⚛️ general relativity

Generalized Regular Black Holes with Tunable Cores and Geodesic Properties

This paper introduces a unified family of static, spherically symmetric regular black hole geometries characterized by a tunable core index and deformation parameter, systematically analyzing their horizon thermodynamics, null and timelike geodesic properties, and gravitational lensing effects to explore how core regularization and exterior deformations influence black hole behavior.

Original authors: Hassan Hassanabadi

Published 2026-09-23

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Hassan Hassanabadi

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 heart of every black hole, according to the standard rules of gravity, lies a point of infinite density where the laws of physics break down. This "singularity" is a mathematical dead end, a place where the smooth fabric of space and time tears apart. For decades, physicists have wondered if this tear is real or merely a sign that our current theories are incomplete. One promising idea is that the singularity might be replaced by a smooth, ultra-dense region that avoids the infinite crash, creating what are known as "regular black holes." These objects look like normal black holes from a distance, with an event horizon that traps light, but their centers are finite and well-behaved. The question is no longer just whether such objects can exist, but how their hidden internal structure affects the world we can observe, from the heat they emit to the paths light takes as it skims past them.

A new study by physicist Hassan Hassanabadi introduces a flexible family of these regular black holes designed to test exactly how the center of the object shapes its behavior. Instead of assuming a single, fixed type of core, the researcher built a model where the center can be tuned like a dial. In this family of solutions, the center can be a dense, ball-like region similar to the inside of a star, or it can be a depleted zone where the density drops to zero right at the middle, creating a Minkowski-like center with vanishing curvature. The model also includes a separate control that changes how the black hole's gravity fades away at large distances, mimicking the effect of an electric charge without actually requiring the black hole to be charged. By adjusting these two knobs independently, the study maps out how a changing interior and a shifting exterior influence the black hole's temperature, its stability, and the shadows it casts against the background of the universe.

The researchers began by constructing a mathematical description of the black hole's interior, starting with a specific pattern for how matter is distributed. They found that by changing a single integer value in their equations, they could switch between a core that has a solid, finite density at the very center and cores that become increasingly depleted as you move toward the middle. For the first type, the center behaves like a tiny, smooth bubble of space that pushes outward, preventing a collapse. For the higher types, the density vanishes completely at the center, creating a region that is surrounded by a ring of matter before fading out. This ability to dial the "emptiness" of the core allowed the team to see how the internal structure dictates the black hole's overall size and shape. They discovered that as the core becomes more depleted, the black hole requires a larger mass to maintain a stable event horizon. If the core is too empty relative to the total mass, the horizon disappears, and the object ceases to be a black hole at all.

This relationship between the core and the horizon has direct consequences for the black hole's temperature. Just as a star cools down as it expands, these regular black holes have a specific temperature that depends on the size of their event horizon. The study shows that black holes with more depleted, hollowed-out centers have a lower maximum temperature than those with dense, solid cores. Furthermore, the point at which the black hole becomes "extremal"—meaning it is as cold as possible and has a single, merged horizon—shifts outward as the core empties. This means that the internal architecture of the black hole leaves a clear thermal fingerprint. The researchers also derived a precise thermodynamic rule for these objects, showing that the energy contained within the horizon is balanced by the temperature and the pressure of the matter at the edge, a relationship that holds true even when the matter outside the horizon is taken into account.

Beyond heat and size, the study explored how light and matter move around these objects. When light passes a black hole, it can get trapped in a circular orbit just outside the event horizon, creating a bright ring that defines the edge of the black hole's shadow. The researchers calculated the size of this shadow and found that while the core's emptiness changes the shadow's size, it does so in a subtle, quadratic way. However, a different parameter in their model, which controls how the gravity behaves far away, changes the shadow size much more dramatically and immediately. This distinction allows astronomers to tell the difference between a change in the core's structure and a change in the distant gravitational field. More importantly, the team analyzed how unstable these light orbits are. They found that while the size of the shadow might look similar for different core types, the speed at which light circles the black hole and the speed at which it falls away from that orbit change in opposite directions. A black hole with a more depleted core causes light to orbit faster, yet the orbit becomes less unstable relative to its orbital period, meaning the light stays in the ring longer per cycle before falling in. This ratio of orbital speed to instability is a new, observable signature that cannot be deduced from the shadow's size alone.

The study also looked at how massive particles, like stars or gas clouds, move in these gravitational fields. They calculated the closest safe orbit a particle can have before it inevitably spirals into the black hole, known as the innermost stable circular orbit. They found that the position of this orbit shifts depending on the core's structure and the distant gravitational field, but again, the two factors leave different marks. The changes caused by the distant field appear immediately and linearly, while the changes caused by the core's internal emptiness appear more slowly. This separation of effects suggests that by carefully measuring the orbits of stars or the light from gas clouds near a black hole, scientists could potentially distinguish between a black hole with a solid core and one with a hollow center, even if they look identical from a great distance.

Ultimately, this work provides a unified framework for understanding how the hidden interior of a black hole influences the observable universe. It demonstrates that the "regular" nature of these objects is not a single, fixed state but a spectrum of possibilities. By separating the effects of the core from the effects of the distant gravity, the study offers a clear path for future observations. It suggests that the next generation of black hole imaging and gravitational wave detection could do more than just confirm the existence of event horizons; they could potentially probe the very nature of the matter that prevents the singularity from forming. The model remains a theoretical construction based on effective physics rather than a specific microscopic theory of matter, but it serves as a precise map for what to look for. It shows that the universe may offer us a way to see the invisible, not by looking directly at the center, but by watching how the light and matter around it dance to the tune of a hidden, tunable core.

Technical Summary: Generalized Regular Black Holes with Tunable Cores and Geodesic Properties

Problem and Motivation
Regular black holes (RBHs) serve as phenomenological models to investigate whether the central singularities of classical Schwarzschild and Kerr geometries can be replaced by nonsingular, high-curvature regions while preserving standard black-hole behavior at large distances. While previous constructions have explored de Sitter cores (e.g., Bardeen, Hayward) and asymptotically Minkowski cores, a systematic framework for continuously or discretely controlling the central structure independently of the exterior deformation has been lacking. Existing models often conflate parameters governing the core's regularization with those controlling the asymptotic falloff. This paper addresses the need to isolate these effects to understand how specific interior modifications propagate to horizon structure, thermodynamics, and geodesic observables.

Methodology
The author constructs a family of static, spherically symmetric regular black-hole geometries using a "source-driven" strategy within Einstein gravity (geometrized units $G=c=\hbar=k_B=1$ ). Rather than prescribing the lapse function directly, the model begins with a normalized, positive effective energy density profile $\rho(r)$ characterized by three parameters:

Discrete core index ( $n$ ): Controls the order of central depletion.
Regularization scale ( $a$ ): Sets the characteristic length of the core.
Dimensionless deformation parameter ( $\eta$ ): Governs the asymptotic falloff of the source.

The density profile is defined as:
$\rho(r) = \frac{M(2n + 1)a (a + \eta s) r^{2n-2}}{4\pi s (s + \eta a)^{2n+2}}, \quad s(r) = \sqrt{r^2 + a^2}$
where $M$ is the total ADM mass. The $tt$-component of Einstein's equations is integrated to derive the Misner–Sharp mass function $m(r)$ and the lapse function $f(r) = 1 - 2m(r)/r$ . The remaining Einstein equations determine the anisotropic pressures ( $p_r, p_t$ ), ensuring the conservation equation $\nabla_\mu T^{\mu r} = 0$ is satisfied.

The study systematically analyzes:

Regularity and Energy Conditions: Examining curvature invariants and the Null, Weak, Strong, and Dominant Energy Conditions (NEC, WEC, SEC, DEC).
Horizon Structure: Solving for event horizons, extremality conditions, and the parameter space constraints on $n$ .
Thermodynamics: Deriving Hawking temperature, entropy, and a horizon thermodynamic identity derived from the radial Einstein equation.
Geodesic Dynamics: Analyzing null geodesics (photon sphere, shadow radius, critical impact parameter, Lyapunov exponent, orbital frequency) and timelike geodesics (marginally bound orbits, innermost stable circular orbits/ISCO).
Weak Lensing: Calculating deflection angles and magnification using a sector-isolated expansion of the asymptotic metric.

Key Contributions and Results

Unified Framework for Core and Exterior:
The model successfully decouples the central geometry from the exterior deformation.
- Core Control ( $n$ ): For $n=1$ , the geometry possesses a finite-density de Sitter-type core. For $n > 1$ , the central density vanishes ( $\rho(0)=0$ ), creating progressively depleted, Minkowski-like cores where curvature invariants vanish at the center.
- Exterior Control ( $\eta$ ): The parameter $\eta$ modifies the asymptotic decay of the density from $r^{-5}$ (Bardeen-like, $\eta=0$ ) to $r^{-4}$ ( $\eta > 0$ ). This generates an effective asymptotic charge scale $Q_{eff}^2 = 2M(2n+1)\eta a$ , introducing Reissner–Nordström (RN)-like corrections to the metric without requiring a full RN solution.
Horizon Existence Constraints:
Regularity alone does not guarantee a black hole for arbitrary $n$ . The existence of an event horizon imposes an upper bound $n_{max}$ on the core index for fixed $M/a$ and $\eta$ . The extremal mass-to-length ratio $(M/a)_{ext}$ grows monotonically with $n$ and steepens significantly with $\eta$ . Consequently, highly depleted cores ( $n \gg 1$ ) require larger mass-to-scale ratios to maintain a black-hole configuration.
Thermodynamic Properties:
- Temperature: The Hawking temperature $T_H$ vanishes at the extremal radius. Increasing $n$ shifts the extremal configuration to larger radii and lowers the peak temperature.
- Entropy: Follows the standard Bekenstein–Hawking area law $S = \pi r_+^2$ .
- Thermodynamic Identity: The author derives a local horizon identity $dE_+ = T_H dS - P_+ dV_+$ , where $E_+ = r_+/2$ is the quasi-local horizon energy and $P_+ = p_r(r_+)$ is the radial pressure. This is distinct from the global ADM mass variation, which includes contributions from the effective source outside the horizon.
Geodesic and Optical Observables:
- Shadow and Photon Sphere: Analytic weak-deformation approximations reveal that for $\eta=0$ (pure core deformation), corrections to the photon radius $r_{ph}$ and critical impact parameter $b_c$ appear at order $(a/M)^2$ . For $\eta > 0$ , corrections appear at linear order $(a/M)$ , mimicking the RN behavior.
- Instability vs. Frequency: A novel finding is the decoupling of the photon-orbit angular frequency $\Omega_{ph}$ $Ω_{p h}$ and the Lyapunov exponent $\lambda_{ph}$ $λ_{p h}$ . The ratio $R_{ph} = \lambda_{ph}/\Omega_{ph}$ $R_{p h} = λ_{p h} / Ω_{p h}$ , which measures instability relative to orbital motion, varies non-trivially with $n$ $n$ and $\eta$ $η$ .
  - Increasing $n$ (with $\eta=0$ ) increases $\Omega_{ph}$ but decreases $\lambda_{ph}$ , reducing $R_{ph}$ .
  - Increasing $\eta$ (with $n=1$ ) significantly increases $\Omega_{ph}$ while $\lambda_{ph}$ changes mildly, also reducing $R_{ph}$ .
- Timelike Geodesics: The marginally bound orbit ( $r_{mb}$ ) and ISCO ( $r_{ISCO}$ ) recover Schwarzschild values ( $4M, 6M$ ) as $a \to 0$ . The leading shifts are linear in $\eta$ (RN-like) and quadratic in $a/M$ (Bardeen-like), with nonlinear mixing terms proportional to $\eta^2 (a/M)^2$ .
Weak Lensing:
The deflection angle is decomposed into a Schwarzschild baseline, an RN-like term proportional to $Q_{eff}^2/b^2$ , and a core-sensitive term proportional to $C_3/b^3$ . This sector-isolated expansion highlights how different physical mechanisms (asymptotic charge vs. core structure) contribute to lensing observables.

Significance and Claims
The paper claims that the primary significance of this work lies not merely in constructing another regular metric, but in providing a controlled phenomenological framework to distinguish between the effects of the central regularization mechanism and the asymptotic exterior deformation.

Differentiation of Effects: The model demonstrates that the "core" ( $n$ ) and "exterior" ( $\eta$ ) parameters affect observables differently. For instance, the order of the leading correction to the shadow radius distinguishes between a pure core deformation (quadratic) and an exterior deformation (linear).
Dynamic Separation: The analysis of the ratio $R_{ph} = \lambda_{ph}/\Omega_{ph}$ provides information not contained in the shadow radius alone, offering a potential observational signature to distinguish between different regular-core profiles even if their asymptotic behaviors are similar.
Physical Constraints: The work establishes that horizon existence is a dynamical constraint on the core index, implying that highly depleted cores are thermodynamically and geometrically distinct from finite-density cores.

The author concludes that while the effective stress tensor is treated phenomenologically, the model offers a unified basis for investigating how tunable regular cores and RN-like exteriors influence black-hole thermodynamics and geodesic properties, setting the stage for future investigations into microscopic realizations and quasinormal mode spectra.

Technical Summary: Generalized Regular Black Holes with Tunable Cores and Geodesic Properties

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