Ghosh-Kumar Black Holes: Regular Deformations

• Ghosh-Kumar black holes are regularized solutions that modify classical metrics by integrating scalar, electromagnetic, and higher-curvature deformations.
• They employ methods such as nonlinear electrodynamics and exponential mass-function corrections to adjust shadow sizes, photon rings, and geodesic structures.
• Their unique optical signatures and accretion dynamics offer actionable predictions for high-resolution observations like those from the Event Horizon Telescope.

The Ghosh-Kumar black hole designates a family of regular or deformed black hole solutions that extend the general relativistic description of compact objects by incorporating additional scalar, electromagnetic, or higher-curvature fields. These geometries frequently exploit nonlinear electrodynamics (NED), scalar field (e.g., quintessence), or higher-derivative corrections—each introducing a distinct deformation to the “vacuum” solutions. These models possess nonsingular cores (i.e., regular at $r=0$), feature additional parameters characterizing the deformation or “hair,” and have been studied extensively for their impact on geodesics, photon rings, shadow images, and accretion phenomena. They have also become central objects for theoretical investigation by combining the mathematical tractability of Kerr(-like) metrics with extensions motivated by cosmological observations and quantum gravity considerations.

1. Metric Structures and Defining Parameters

Variants termed “Ghosh-Kumar black holes” share the property of regularizing the classical singularity at $r=0$ via a deformation of the mass function or via modified lapse functions. Key representative metrics include:

• Magnetically Charged NED Deformation (Minimal Deformation Model):

$$f^{(\mathrm{GK})}(r) = 1 - \frac{2M}{\sqrt{r^2 + a^2}}$$

where $a$ encodes the NED-induced magnetic charge. The event horizon lies at $r_h^{(\mathrm{GK})} = \sqrt{4M^2 - a^2}$; for $a \rightarrow 0$, Schwarzschild is recovered. This structure yields a contracted event horizon and photon sphere relative to Schwarzschild, with all curvature invariants regular at $r=0$ (Gong et al., 28 Aug 2025).

• Kerr-Like Regular Rotators:

$$ds^2 = -\left[1 - \frac{2m(r)r}{\Sigma}\right] dt^2 - \frac{4 a m(r) r \sin^2\theta}{\Sigma} dt d\phi + \Sigma\left(\frac{dr^2}{\Delta} + d\theta^2\right) + \frac{\mathcal{A}}{\Sigma}\sin^2\theta d\phi^2$$

with $m(r) = M e^{-k/r}$, $\Sigma = r^2 + a^2 \cos^2\theta$, $\Delta = r^2 + a^2 - 2 m(r) r$, and $$\mathcal{A} = (r^2 + a^2)^2 - a^2\Delta \sin^2\theta$$. The parameter $k > 0$ controls the deviation from Kerr and regularizes curvature (Fauzi et al., 10 Mar 2025, Alam et al., 26 Apr 2024).

• Kerr-Like Black Hole Embedded in Quintessence:

$$\Delta = r^2 + a^2 + 2Mr - \frac{\mathcal{A}_q}{r^{3\omega_q - 1}}$$

where $(\mathcal{A}_q,\omega_q)$ parametrize the strength and equation of state of a quintessence field. Here, $a$ is the spin (as in Kerr), and the quintessence term can cause repulsive or attractive modifications depending on $\omega_q$ (Sarkar et al., 2019).

These metrics admit the separation of the geodesic equations and support the analysis of photon orbits, shadows, and accretion models.

2. Geodesics, Spherical Orbits, and Stability Analysis

The geodesic structure of Ghosh-Kumar black holes mirrors that of the Kerr and Kerr-Newman geometries but includes distinctive features arising from the additional parameters (such as $k$ or $a$ representing NED charge). For instance, considering the rotating regular metric $m(r) = M e^{-k/r}$:

• The radial and polar geodesic equations result from the Hamilton–Jacobi separation, admitting Carter’s constant and a clear analytic treatment (notably via the Mino time formalism)
• Spherical orbits (constant $r$) are obtained by solving $R(r) = 0$ and $dR/dr = 0$, leading to analytical solutions involving Jacobi elliptic functions.
• As $k$ increases, the amplitude of latitudinal (polar) oscillations of spherical photon orbits increases, altering the observable structure of photon trajectories, rings, and shadows.

Stability: For null (photon) orbits, all spherical photon orbits outside the horizon are unstable ($d^2R/dr^2 > 0$), aligning with the general behavior in the Kerr family. The innermost stable spherical orbits (ISSOs) demonstrate model-dependent branching structures: in Ghosh geometries, branches can become complex at finite Carter constant values due to the transcendental structure of the mass function, in contrast to the merging observed in Kerr-Newman black holes (Alam et al., 26 Apr 2024).

3. Optical Appearance: Shadows, Photon Rings, and Accretion Disk Imaging

The appearance of Ghosh-Kumar black holes, as elucidated via backward ray-tracing simulations and analytic shadow computation, differs predictably from that of Kerr and standard Schwarzschild black holes:

• Shadow Structure: The shadow’s outline is sensitive to the spin $a$, deformation/magnetic charge $q$ (or $a^{(\mathrm{GK})}$), and observer inclination $\theta_\text{obs}$. Increasing $a$ shifts and deforms (flattens) the shadow; increasing $q$ or $a^{(\mathrm{GK})}$ reduces the shadow size and induces “oval” distortions. At high deformation/magnetic charge, the Einstein ring transitions from a full circle to an arc, and the center of the shadow is displaced (Yang et al., 18 Nov 2024, Gong et al., 28 Aug 2025).
• Photon and Lensed Rings: In the thin disk models, the widths of photon and lensed rings in Ghosh-Kumar black holes are typically broader than in Schwarzschild and quantum-corrected Kazakov–Solodukhin (KS) metrics. For these NED or regular configurations, the direct emission from the disk dominates the brightness, while higher-order rings are enhanced in width but not in intensity.
• Accretion Disk Appearance: The projected accretion disk image is subject to both gravitational redshift and Doppler effects, producing structure that transitions from disk-like to hat-like as the inclination increases. Prograde disk flows amplify the blue-shifted side, whereas retrograde flows are dimmer and less centrally peaked. The redshift/blueshift distribution across the observer’s screen can be substantially altered by varying $a$ and $q$.

The size of the shadow can be directly linked to the event horizon and photon sphere radii: for the GK deformation, both shrink with increasing $a^{(\mathrm{GK})}$, with the critical impact parameter $b_p$ correspondingly reduced. These features are independent of the details of the accretion model for the central shadow radius (Gong et al., 28 Aug 2025).

4. Pseudo-Newtonian Potentials, Effective Forces, and Accretion Physics

Pseudo-Newtonian potentials (PNPs) have been developed for Ghosh-Kumar black holes to describe particle motion and accretion disk dynamics without resorting to full general relativity:

• The method generalizes the Paczyński–Wiita and Mukhopadhyay approaches to rotating, deformed backgrounds, incorporating an effective potential derived from the Lagrangian formulation in the symmetry plane ($\theta = \pi/2$) (Sarkar et al., 2019).
• The PNPs reproduce general relativistic results for binding energies and angular momenta with less than $4.95\%$ error and can accommodate the influence of dark energy (for example, via a quintessence field).
• For configurations with significant quintessence or NED charge, the effective gravitational force as seen by disk matter may become repulsive at large radii or for low rotation, implying both astrophysical phenomenology (modified disk structure, new stability points) and links to cosmic acceleration.

In higher-curvature gravity (e.g., Einstein–Gauss–Bonnet), the inclusion of coupling parameters (e.g., $\alpha$ in 4D EGB) further shifts the innermost stable circular orbit (ISCO) inward (for positive $\alpha$), increasing disk temperature and radiative efficiency compared to Kerr black holes, and ultimately altering the observed spectra and luminosity profiles (Heydari-Fard et al., 2021).

5. Comparison with Kerr, Kerr-Newman, and Other Deformations

Contrasting Ghosh-Kumar black holes with other black hole families reveals critical differences:

Geometry	Horizon/Photon Sphere Shift	Shadow Size vs. Schwarzschild	Ring Width (Thin Disk)	Core Regularity	Deformation Parameter
Schwarzschild	Baseline	Baseline	Baseline	Singular at $r=0$	—
Kerr	$a$ shifts ISCO in/out	Slight decrease/increase with $a$	Slightly broadened	Singular at $r=0$	$a$ (spin)
Kerr-Newman	$Q$ pushes radii outward	Larger ($Q^2$ effect)	Similar to Kerr	Singular at $r=0$	$Q$ (charge)
Kazakov–Solodukhin (KS)	All increase	Larger	Narrower	Regular	$a^{(\mathrm{KS})}$
Ghosh-Kumar (GK), NED	All decrease	Smaller	Broader	Regular	$a^{(\mathrm{GK})}$
Ghosh-Kumar, Exponential	All decrease as $k$ increases	Smaller	Enhanced oscillations	Regular	$k$

The key optical effect is that the Ghosh-Kumar solutions with “magnetic hair” or exponential regularization reduce the observable shadow and photon sphere compared to both Schwarzschild and KS solutions, while modifying the width/intensity of photon/lensed rings.

6. Astrophysical and Observational Constraints

The phenomenology of the Ghosh-Kumar black hole has direct implications for tests with current or planned astronomical instrumentation:

• Event Horizon Telescope (EHT): By matching the observed shadow size to model-derived $b_p$ values for Sagittarius A*, the NED charge $a^{(\mathrm{GK})}$ is constrained to $a^{(\mathrm{GK})}/M \lesssim 1.38$ ($1\sigma$), while the KS parameter is more tightly constrained ($a^{(\mathrm{KS})}/M \lesssim 0.23$) (Gong et al., 28 Aug 2025).
• Shadow and Intensity Mapping: The smaller (for GK) or larger (for KS) shadow radius, and the broader (GK) or narrower (KS) photon/lensed rings, enable the discrimination of these deformations at high angular resolution.
• Accretion Disk Imaging: The inner edge location (ISCO or photon sphere) shifts, spectral hardening, and brightness trends are diagnostic of the underlying metric and deformation parameters.
• Superspinars and Transition Phenomena: Regular black holes with spin exceeding the extremal limit ($a > a_c$) transition into “superspinars,” lacking horizons. The timescale for the appearance of new photon ring features after horizon destruction can be $\sim$ two weeks for M87*-mass black holes, rendering this process observable with future high-cadence imaging (Fauzi et al., 10 Mar 2025).

7. Theoretical Extensions, Quantum Gravity, and Stability

Ghosh-Kumar black holes also appear in frameworks including quadratic gravity and generalized uncertainty principle corrections:

• In higher-derivative (quadratic) gravity, the Hawking evaporation process is modified. With suitable tuning of the free parameters and GUP parameter $\beta$, the evaporation stops at a critical mass, forming absolutely stable remnants with $M_0 \sim \mathcal{O}(\mathrm{few}) m_p$—in contrast to the standard infinite temperature endpoint of Schwarzschild evaporation. These models are considered as potential resolutions to the information loss problem and as dark matter candidates (Kuntz et al., 2019).
• Quintessence or dark energy fields included in the metric (e.g., via $\mathcal{A}_q/r^{3\omega_q-1}$) cause novel modifications to particle and photon dynamics: for slowly rotating or non-rotating objects, the gravitational attraction can turn into repulsion at large radii, a reflection of cosmic acceleration induced by dark energy. This suggests an intriguing link between local strong-field phenomena and cosmological behavior (Sarkar et al., 2019).

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The Ghosh-Kumar black hole paradigm demonstrates that regularized and deformed compact object metrics, characterized by additional “hair” and alternative source fields, yield testable predictions for shadows, photon rings, accretion dynamics, and horizon phenomenology. Their distinctive optical and astrophysical signatures serve as frameworks both for probing the structure of gravity in the strong-field regime and for constraining extensions to general relativity through ever-improving astronomical data.