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11 pages, 317 KiB

Open AccessArticle

Phenomenological Charged Extensions of the Quantum Oppenheimer–Snyder Collapse Model

by S. Habib Mazharimousavi

Universe 2025, 11(8), 257; https://doi.org/10.3390/universe11080257 - 4 Aug 2025

Viewed by 115

This work presents a semi-classical, quantum-corrected model of gravitational collapse for a charged, spherically symmetric dust cloud, extending the classical Oppenheimer–Snyder (OS) framework through loop quantum gravity effects. Our goal is to study phenomenological quantum modifications to geometry, without necessarily embedding them within [...] Read more.

This work presents a semi-classical, quantum-corrected model of gravitational collapse for a charged, spherically symmetric dust cloud, extending the classical Oppenheimer–Snyder (OS) framework through loop quantum gravity effects. Our goal is to study phenomenological quantum modifications to geometry, without necessarily embedding them within full loop quantum gravity (LQG). Building upon the quantum Oppenheimer–Snyder (qOS) model, which replaces the classical singularity with a nonsingular bounce via a modified Friedmann equation, we introduce electric and magnetic charges concentrated on a massive thin shell at the boundary of the dust ball. The resulting exterior spacetime generalizes the Schwarzschild solution to a charged, regular black hole geometry akin to a quantum-corrected Reissner–Nordström metric. The Israel junction conditions are applied to match the interior APS (Ashtekar–Pawlowski–Singh) cosmological solution to the charged exterior, yielding constraints on the shell’s mass, pressure, and energy. Stability conditions are derived, including a minimum radius preventing full collapse and ensuring positivity of energy density. This study also examines the geodesic structure around the black hole, focusing on null circular orbits and effective potentials, with implications for the observational signatures of such quantum-corrected compact objects. Full article

(This article belongs to the Special Issue Loop Quantum Gravity and Non-Perturbative Approaches to Quantum Cosmology, Second Edition)

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14 pages, 3378 KiB

Open AccessArticle

The pcGR Within the Hořava-Lifshitz Gravity and the Wheeler-deWitt Quantization

by Peter O. Hess, César A. Zen Vasconcellos and Dimiter Hadjimichef

Galaxies 2025, 13(4), 85; https://doi.org/10.3390/galaxies13040085 (registering DOI) - 1 Aug 2025

Viewed by 173

Abstract

We investigate pseudo-complex General Relativity (pcGR)—a coordinate-extended formulation of General Relativity (GR)—within the framework of Hořava-Lifshitz gravity, a regularized theory featuring anisotropic scaling. The pcGR framework bridges GR with modified gravitational theories through the introduction of a minimal length scale. Focusing on Schwarzschild [...] Read more.

We investigate pseudo-complex General Relativity (pcGR)—a coordinate-extended formulation of General Relativity (GR)—within the framework of Hořava-Lifshitz gravity, a regularized theory featuring anisotropic scaling. The pcGR framework bridges GR with modified gravitational theories through the introduction of a minimal length scale. Focusing on Schwarzschild black holes, we derive the Wheeler-deWitt equation, obtaining a quantized description of pcGR. Using perturbative methods and semi-classical approximations, we analyze the solutions of the equations and their physical implications. A key finding is the avoidance of the central singularity due to nonlinear interaction terms in the Hořava-Lifshitz action. Notably, extrinsic curvature (kinetic energy) contributions prove essential for singularity resolution, even in standard GR. Furthermore, the theory offers new perspectives on dark energy, proposing an alternative mechanism for its accumulation. Full article

(This article belongs to the Special Issue Cosmology and the Quantum Vacuum—2nd Edition)

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15 pages, 1420 KiB

Open AccessArticle

Spectral Dimensionality of Spacetime Around a Radiating Schwarzschild Black-Hole

by Mauricio Bellini, Juan Ignacio Musmarra, Pablo Alejandro Sánchez and Alan Sebastián Morales

Universe 2025, 11(8), 243; https://doi.org/10.3390/universe11080243 - 24 Jul 2025

Viewed by 143

Abstract

In this work we study the spectral dimensionality of spacetime around a radiating Schwarzschild black hole using a recently introduced formalism of quantum gravity, where the alterations of the gravitational field produced by the radiation are represented on an extended manifold, and describe [...] Read more.

In this work we study the spectral dimensionality of spacetime around a radiating Schwarzschild black hole using a recently introduced formalism of quantum gravity, where the alterations of the gravitational field produced by the radiation are represented on an extended manifold, and describe a non-commutative and nonlinear quantum algebra. The relation between classical and quantum perturbations of spacetime can be measured by the parameter

z \geq 0

. In this work we have found that when

z = (1 + \sqrt{3}) / 2 ≃ 1.3660

, a relativistic observer approaching the Schwarzschild horizon perceives a spectral dimension

N (z) = 4 [θ (z) - 1] ≃ 2.8849

, which is related to quantum gravitational interference effects in the environment of the black hole. Under these conditions, all studied Schwarzschild black holes with masses ranging from the Planck mass to

10^{46}

times the Planck mass present the same stability configuration, which suggests the existence of a universal property of these objects under those particular conditions. The difference from the spectral dimension previously obtained at cosmological scales leads to the conclusion that the spacetime dimensionality is scale-dependent. Another important result presented here is the fundamental alteration of the effective gravitational potential near the horizon due to Hawking radiation. This quantum phenomenon prevents the potential from diverging to negative infinity as the observer approaches the Schwarzschild horizon. Full article

(This article belongs to the Special Issue Extensions of General Relativity and Applications to Cosmology and Gravitation)

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12 pages, 5751 KiB

Open AccessArticle

Chaos of Charged Particles in Quadrupole Magnetic Fields Under Schwarzschild Backgrounds

by Qihan Zhang and Xin Wu

Universe 2025, 11(7), 234; https://doi.org/10.3390/universe11070234 - 16 Jul 2025

Viewed by 172

Abstract

A four-vector potential of an external test electromagnetic field in a Schwarzschild background is described in terms of a combination of dipole and quadrupole magnetic fields. This combination is an interior solution of the source-free Maxwell equations. Such external test magnetic fields cause [...] Read more.

A four-vector potential of an external test electromagnetic field in a Schwarzschild background is described in terms of a combination of dipole and quadrupole magnetic fields. This combination is an interior solution of the source-free Maxwell equations. Such external test magnetic fields cause the dynamics of charged particles around the black hole to be nonintegrable, and are mainly responsible for chaotic dynamics of charged particles. In addition to the external magnetic fields, some circumstances should be required for the onset of chaos. The effect of the magnetic fields on chaos is shown clearly through an explicit symplectic integrator and a fast Lyapunov indicator. The inclusion of the quadrupole magnetic fields easily induces chaos, compared with that of the dipole magnetic fields. This result is because the Lorentz forces from the quadrupole magnetic fields are larger than those from the dipole magnetic fields. In addition, the Lorentz forces act as attractive forces, which are helpful for bringing the occurrence of chaos in the nonintegrable case. Full article

(This article belongs to the Special Issue Celebrating the 110th Anniversary of General Relativity: Advances, Challenges and Perspectives)

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1 pages, 137 KiB

Open AccessCorrection

Correction: Bianconi, G. The Quantum Relative Entropy of the Schwarzschild Black Hole and the Area Law. Entropy 2025, 27, 266

by Ginestra Bianconi

Entropy 2025, 27(7), 724; https://doi.org/10.3390/e27070724 - 4 Jul 2025

Viewed by 198

Abstract

The new version of [...] Full article

(This article belongs to the Special Issue Entropy in Quantum Systems and Quantum Field Theory (QFT): Gravity, 3rd Edition)

14 pages, 796 KiB

Open AccessArticle

Tidal Forces Around Black-Bounce-Reissner–Nordström Black Hole

by Rashmi Uniyal

Universe 2025, 11(7), 221; https://doi.org/10.3390/universe11070221 - 2 Jul 2025

Viewed by 429

Abstract

The central singularity present in black hole (BH) spacetimes arising in the general theory of relativity (GR) can be avoided by using various methods. In the present work we have investigated the gravitational effect of one of such spacetime known as a black-bounce-Reissner–Nordström [...] Read more.

The central singularity present in black hole (BH) spacetimes arising in the general theory of relativity (GR) can be avoided by using various methods. In the present work we have investigated the gravitational effect of one of such spacetime known as a black-bounce-Reissner–Nordström spacetime. We revisited its horizon structure along with first integrals of its geodesic equations. We derived the expressions for Newtonian radial acceleration for freely infalling neutral test particles. For the description of tidal effects, the geodesic deviation equations are derived and solved analytically as well as numerically. To be specific, in the numerical approach, we have opted for two initial conditions to elaborate on the evolution of geodesic deviation vectors in radial and angular directions. The corresponding nature of geodesic deviation vectors in radial and angular directions is then compared with the standard results such as Schwarzschild and Reissner–Nordström BHs in order to figure out the differences. Full article

(This article belongs to the Special Issue Recent Advances in Gravitational Lensing and Galactic Dynamics)

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21 pages, 2916 KiB

Open AccessArticle

Reissner–Nordström and Kerr-like Solutions in Finsler–Randers Gravity

by Georgios Miliaresis, Konstantinos Topaloglou, Ioannis Ampazis, Nefeli Androulaki, Emmanuel Kapsabelis, Emmanuel N. Saridakis, Panayiotis C. Stavrinos and Alkiviadis Triantafyllopoulos

Universe 2025, 11(7), 201; https://doi.org/10.3390/universe11070201 - 20 Jun 2025

Viewed by 240

Abstract

In a previous study we investigated the spherically symmetric Schwarzschild and Schwarzschild–de Sitter solutions within a Finsler–Randers-type geometry. In this work, we extend our analysis to charged and rotating solutions, focusing on the Reissner–Nordström and Kerr-like metrics in the Finsler–Randers gravitational framework. In [...] Read more.

In a previous study we investigated the spherically symmetric Schwarzschild and Schwarzschild–de Sitter solutions within a Finsler–Randers-type geometry. In this work, we extend our analysis to charged and rotating solutions, focusing on the Reissner–Nordström and Kerr-like metrics in the Finsler–Randers gravitational framework. In particular, we extract the modified gravitational field equations and we examine the geodesic equations, analyzing particle trajectories and quantifying the deviations from their standard counterparts. Moreover, we compare the results with the predictions of general relativity, and we discuss how potential deviations from Riemannian geometry could be reached observationally. Full article

(This article belongs to the Special Issue Celebrating the 110th Anniversary of General Relativity: Advances, Challenges and Perspectives)

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13 pages, 243 KiB

Open AccessArticle

Complex Riemannian Spacetime and Singularity-Free Black Holes and Cosmology

by John W. Moffat

Axioms 2025, 14(6), 459; https://doi.org/10.3390/axioms14060459 - 12 Jun 2025

Viewed by 617

Abstract

An approach is presented to address singularities in general relativity using a complex Riemannian spacetime extension. We demonstrate how this method can be applied to both black hole and cosmological singularities, specifically focusing on the Schwarzschild and Kerr black holes and the Friedmann–Lemaître–Robertson–Walker [...] Read more.

An approach is presented to address singularities in general relativity using a complex Riemannian spacetime extension. We demonstrate how this method can be applied to both black hole and cosmological singularities, specifically focusing on the Schwarzschild and Kerr black holes and the Friedmann–Lemaître–Robertson–Walker (FLRW) Big Bang cosmology. By extending the relevant coordinates into the complex plane and carefully choosing integration contours, we show that it is possible to regularize these singularities, resulting in physically meaningful, singularity-free solutions when projected back onto real spacetime. The removal of the singularity at the Big Bang allows for a bounce cosmology. The approach offers a potential bridge between classical general relativity and quantum gravity effects, suggesting a way to resolve longstanding issues in gravitational physics without requiring a full theory of quantum gravity. Full article

(This article belongs to the Special Issue Complex Variables in Quantum Gravity)

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24 pages, 1157 KiB

Open AccessArticle

New Perspectives on the Irregular Singular Point of the Wave Equation for a Massive Scalar Field in Schwarzschild Space-Time

by Giampiero Esposito and Marco Refuto

Symmetry 2025, 17(6), 922; https://doi.org/10.3390/sym17060922 - 10 Jun 2025

Viewed by 397

Abstract

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, [...] Read more.

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, horizon and point at infinity, and then we relate the radial with the Heun equation, obtaining local solutions at the regular singular points. Moreover, we obtain the full asymptotic expansion of the local solution in the neighborhood of the irregular singular point at infinity. We also obtain and study the associated integral representation of the massive scalar field. Eventually, the technique developed for the irregular singular point is applied to the homogeneous equation associated with the inhomogeneous Zerilli equation for gravitational perturbations in a Schwarzschild background. Full article

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18 pages, 251 KiB

Open AccessArticle

Complex Riemannian Spacetime: Removal of Black Hole Singularities and Black Hole Paradoxes

by John W. Moffat

Axioms 2025, 14(6), 440; https://doi.org/10.3390/axioms14060440 - 4 Jun 2025

Viewed by 351

Abstract

An approach is presented to resolve key paradoxes in black hole physics through the application of complex Riemannian spacetime. We extend the Schwarzschild metric into the complex domain, employing contour integration techniques to remove singularities while preserving the essential features of the original [...] Read more.

An approach is presented to resolve key paradoxes in black hole physics through the application of complex Riemannian spacetime. We extend the Schwarzschild metric into the complex domain, employing contour integration techniques to remove singularities while preserving the essential features of the original solution. A new regularized radial coordinate is introduced, leading to a singularity-free description of black hole interiors. Crucially, we demonstrate how this complex extension resolves the long-standing paradox of event horizon formation occurring only in the infinite future of distant observers. By analyzing trajectories in complex spacetime, we show that the horizon can form in finite complex time, reconciling the apparent contradiction between proper and coordinate time descriptions. This approach also provides a framework for the analytic continuation of information across event horizons, resolving the Hawking information paradox. We explore the physical interpretation of the complex extension versus its projection onto real spacetime. The gravitational collapse of a dust sphere with negligible dust is explored in the complex spacetime extension. The approach offers a mathematically rigorous framework for exploring quantum gravity effects within the context of classical general relativity. Full article

(This article belongs to the Special Issue Complex Variables in Quantum Gravity)

24 pages, 541 KiB

Open AccessArticle

New Black Hole Solution in f(R) Theory and Its Related Physics

by G. G. L. Nashed and Ali Eid

Universe 2025, 11(6), 175; https://doi.org/10.3390/universe11060175 - 30 May 2025

Cited by 1 | Viewed by 1253

Abstract

Recent observations suggest that General Relativity (GR) faces challenges in fully explaining phenomena in regimes of strong gravitational fields. A promising alternative is the

f (R)

theory of gravity, where R denotes the Ricci scalar. This modified theory aims to address [...] Read more.

Recent observations suggest that General Relativity (GR) faces challenges in fully explaining phenomena in regimes of strong gravitational fields. A promising alternative is the

f (R)

theory of gravity, where R denotes the Ricci scalar. This modified theory aims to address the limitations observed in standard GR. In this study, we derive a black hole (BH) solution without introducing nonlinear electromagnetic fields or imposing specific constraints on R or the functional form of

f (R)

gravity. The BH solution obtained here is different from the classical Schwarzschild solution in GR and, under certain conditions, reduces to the Schwarzschild (A)dS solution. This BH is characterized by the gravitational mass of the system and an additional parameter, which distinguishes it from GR BHs, particularly in the asymptotic regime. We show that the curvature invariants of this solution remain well defined at both small and large values of r. Furthermore, we analyze their thermodynamic properties, demonstrating consistency with established principles such as Hawking radiation, entropy, and quasi-local energy. This analysis supports their viability as alternative models to classical GR BHs. Full article

(This article belongs to the Special Issue Origins and Natures of Inflation, Dark Matter and Dark Energy, 2nd Edition)

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16 pages, 1077 KiB

Open AccessArticle

Non-Singular “Gauss” Black Hole from Non-Locality

by Jens Boos

Universe 2025, 11(4), 112; https://doi.org/10.3390/universe11040112 - 29 Mar 2025

Cited by 1 | Viewed by 309

Abstract

Cutting out an infinite tube around

r = 0

formally removes the Schwarzschild singularity, but without a physical mechanism, this procedure seems ad hoc and artificial. In this paper, we provide justification for such a mechanism by means of non-locality. Motivated by the [...] Read more.

Cutting out an infinite tube around

r = 0

formally removes the Schwarzschild singularity, but without a physical mechanism, this procedure seems ad hoc and artificial. In this paper, we provide justification for such a mechanism by means of non-locality. Motivated by the Gauss law, we define a suitable radius variable as the inverse of a regular non-local potential, and use this variable to model a non-singular black hole. The resulting geometry has a de Sitter core, and for generic values of the regulator, there is no inner horizon, saving this model from potential issues via mass inflation. An outer horizon only exists for masses above a critical threshold, thereby reproducing the conjectured “mass gap” for black holes in non-local theories. The geometry’s density and pressure terms decrease exponentially, thereby rendering it an almost-exact vacuum solution of the Einstein equations outside of astrophysical black holes. Its thermodynamic properties resemble those of the Hayward black hole, with the notable exception that for critical mass, the horizon radius is zero. Full article

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25 pages, 3100 KiB

Open AccessArticle

Black Holes and de Sitter Space as Time Mirrors

by Anatoly Svidzinsky

Universe 2025, 11(4), 109; https://doi.org/10.3390/universe11040109 - 25 Mar 2025

Viewed by 682

Abstract

It is usually assumed that matter disappears together with the spacetime at the center of a Schwarzschild black hole (BH). Here, we find that if we impose a boundary condition that the field does not disappear at the BH center (that is, field [...] Read more.

It is usually assumed that matter disappears together with the spacetime at the center of a Schwarzschild black hole (BH). Here, we find that if we impose a boundary condition that the field does not disappear at the BH center (that is, field flux into the singularity vanishes), the BH acts as a time mirror that totally reflects the infalling light and matter outside the BH. Namely, the reflected field propagates backward in time, passes the event horizon and moves away from the BH. In this case, a BH can be used as a time machine that allows us to send a signal into the past. We also show that de Sitter spacetime acts as a time mirror provided particles do not disappear from the spacetime at

r = \infty

. Full article

(This article belongs to the Collection Open Questions in Black Hole Physics)

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10 pages, 586 KiB

Open AccessArticle

The Quantum Relative Entropy of the Schwarzschild Black Hole and the Area Law

by Ginestra Bianconi

Entropy 2025, 27(3), 266; https://doi.org/10.3390/e27030266 - 4 Mar 2025

Cited by 1 | Viewed by 1575 | Correction

Abstract

The area law obeyed by the thermodynamic entropy of black holes is one of the fundamental results relating gravity to statistical mechanics. In this work, we provide a derivation of the area law for the quantum relative entropy of the Schwarzschild black hole [...] Read more.

The area law obeyed by the thermodynamic entropy of black holes is one of the fundamental results relating gravity to statistical mechanics. In this work, we provide a derivation of the area law for the quantum relative entropy of the Schwarzschild black hole for an arbitrary Schwarzschild radius. The quantum relative entropy between the metric of the manifold and the metric induced by the geometry and the matter field has been proposed in G. Bianconi as the action for entropic quantum gravity leading to modified Einstein equations. The quantum relative entropy generalizes Araki’s entropy and treats the metrics between zero-forms, one-forms, and two-forms as quantum operators. Although the Schwarzschild metric is not an exact solution of the modified Einstein equations of the entropic quantum gravity, it is an approximate solution valid in the low-coupling, small-curvature limit. Here, we show that the quantum relative entropy associated to the Schwarzschild metric obeys the area law for a large Schwarzschild radius. We provide a full statistical mechanics interpretation of the results. Full article

(This article belongs to the Special Issue Entropy in Quantum Systems and Quantum Field Theory (QFT): Gravity, 3rd Edition)

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15 pages, 290 KiB

Open AccessArticle

Quasi-Homogeneous Black Hole Thermodynamics in Non-Commutative Geometry

by Hernando Quevedo and María N. Quevedo

Universe 2025, 11(3), 79; https://doi.org/10.3390/universe11030079 - 27 Feb 2025

Viewed by 657

Abstract

We study the thermodynamic properties of a black hole that takes into account the effects of non-commutative geometry. To emphasize the role of new effects, we have chosen a specific modified Schwarzschild black hole inspired by non-commutative geometry. We show that, in order [...] Read more.

We study the thermodynamic properties of a black hole that takes into account the effects of non-commutative geometry. To emphasize the role of new effects, we have chosen a specific modified Schwarzschild black hole inspired by non-commutative geometry. We show that, in order to apply the laws of quasi-homogeneous thermodynamics using the formalism of geometrothermodynamics, it is necessary to consider the non-commutative parameter as an independent thermodynamic variable. As a result, the properties of the black hole change drastically, leading to phase transitions that are directly related to the value of the non-commutative parameter. We also prove that an unstable commutative black hole can become stable in non-commutative geometry for particular values of the non-commutative parameter. Full article

(This article belongs to the Collection Open Questions in Black Hole Physics)

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