Frolov Black Hole: Regular Charged Geometry

• Frolov black hole is a regular, charged spacetime model that replaces the central singularity with a finite de Sitter-like core controlled by a length parameter.
• It extends both Hayward and Reissner–Nordström solutions, showing modified thermodynamics, shadow properties, and stability features in its geometric and causal structure.
• Analyses of geodesics, wave scattering, and quasinormal modes provide practical insights into gravitational lensing, phase transitions, and evaporation in regular black hole scenarios.

Searching arXiv for papers on the Frolov black hole and closely related variants. arXiv search query: "Frolov black hole regular Reissner-Nordström" The Frolov black hole is a static, spherically symmetric, charged regular black-hole geometry that can be viewed both as a charged extension of the Hayward black hole and as a regularization of the Reissner–Nordström spacetime. Its defining feature is the replacement of the central curvature singularity by a finite core controlled by a length parameter denoted in the literature by $\alpha_0$, $g$, $\ell$, $l$, or $\alpha$. In the asymptotically flat case it reduces to Reissner–Nordström when the regularization scale is sent to zero, while several recent works have examined its thermodynamics, null and timelike geodesics, shadows, quasinormal spectra, wave scattering, and extensions by quintessence, topological defects, string clouds, and AdS asymptotics (Gohain et al., 2024, Tang et al., 27 Jan 2026, Sueto et al., 2023).

1. Geometric definition and regular core

In Schwarzschild-like coordinates, the standard Frolov metric is written as

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$

with

$$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$

Equivalent notations used in the literature replace $q$ by $Q$ and $\alpha_0$ by $g$0, $g$1, $g$2, or $g$3 (Song et al., 2024, Kala et al., 25 Mar 2025).

The parameters are the ADM mass $g$4, an electric-charge parameter $g$5 or $g$6, and the regularization scale $g$7. In the limit $g$8, the lapse function reduces to

$g$9

so the geometry returns to Reissner–Nordström; if in addition $\ell$0, one recovers Schwarzschild. For $\ell$1, the metric reduces to a neutral Hayward/Frolov black hole (Gohain et al., 2024, Song et al., 2024).

Its regularity is encoded in the small-$\ell$2 behavior. One analysis gives

$\ell$3

while another describes the central region as a de Sitter-like core with effective cosmological constant $\ell$4. These statements are consistent with the absence of a central curvature singularity for $\ell$5 (Tang et al., 27 Jan 2026, Song et al., 2024).

In the black-hole branch, one paper states the bound

$\ell$6

beyond which horizons disappear. This bound is used repeatedly in analyses of perturbations and quintessence-dressed solutions (Song et al., 2024, Gohain et al., 2024).

2. Horizons, causal structure, and deformations of the lapse function

For the asymptotically flat Frolov geometry, horizons are the positive real roots of

$\ell$7

No simple closed-form solution exists in general. In the $\ell$8 limit, the roots recover the Reissner–Nordström radii

$\ell$9

Numerically, increasing $l$0 or $l$1 tends to shrink the outer horizon $l$2 while pushing the inner horizon outward (Gohain et al., 2024, Ahmed et al., 17 Apr 2025).

A widely studied generalization adds a Kiselev-type quintessence contribution,

$l$3

where $l$4 is the quintessence normalization and $l$5 is its equation-of-state parameter. In that case the positive real roots of $l$6 can include an inner Cauchy horizon $l$7, a black-hole horizon $l$8, and a cosmological-type horizon $l$9. Numerically, increasing $\alpha$0 tends to enlarge the outermost root $\alpha$1 (Gohain et al., 2024, Gohain et al., 2024).

The regular core does not by itself fix the global causal structure under dynamical evolution. In the charged Vaidya implementation of the Hayward–Frolov scenario, two regularization schemes were examined. For the constant-scale regularization,

$\alpha$2

all curvature invariants remain finite at $\alpha$3 throughout collapse and evaporation. Numerical integration of outgoing null geodesics then yields a stronger statement: no outgoing null geodesic remains trapped forever, so no event horizon forms, and the final causal diagram is topologically the same as Minkowski space with a temporary trapped region (Sueto et al., 2023).

This places the Frolov geometry at the intersection of two research programs: regular black-hole model building and causal alternatives to classical evaporating black holes. A plausible implication is that the near-core regularization is being used not merely as a local curvature modification, but as an organizing principle for globally nonsingular collapse-and-evaporation scenarios.

3. Thermodynamics and stability structure

For the quintessence-dressed Frolov black hole at fixed $\alpha$4, the mass, temperature, entropy, heat capacity, and Helmholtz free energy are defined by standard horizon thermodynamics. If $\alpha$5 denotes the largest black-hole horizon root of $\alpha$6, then

$\alpha$7

The reported phase structure is sharp: $\alpha$8 for the small-black-hole branch $\alpha$9, while $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$0 for $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$1. Thus the smaller black hole is locally thermodynamically stable, yet the free energy satisfies $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$2 for all allowed radii, so the system is globally thermodynamically unstable and exhibits no Hawking–Page transition to a lower-energy phase (Gohain et al., 2024).

The Frolov scale modifies this phase diagram in a specific way. It introduces a finite-curvature core, lowers the peak Hawking temperature, and drives the Davies-point instability to larger $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$3. By contrast, the quintessence parameter only weakly modifies the thermodynamics: the peak $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$4 and free energy are nearly unchanged, and the global instability persists irrespective of $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$5, $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$6, or $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$7 (Gohain et al., 2024).

For the asymptotically flat static Frolov black hole written with parameter $$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$8, one paper gives the explicit temperature

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$9

That analysis states that the Frolov temperature is lower than the Reissner–Nordström value for the same $$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$0 (Kala et al., 25 Mar 2025).

Not all extensions preserve the area law. In the Frolov black hole with global monopole and cosmic string, the entropy obtained from $$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$1 contains both rational corrections and a logarithmic term,

$$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$2

so the Bekenstein–Hawking law is explicitly violated by the combined effect of defects and core regularization (Ahmed et al., 17 Apr 2025).

4. Geodesics, lensing, and black-hole shadow

The geodesic sector is controlled by the Killing integrals

$$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$3

and the radial equation

$$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$4

For null motion,

$$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$5

and circular photon orbits satisfy

$$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$6

For timelike motion,

$$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$7

and bound orbits lie between periastron $$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$8 and apastron $$f(r)=1-\frac{(2Mr-q^2)\,r^2}{r^4+(2Mr+q^2)\,\alpha_0^2}.$$9, with precession

$q$0

Numerical integration shows that increasing quintessence strength $q$1 pushes $q$2 outward and repels null rays, whereas $q$3 and $q$4 tend to draw $q$5 inward; timelike perihelion precession is only weakly affected by $q$6 (Gohain et al., 2024).

For weak gravitational lensing by the quintessence-dressed solution, the deflection angle decreases monotonically with the impact parameter $q$7. Increasing $q$8 makes the deflection decrease more rapidly at large $q$9, while variations of $Q$0 or $Q$1 have negligible impact in the weak-deflection regime (Gohain et al., 2024). For the bare static Frolov metric, the weak-field expansion reads

$Q$2

so the regularization scale $Q$3 reduces the bending of light (Kala et al., 25 Mar 2025).

The shadow is determined by the same photon-sphere data. For a distant static observer,

$Q$4

Across several models, the same qualitative trends recur. In the quintessence setting, $Q$5 increases with $Q$6, decreases with $Q$7 or $Q$8, and depends only weakly on $Q$9 once $\alpha_0$0 is fixed (Gohain et al., 2024). In the topological-defect extension, increasing $\alpha_0$1 or $\alpha_0$2 makes the shadow smaller, whereas increasing the cosmic-string parameter $\alpha_0$3 or the global-monopole parameter $\alpha_0$4 enlarges it, sometimes dramatically (Ahmed et al., 17 Apr 2025). In the asymptotically flat static model, the small-$\alpha_0$5 expansion gives

$\alpha_0$6

again showing that the Frolov scale shrinks the shadow (Kala et al., 25 Mar 2025).

Several observational bounds have been reported from EHT-inspired shadow analyses. For the quintessence model with $\alpha_0$7, one paper quotes

$\alpha_0$8

at $\alpha_0$9 (Gohain et al., 2024). A separate static-shadow study for Sgr A$g$00 uses

$g$01

and infers

$g$02

with representative constraints $g$03 at $g$04, and $g$05 at $g$06 (Kala et al., 25 Mar 2025). This suggests that current shadow bounds are sensitive to the precise extension under consideration, particularly to whether quintessence is included.

5. Perturbations, quasinormal spectra, greybody factors, and wave scattering

For a massless scalar field, separation of variables and the tortoise coordinate $g$07 reduce the dynamics to

$g$08

Using the Chebyshev-pseudospectral method in the frequency domain, one study found that all scalar quasinormal frequencies satisfy $g$09, with time-domain evolution showing an initial burst, quasinormal ringing, and a late-time power-law tail

$g$10

For the monopole $g$11 mode, $g$12 depends non-monotonically on $g$13, while $g$14 decreases monotonically; for $g$15, both $g$16 and $g$17 decrease monotonically as $g$18 increases, indicating that angular momentum dominates the quantum-correction effect in higher multipoles (Song et al., 2024).

In the quintessence-dressed case, scalar quasinormal modes were also computed with a WKB treatment. There the parameter dependence differs: increasing $g$19 decreases both $g$20 and $g$21, whereas increasing $g$22 or $g$23 increases $g$24 and decreases $g$25. The same study introduces a rigorous lower bound on the greybody factor,

$g$26

and reports that larger $g$27 suppresses transmission, smaller $g$28 makes the barrier steeper, and quintessence only mildly enhances low-frequency transmission (Gohain et al., 2024).

Electromagnetic and Dirac axial perturbations have been studied using the $g$29-order WKB method with Padé resummation. The effective potentials are

$g$30

and

$g$31

Relative to Reissner–Nordström, turning on $g$32 raises $g$33 by roughly $g$34 and slightly reduces $g$35, while low-frequency greybody transmission is suppressed by a few percent in the examples reported for $g$36, $g$37 (Pathrikar, 1 Oct 2025).

The wave-scattering sector provides a complementary high-frequency description. For massless scalar waves, the photon sphere radius $g$38, critical impact parameter

$g$39

and geometric capture cross section

$g$40

control the high-frequency limit. The total absorption cross section obeys

$g$41

and at high frequency follows the sinc approximation around $g$42. Backward scattering admits the glory form

$g$43

The striking numerical result is an iso-impact-parameter degeneracy: when Frolov, Reissner–Nordström, and Hayward black holes are chosen to have identical $g$44 or identical $g$45, their absorption and scattering curves overlap to high precision over the full frequency or angular range. The paper interprets this as evidence that photon-sphere data dominate scalar-wave observables, while the detailed core regularization plays a secondary role (Tang et al., 27 Jan 2026).

6. Extended families and broader theoretical role

The Frolov metric has generated a substantial family of deformations. With a global monopole and a cosmic string, the metric function becomes

$g$46

In this model, increasing $g$47 or $g$48 shrinks the outer horizon and the shadow, whereas increasing $g$49 or $g$50 enlarges both. Scalar and electromagnetic quasinormal modes computed with the $g$51-order WKB approximation have negative imaginary parts throughout the reported parameter ranges, confirming linear stability; increasing $g$52 or $g$53 decreases both $g$54 and $g$55 in magnitude, while increasing $g$56 or $g$57 increases $g$58 and decreases $g$59 in magnitude (Ahmed et al., 17 Apr 2025).

A different deformation adds a cloud of strings through the simple shift

$g$60

with $g$61. The parameter $g$62 arises from the Letelier cloud-of-strings stress tensor and produces an effective solid-angle deficit $g$63. Near the origin one has

$g$64

so the proper-time integral for radial timelike geodesics remains finite provided $g$65, which the paper interprets as a sign of geodesic completeness. At the same time, it explicitly notes that the Kretschmann scalar diverges when $g$66, so this extension does not preserve the full regularity of the original Frolov core even though several geodesic properties remain close to the $g$67 case (Nascimento et al., 13 Jan 2026).

In the AdS generalization surrounded by a fluid of strings, the metric function is

$g$68

with the string-fluid profile labeled by $g$69. Only the range $g$70 preserves both the nonsingular Frolov core at $g$71 and AdS asymptotics at infinity, with $g$72 singled out as the simplest representative. For that case the Kretschmann scalar remains finite at both $g$73 and $g$74, radial geodesics can be smoothly extended through $g$75, and the heat capacity remains finite, changing sign only once. By contrast, the $g$76 branch is singular and exhibits a Davies-type divergence of $g$77 (Nascimento et al., 10 May 2025).

Finally, the evaporation analysis of regularized charged black holes gives the Frolov construction a broader theoretical role. In the constant-$g$78 charged Vaidya model, there is no $g$79 curvature singularity, no persistent trapped region, and no event or Cauchy horizon; the past of future null infinity covers all of spacetime. The authors explicitly present this as a non-singular, unitary alternative to the classical information-loss scenario, albeit within spherical symmetry and an ad hoc regularization ansatz (Sueto et al., 2023).

Taken together, these results situate the Frolov black hole as more than a single metric. It functions as a regularization template for charged black holes, a platform for testing how near-core modifications propagate into photon-sphere observables, and a model space in which thermodynamic instability, wave propagation, shadow phenomenology, and horizonless evaporation can be studied within a common geometric framework.