekenstein–Hawking entropy are incorporated via a saddle-point expansion of the partition function, introducing coefficients $\beta_1$ and $\beta_2$ to model quantum-gravitational effects.
Stability Analysis: A linear scalar-perturbation analysis is performed on a minimally coupled massless test field. The wave equation is reduced to a Schrödinger-like form to examine the effective radial potential ( $V_\ell$ ) for monopole ( $\ell=0$ ) perturbations.
Radiative Properties: Using the Novikov–Thorne model for geometrically thin, optically thick accretion disks, the authors compute the electromagnetic flux, effective disk temperature, and spectral luminosity. This involves determining the Innermost Stable Circular Orbit (ISCO) and integrating the flux over the disk area.

Key Contributions and Results

Thermodynamic Phase Transitions: For the black hole branch, the heat capacity exhibits a divergence at a critical radius $r^*_+$ , signaling a second-order phase transition between thermally unstable (large $r_+$ ) and stable (small $r_+$ ) phases. Increasing the MOG coupling $\alpha$ shifts this transition to larger radii and widens the stable window, whereas increasing the bounce parameter $l$ narrows the stable region.
Distinct Wormhole Thermodynamics: For the wormhole branch, the Lyapunov temperature $T_{Lyap}$ decreases monotonically as the black-bounce parameter increases, vanishing when the photon sphere disappears. The associated heat capacity $C_{Lyap}$ remains negative throughout its existence, showing no analogue to the black hole's second-order phase transition. The authors emphasize that $T_{Lyap}$ and $T_H$ are conceptually distinct and do not match continuously at the transition point $l = l_{cr}$ .
Perturbative Stability: The linear scalar-perturbation analysis reveals that the effective radial potential $V_{\ell=0}(r)$ remains non-negative across both the black hole exterior and the wormhole throat for all studied parameters. This indicates no evidence of linear instability against monopole scalar perturbations, independent of the thermodynamic stability inferred from heat capacity.
Quantum-Corrected Entropy: The inclusion of logarithmic corrections ( $\beta_1 \ln S_0$ and $\beta_2/S_0$ ) modifies the entropy significantly at small horizon radii. A key result is that the non-zero bounce parameter $l$ prevents the horizon area from vanishing, ensuring the corrected entropy remains bounded from below, unlike in singular Schwarzschild geometries where such corrections would diverge.
Accretion Disk Signatures:
- MOG Coupling ( $\alpha$ ): Increasing $\alpha$ enlarges the ISCO and the peak electromagnetic flux but lowers the peak disk temperature and softens the spectral luminosity (shifting the peak to lower frequencies). This occurs because the increased dissipation is spread over a larger disk area.
- Bounce Parameter ( $l$ ): Increasing $l$ suppresses the peak flux and broadens the spectral profile. For wormholes, the disk extends to the throat ( $r=0$ ), creating a distinctive central emission region absent in black hole geometries. However, the total spectral luminosity of wormholes is orders of magnitude lower than that of black holes due to the lack of an event horizon and weaker energy dissipation.

Significance and Claims
The paper claims to provide the first comprehensive study of the thermodynamic and radiative properties of SV-MOG compact objects. The primary significance lies in demonstrating how the interplay between the MOG coupling and the black-bounce parameter creates a rich phenomenology that distinguishes these objects from their Schwarzschild and pure SV counterparts.

Specifically, the authors argue that:

The SV-MOG framework successfully removes the central singularity while retaining the observational signatures of modified gravity.
The thermodynamic behavior of the wormhole branch is fundamentally different from the black hole branch, characterized by a lack of phase transitions and a distinct temperature definition based on photon sphere instability rather than horizon surface gravity.
Observational signatures, such as the shift in spectral peak frequency, the modification of the disk temperature profile, and the presence of a bright central emission region in wormhole configurations, could potentially be constrained by current and next-generation X-ray telescopes (e.g., NICER).

The authors maintain a modest stance, noting that their results are qualitative predictions based on an idealized, non-rotating Novikov–Thorne framework. They explicitly state that a quantitative comparison with real X-ray spectra requires future work incorporating black hole spin, non-equatorial disk structures, and full radiative transfer. Furthermore, the stability analysis is restricted to linear monopole scalar perturbations, and a general dynamical stability proof requires further investigation into higher multipoles and coupled gravitational/electromagnetic perturbations.

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