The starting point is the scalar-tensor-vector gravity theory, also known as MOG, proposed by Moffat. This framework introduces a dynamical gravitational coupling, a massive vector field and an associated scalar field, and its unique asymptotically flat, spherically symmetric vacuum solution is the Schwarzschild-MOG black hole, which carries a vector-field charge proportional to the square root of the MOG coupling times the mass. The theory was designed to reproduce galaxy rotation curves, cluster dynamics and cosmological observations without invoking dark matter. Yet even this modified solution still harbours a curvature singularity at its origin. The Simpson–Visser prescription removes that singularity by substituting the radial coordinate squared with the sum of the radial coordinate squared and the bounce parameter squared, yielding a lapse function that involves both the MOG coupling and the bounce scale.

The horizon structure follows from the zeros of this modified lapse function. A real outer horizon exists only when the bounce parameter stays below a critical value that grows monotonically with the MOG coupling, because a stronger vector-field charge enlarges the black-hole region of parameter space. Below the critical bounce, the classical singularity is replaced by a smooth minimal-area two-sphere and the spacetime is geodesically complete. Exactly at the critical value the two horizons coalesce into a one-way wormhole with vanishing surface gravity, and beyond it the throat at the origin has a finite areal radius accessible to two-way timelike geodesics. In the limit of vanishing MOG coupling the construction reduces to the standard Simpson–Visser geometry, while with no bounce it recovers the Schwarzschild-MOG black hole.

The thermodynamic analysis draws a careful distinction between the two branches. For the black-hole branch, the authors derive the Hawking temperature from the surface gravity at the outer horizon and compute the heat capacity at fixed bounce and coupling. The temperature behaves non-monotonically as a function of the horizon radius: it rises from zero at the extremal radius, peaks at a critical radius and then falls back toward zero at large radii. The heat capacity diverges exactly at that peak, which the authors interpret as a second-order phase transition. Below the critical radius the heat capacity turns positive and the black hole becomes thermally stable, a stability window the paper notes is absent in both Schwarzschild and Reissner–Nordström geometries. Increasing the MOG coupling shifts the transition to larger radii and widens the stable branch, while increasing the bounce narrows it.

For the horizonless wormhole branch, no causal horizon exists and therefore no genuine Hawking radiation in the usual sense. Rather than formally continuing the black-hole surface-gravity formula, which the authors explicitly reject as physically meaningless there, they construct an effective temperature from the Lyapunov exponent of the unstable photon sphere, the light ring whose orbital instability governs phenomena such as photon echoes and early ringdown in horizonless ultracompact objects. This Lyapunov temperature is a genuine geometric invariant requiring no horizon. Numerically, an unstable photon sphere exists throughout the black-hole branch and survives into a finite window of the wormhole branch; for vanishing MOG coupling that window runs from roughly two to about three times the mass, and it widens and shifts to larger bounce values as the coupling grows. Notably, the Lyapunov temperature does not vanish as the extremal point is approached from the wormhole side, in contrast to the Hawking temperature on the black-hole side, a discontinuity the authors regard as physically informative because the two quantities encode different physics.

The associated heat capacity built from the Lyapunov temperature behaves quite differently from its black-hole counterpart. It remains negative throughout its entire window of existence for every coupling value studied, dipping to a shallow negative minimum and then rising monotonically toward zero as the photon sphere disappears. It shows no divergence and no sign change, so the wormhole branch displays no analogue of the black hole’s second-order phase transition. The authors stress that the two heat capacities need not match at the transition, precisely because the underlying constructions describe distinct processes, and they no longer claim such a match.

Quantum-gravitational corrections enter through logarithmic terms in the entropy, parameterised by two dimensionless coefficients whose theoretically preferred magnitudes the authors discuss. Loop-quantum-gravity state counting favours a leading coefficient of negative one half, related schemes yield values as large in magnitude as negative three halves, string-theory D-brane counting produces order-unity coefficients of either sign, and generalised-uncertainty-principle constructions generically give positive values. The authors therefore treat both coefficients as free phenomenological parameters of order unity. The corrected entropy reduces to the Bekenstein–Hawking area law at large horizon radii, while deviations become significant at small horizon sizes. A key advantage of the regularisation is that the non-zero bounce prevents the horizon area from vanishing, so the corrected entropy stays bounded from below, whereas in unregularised geometries such as Schwarzschild the inverse-area term diverges. The authors also identify a genuine limitation of the perturbative ansatz near extremality, where the Hawking temperature vanishes and the logarithmic term diverges, and they restrict the corrected-entropy analysis to a finite neighbourhood of the semiclassical regime.

Independently of the thermodynamic picture, the authors examine linear scalar perturbations. Decomposing a minimally coupled massless test field into spherical harmonics reduces the wave equation to a Schrödinger-like form with an effective radial potential that includes curvature corrections from the non-trivial areal-radius function. For the monopole channel, the most dangerous case because no centrifugal barrier assists stability, the potential remains non-negative throughout the black-hole exterior and across the wormhole throat for every parameter combination shown, vanishing only at the horizon or, in the near-extremal limit, at the throat itself. No negative potential well capable of supporting a growing mode is found. The authors caution that this single-channel, linearised, test-field analysis is a necessary but not sufficient condition for stability and should not be read as a general stability proof; higher multipoles and coupled gravitational and electromagnetic perturbations are left for future work.

The radiation sector uses the Novikov–Thorne model of a steady, geometrically thin, optically thick accretion disk whose inner edge sits at the innermost stable circular orbit. Increasing the MOG coupling at fixed bounce enlarges the event horizon and the ISCO, raises the peak electromagnetic flux nearly sevenfold in the illustrated case, yet lowers the peak disk temperature from roughly 3.5 million to about 2 million kelvin, because the larger dissipated power spreads over a bigger disk area. Increasing the bounce instead suppresses and broadens the flux while only weakly affecting the peak temperature. In the spectral luminosity, stronger coupling softens the high-frequency emission and shifts the peak to lower frequencies, while a larger bounce lowers the peak luminosity. Strikingly, the wormhole branch’s spectral luminosity sits several orders of magnitude below the black-hole case, and its disk extends inward to the throat, producing a distinctive bright central emission region absent around standard black holes. The authors emphasise these are qualitative, model-level predictions of an idealised thin-disk framework, and that quantitative comparison with real X-ray spectra from missions such as NICER would require relaxing assumptions about spin, inclination and radiative transfer.

Subject of Research: Theoretical study of thermodynamics and accretion-disk radiation of Simpson–Visser regularised Schwarzschild-MOG compact objects

Article Title: Corrected thermodynamics and radiation predictions of modified black bounce compact objects

Article References: Jumaniyozov, S., Rayimbaev, J., Sekhmani, Y., Palvanov, S., Tursunboyev, O., & Karshiev, D. (2026). Corrected thermodynamics and radiation predictions of modified black bounce compact objects. The European Physical Journal C, 86(10), Article 1241. https://doi.org/10.1140/epjc/s10052-026-16493-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16493-3

Keywords: black bounce, wormhole, MOG, Simpson–Visser, Hawking temperature, entropy corrections, accretion disk, Lyapunov exponent, phase transition, modified gravity, photon sphere, SV-MOG