Charged Torus-Like Black Holes

• Charged torus-like black holes are defined by toroidal event horizon topologies that extend traditional Reissner–Nordström and Kerr–Newman solutions through additional field interactions and modified gravity effects.
• They exhibit complex thermodynamic behavior with multiple phase transitions, entropy corrections, and distinctive features revealed by analyses of heat capacity and Ruppeiner geometry.
• Quantum corrections, rotational dynamics, and higher-dimensional generalizations further enrich their structure, offering insights into black hole microphysics and observational signatures.

Charged torus-like black holes are solutions to the Einstein equations (with or without matter couplings) whose event horizon topology is toroidal or “torus-like,” as opposed to the more familiar spherical topology. When electric or other charges are present, these solutions generalize the Reissner–Nordström or Kerr–Newman families, potentially incorporating additional field interactions, higher-dimension effects, or quantum corrections. Their theoretical interest lies in both gravitational thermodynamics and the astrophysics of accretion, and such objects serve as important laboratories for probing black hole microphysics, entropy corrections, horizon structure, and observational signatures.

1. Geometric Structure and Theoretical Frameworks

Charged torus-like black holes have horizon cross-sections homeomorphic to $T^{d-1}$ in $(d+1)$-dimensional spacetime or $S_1 \times S_1 \times \mathbb{R}$ in the four-dimensional case (Sharifian et al., 2017, Panotopoulos et al., 2018). In Einstein–Hilbert and modified gravity models, such as $f(\mathbb{Q})$ non-metricity gravity, the metric takes the form

$$ds^2 = -N^2(r) dt^2 + \frac{dr^2}{N^2(r)} + r^2 \sum_{i=1}^{d-1} d\phi_i^2$$

where each $\phi_i$ (for $i = 1, ..., d-1$) is a periodic coordinate. The lapse $N(r)$ and matter profiles are determined by the choice of matter coupling (Maxwell, non-linear electrodynamics, Skyrme fields, etc.) and the dimension (Henríquez-Baez et al., 16 Dec 2024, Nashed, 2023, Panotopoulos et al., 2018, Sharifian et al., 2017).

Special cases include:

• Einstein–Maxwell theory: charge is introduced via the standard Maxwell action; the resulting solutions reduce to a Reissner–Nordström-like toroidal geometry in certain parameter regimes (Sharifian et al., 2017).
• Non-linear electrodynamics or $f(\mathbb{Q})$ quadratic gravity: the horizon structure, electric potential, and curvature singularities are dramatically altered (Nashed, 2023).
• Einstein–Skyrme systems: additional non-Abelian “hair” is supported, and rotation is incorporated via improper coordinate transformations, fundamentally altering the horizon geometry and thermodynamics (Henríquez-Baez et al., 16 Dec 2024).

2. Thermodynamics and Entropy Corrections

These black holes exhibit thermodynamic properties that can be described via standard as well as generalized entropy frameworks.

Key Thermodynamic Quantities

For a horizon at $r_+$,

• Mass (in suitable units): $M \propto ([2P S^2 + 3 Q^2]/\sqrt{S})$ for the Bekenstein–Hawking entropy (Zafar et al., 1 Sep 2025)
• Temperature: $$T = \left(\frac{\partial M}{\partial S}\right)_{P,Q}$$
• Entropy: multiple choices, including the Bekenstein–Hawking form $S_{\mathrm{BH}} = A/4$, the exponential corrected entropy $S_{\mathrm{exp}} = e^{-\pi r_+^2} + \pi r_+^2$, and the Rénnyi entropy (Zafar et al., 1 Sep 2025)

Phase Structure and Ruppeiner Geometry

When employing exponential corrected entropy, the heat capacity $C_P$ can vanish at two points, indicating multiple phase transitions (stable and unstable regimes may intervene). The points of vanishing $C_P$ coincide with divergences of the Ruppeiner Ricci scalar, calculated from the thermodynamic (Ruppeiner or Weinhold) metric $$g^{\mathrm{Rup}}_{\nu \omega} = \partial^2 M / \partial x^\nu \partial x^\omega$$ (scaled by $1/T$) (Zafar et al., 1 Sep 2025). This alignment of geometric and thermodynamic criticality strongly supports the interpretation of these points as true phase transitions.

Comparison table of entropy models (as in (Zafar et al., 1 Sep 2025)):

Entropy Model	Zeros of $C_P$	Ricci Scalar Divergence	Phase Transitions
Bekenstein–Hawking	1	1	Single
Exponential Corrected	2	2	Multiple
Rénnyi	1	1	Single

The presence of multiple zero points in $C_P$ and associated geometric singularities in the exponential corrected case signals a richer phase structure and increased sensitivity to underlying microstructure relative to the Hawking–Bekenstein or Rénnyi cases (Zafar et al., 1 Sep 2025).

Equation of State and Absence of Critical Point

A key thermodynamic feature is the absence of a critical point in the $P$–$V$ diagram, in contrast to Van der Waals or charged AdS black holes. The equation of state typically takes the form

$$P = \frac{\pi^{2/3}}{3 \sqrt[3]{6}} \left[\frac{4Q^2}{V^{4/3}} + \frac{3T}{V^{1/3}}\right]$$

and neither $\partial P/\partial V$ nor its second derivative vanish simultaneously, precluding gas/liquid-type phase transitions (Feng et al., 2021, Zafar et al., 1 Sep 2025).

3. Quantum, Rotational, and Higher-Dimensional Corrections

Quantum corrections, rotation, and higher-dimensional generalizations further enrich the charged torus-like black hole landscape.

• Quantum corrections: The entropy receives logarithmic and inverse-area corrections, indicating that the area law is only leading order. For arbitrary horizon topology, including toroidal cases, quantum tunneling analyses predict

$$S = \frac{A}{4\hbar} + (\text{constant}) \ln A + \sum_{n>1} (\text{const})_n\,A^{-n}\,.$$

This structure is universal across static, rotating, or toroidal horizons (Akbar et al., 2010).

• Rotating solutions: Techniques for constructing exact spinning metrics include improper coordinate transformations that impart angular momentum while preserving horizon topology and Skyrme hair (Henríquez-Baez et al., 16 Dec 2024). In higher dimensions or with equal angular momenta, effective large-$D$ expansions reveal phase diagrams with instabilities, including ultraspinning and charge-induced branches, directly connecting torus-like instability to ultraspinning limits (Tanabe, 2016).
• Modified and higher-dimensional gravities: Quadratic $f(\mathbb{Q})$ theories support torus-like charged black holes only for $N\geq4$ and induce inseparable monopole and quadrupole momenta in the charge distribution, with curvature singularities milder than in Einstein–Maxwell theory (Nashed, 2023). The metric and electric potential in such cases cannot be separated into distinct charge multipoles, constituting a major structural difference.

4. Astrophysical Fluids, Tori, and Observational Implications

Charged torus-like structures also occur as equilibrium configurations for charged or magnetized astrophysical fluids around black holes.

• Stationary fluid tori: Models with negligible conductivity show that small net charge in a dielectric fluid torus around a static charged background modifies the torus’s extent and density maximum; positively charged tori are more extended, negatively charged tori more compact (Kovář et al., 2011).
• Self-gravitating and magnetized tori: Construction of force-balanced, self-gravitating, and magnetized tori demonstrates how net charge and large-scale fields (including Wald fields) interplay to create stable, stratified, or vertically-lobed configurations. The polytropic index, rotation profile, and external field control the topology and stability regime (Trova et al., 2016, Kovář et al., 2014).
• Observational signatures: Theoretical models compute redshift, blueshift, and gravitational shift profiles, revealing that torus-like geometries generically admit frequency shifts (for photons emitted from the ISCO or the photon sphere) compatible with current megamaser disk observations (e.g., NGC 4258, UGC 3789) under specific charge and cosmological constant values (Zafar et al., 1 Sep 2025). Violations of the standard $r_{\rm ph} < r_{\rm ISCO}$ relation become possible at high charge, distinguishing toroidal from spherical cases.

5. Black Hole Radiation, Heat Engines, and Extended Thermodynamics

Hawking Radiation and Sparsity

The emission rate and sparsity of Hawking radiation is entwined with horizon geometry and quantum/statistical entropy corrections. For exponential corrected entropy, the sparsity parameter $\eta$ can be expressed as

$$\eta_\mathrm{Exp} = \frac{16\mathcal{C}\pi (S-1)^2}{27\mathcal{G}[Q^2-4P(S-1)]^2}$$

with $\mathcal{C}$ and $\mathcal{G}$ constants determined by geometric optics and the absorption cross-section, and $S$ the entropy (Zafar et al., 1 Sep 2025). Exponential correction models regularize and stabilize emission relative to the baseline Hawking–Bekenstein case.

Heat Engines and Joule–Thomson Expansion

Charged torus-like black holes have been used to model black hole heat engines:

• In Carnot cycles, heat input and rejection along isotherms yield the Carnot efficiency $\eta_C = 1 - (T_L / T_H)$, with explicit entropy–volume relations for toroidal horizons (Feng et al., 2021).
• For rectangular and circular cycles, analytic and numerical efficiency bounds are computed. Upper bounds as $Q \rightarrow Q_{\rm ext}$ (extremality) are saturable and independent of spacetime dimension for vanishing $C_V$: $\eta \leq (2\pi) / (\pi + 4)$ (Feng et al., 2021).
• The Joule–Thomson expansion (isenthalpic process with $\mu = (\partial T/\partial P)_H$) exhibits only a “lower” inversion curve, in contrast to Van der Waals fluids, reflecting the impact of toroidal topology on phase structure and cooling–heating regimes (Liang et al., 2021).

6. Regularity, T-Duality, and Modified Electrodynamics

A class of electrically charged, torus–like black holes arises from gravitational and Maxwell theory deformations inspired by T-duality and minimal-length effects:

• Propagator modification with a zero-point length $l_0$: $$G(k) = -l_0 \sqrt{k^2 + m^2} K_1(l_0 \sqrt{k^2 + m^2})$$ (Gaete et al., 2022).
• In position space, the static potential becomes $A_t = -Q/\sqrt{r^2 + l_0^2}$, regularizing field and metric profiles at $r=0$.
• The physically meaningful limit is achieved by identifying $l_0 \to Q$, which aligns with known regular black hole metrics (Ayón–Beato–García) and produces a de Sitter–like or torus–like core instead of a classical singularity (Gaete et al., 2022).

Such regularizations are of particular interest for string theory phenomenology and suggest that toroidal (or smeared) interiors may underlie the resolution of curvature singularities.

7. Stability, Uniqueness, and Microstructure

• Stability: Thermodynamic and dynamical stabilities are sensitive to the presence of Gauss–Bonnet terms, charge, rotation, and the entropy correction model (e.g., positive heat capacity, sign of Gibbs free energy, and the associated Ruppeiner geometry) (Feng et al., 2021, Nashed, 2023).
• Uniqueness/Nonuniqueness: In certain modified gravity or dilaton-coupled scenarios, nonuniqueness can arise; e.g., multiple black holes with identical $(M, Q, J)$ but different horizon or dilaton profiles (Herdeiro et al., 18 Jun 2025).
• Microstructure: Divergences in the Ruppeiner Ricci scalar at phase transition points, particularly in entropy-corrected models, reveal crucial information about black hole microstructure and indicate repulsive/attractive interaction signatures at the horizon scale (Zafar et al., 1 Sep 2025).

References to Key Theoretical Models and Formulas

Aspect	Key Reference(s)	Representative Expression/Result
Metric	(Sharifian et al., 2017, Panotopoulos et al., 2018)	$$ds^2 = -N^2(r) dt^2 + dr^2/N^2(r) + r^2 \sum d\phi_i^2$$
Entropy Correction	(Akbar et al., 2010, Zafar et al., 1 Sep 2025)	$$S = S_0 + (\text{const}) \ln A + (\text{higher order})$$
Thermodynamic Volume	(Feng et al., 2021)	$V = \frac{4\pi^2 r_+^3}{3}$
Hawking Temperature	(Sharif et al., 2013, Feng et al., 2021)	$$T_H = \frac{1}{2\pi}[-{\Lambda}r_+ /3 + M / (\pi r_+^2) - {4Q^2} / (\pi r_+^3)]$$
Heat Engine Efficiency	(Feng et al., 2021)	Carnot: $\eta_C = 1 - (T_L/T_H)$; Circular bound: $\eta \leq (2\pi)/(\pi+4)$
Ruppeiner Geometry	(Zafar et al., 1 Sep 2025)	$$R^{\mathrm{Rup}} = \frac{6 P S^2 + 17 Q^2}{4 Q^2 S - 8 P S^3}$$ (HBE case)
Joule–Thomson Inversion	(Liang et al., 2021)	$$T_i = V \left(\frac{\partial T}{\partial V}\right)_P$$
T-duality Regularization	(Gaete et al., 2022)	$A_t = -Q/\sqrt{r^2 + l_0^2}$, $l_0 \to Q$

Conclusion

Charged torus-like black holes constitute a diverse class of exact, numerical, or phenomenological solutions whose horizon topology, matter content, and thermodynamics depart from the canonical spherical paradigm. Their theoretical analysis combines elements of classical and quantum gravity, extended thermodynamics, fluid dynamics, and geometric microstructure. Developments in this area have not only advanced the understanding of horizon topology, entropy corrections, and quantum emission but have also opened novel possibilities for observational astrophysics, stability characterization, and the exploration of underlying microscopic physics. These systems continue to serve as theoretically tractable yet phenomenologically rich models in both gravitational theory and relativistic astrophysics.