Accelerating Rotating Black Holes

• Accelerating rotating black holes are exact Einstein solutions featuring linear acceleration, rotation, and conical singularities with multi-horizon structures.
• The analysis employs advanced Plebański–Demiański metrics to explore modified thermodynamics, quantum entropy corrections, and universal particle acceleration effects.
• Implications include observable disk dynamics, holographic dualities, and novel predictions for black hole entropy and astrophysical jet launching.

Accelerating black holes with rotation are exact solutions to Einstein's equations, and, in extended theories, supergravity or Gauss–Bonnet/Lovelock gravity, that describe black holes undergoing both linear acceleration and angular momentum (rotation). The canonical class of such metrics belongs to the Plebański–Demiański family, encompassing the rotating C-metric and its generalizations, and appears in contexts ranging from pure Einstein–Maxwell theory to higher-curvature and string-motivated gravity, with or without a cosmological constant. These solutions display a uniquely intricate interplay of acceleration, frame-dragging, conical singularities, multi-horizon structure, and modified thermodynamics. Recent developments have clarified the geometric, dynamical, and quantum properties of rotating, accelerating black holes, including their universal role as particle accelerators, their holographic duals, extended thermodynamics, and quantum corrections to entropy.

1. Geometric Structure and Metric Forms

Accelerating, rotating black holes in four dimensions are often described by the Plebański–Demiański (PD) metric: $$ds^2 = \frac{1}{H^2} \left\{ -\frac{Q}{\Sigma} \left[dt - a \sin^2\theta\, d\phi\right]^2 + \frac{\Sigma}{Q} dr^2 + \frac{\Sigma}{P} d\theta^2 + \frac{P}{\Sigma} \sin^2\theta \left[a dt - (r^2 + a^2) d\phi\right]^2 \right\},$$ with

$$H = 1 - \alpha r \cos\theta, \quad \Sigma = r^2 + a^2 \cos^2\theta,$$

$$P(\theta) = 1 - 2\alpha m \cos\theta + [\alpha^2(a^2 + e^2 + g^2) - a^2] \cos^2\theta,$$

$$Q(r) = (r^2 - 2m r + a^2 + e^2 + g^2)(1 - \alpha^2 r^2) + (r^2 + a^2)r^2,$$

and various charges (electric $e$, magnetic %%%%1%%%%), rotation parameter $a$, mass $m$, and acceleration parameter $\alpha$. The metric generalizes Kerr–Newman (rotation and charge), C-metric (acceleration), and AdS/dS backgrounds. The PD family includes NUT charge and cosmological constant as additional parameters.

The presence of acceleration ($\alpha \neq 0$) always induces conical singularities at the poles, corresponding physically to strings or struts sourcing the acceleration. For rotating, accelerating black holes in arbitrary cosmological backgrounds, a compact and root-parameterized C-metric form clarifies the horizon and axis structure (Chen et al., 2016).

In higher dimensions, new classes of exact rotating, accelerating black holes have been constructed in five-dimensional Einstein–Gauss–Bonnet gravity and further extended to cubic Lovelock gravity in seven dimensions, where acceleration is incorporated by coupling the lower-dimensional seed solution to extra warped directions (Anabalon et al., 6 Apr 2024).

2. Horizon Structure, Conical Deficits, and Uplift

Acceleration introduces additional horizons (the so-called acceleration or Rindler horizons), whose spacetime locations depend on $\alpha$. For fixed values of cosmological constant, rotation, and mass, the function $Q(r)$ typically admits multiple real roots, corresponding to inner, outer, and acceleration horizons. The axes $\theta=0,\pi$ carry conical singularities unless an explicit quantization (matching) of the period of $\phi$ is imposed.

The conical defects associated with acceleration have a geometric interpretation: the horizon surface (at fixed $t$, $r$) is topologically a spindle $\mathbb{WCP}^1_{[n_-,n_+]}$, a weighted projective line with conical deficits determined by integers $n_+, n_-$ labeling the two poles. When uplifted along a regular Sasaki–Einstein space (as in consistent truncations to eleven-dimensional supergravity), these conical deficits are absorbed into global smoothness conditions for the total space, precisely canceling possible orbifold singularities (Ferrero et al., 2020).

3. Particle Acceleration and the BSW Effect

A universal property of rotating black holes, including those with acceleration, is their efficient acceleration of infalling particles to arbitrarily high center-of-mass energies in the near-horizon region. This Bañados–Silk–West (BSW) effect holds in all spacetimes with a regular rotating horizon, whether "clean" (vacuum Kerr) or "dirty" (with exterior matter/fields) (Zaslavskii, 2010):

• The near-horizon geometry supports conserved energy $E$ and angular momentum $L$.
• Particles with $E - \omega_H L = 0$, where $\omega_H$ is the horizon angular velocity, are termed "critical." Fine-tuning a particle's $L$ to its critical value, a collision with another (generic) particle yields center-of-mass energies

$E_{\text{cm}}^2 \sim \frac{1}{N^2}$

as the lapse $N \to 0$ at the horizon.

• This result is model-independent and holds for accelerating, rotating black holes as long as the near-horizon expansion structure persists (Zaslavskii, 2011, Harada et al., 2014).

Moreover, analogous acceleration effects are seen in modified black hole solutions (e.g., regular Hayward, Bardeen, Einstein–Maxwell–Dilaton, anisotropic matter backgrounds) and in extended topologies (cylindrical or higher-dimensional black holes) (Amir et al., 2015, Pourhassan et al., 2015, Rizwan et al., 2020, Raza et al., 18 Mar 2025, Said et al., 2011).

4. Thermodynamics and Quantum Corrections

The thermodynamics of rotating, accelerating black holes is rich, featuring:

• Modified surface gravity and Hawking temperature, which receive explicit corrections from both acceleration and rotation:

$$\kappa_H = \frac{(r_+ - m)}{r_+^2 + a^2} (1 - \alpha r_+)^3(1 + \alpha r_+)$$

• Horizon area and entropy formulas incorporate the acceleration parameter $\alpha$ and rotation $a$. The entropy remains one quarter of the horizon area per the Bekenstein–Hawking law (Bilal et al., 2010, Anabalon et al., 2018):

$$S = \frac{A}{4} = \frac{\pi (r_+^2 + a^2)}{K(1 - A^2 r_+^2)},$$

with $K$ a regularity factor depending on the string tension (conical deficit).

• The first law and Smarr relation acquire extra terms associated with string tensions $\mu_\pm$ (or, equivalently, the parameters of conical defects), necessitating a "full cohomogeneity" first law when both acceleration and rotation are present (Anabalon et al., 2018):

$$\delta M = T \delta S + \Omega \delta J + \Phi \delta Q - \lambda_+ \delta \mu_+ - \lambda_- \delta \mu_- + V \delta P,$$

where $\lambda_\pm$ are thermodynamic lengths conjugate to the string tensions.

• Rotation imposes a bound on the acceleration parameter. For physical black holes:

$$\alpha < \frac{1}{m + \sqrt{m^2 - a^2 - e^2 - g^2}}$$

to avoid negative temperature and related pathologies (Bilal et al., 2010).

At low temperatures (near extremality), quantum corrections to the black hole entropy due to zero modes in the Euclidean path integral yield universal logarithmic corrections, whose coefficient depends on rotation and charge content:

• With only rotation: $(3/2)\log T$.
• With charge and no rotation: $(3/2 + 1/2 + 1/2)\log T$ (from tensor, vector, and photon zero modes).
• With both: $2\log T$ (Xu, 15 Oct 2025).

These corrections resolve the linear-in-$T$ mass gap and are sensitive to the near-horizon warped/twisted geometry induced by acceleration and frame dragging.

5. Holography, CFT Duals, and Topology

In the extremal limit, the near-horizon geometry of the accelerating, rotating black hole becomes a warped/twisted $AdS_2 \times S^2$ (or its deformed versions), supporting an $SL(2,\mathbb{R}) \times U(1)$ symmetry. The Kerr/CFT correspondence has been extended to these geometries (Astorino, 2016):

• The central charge $c_J$ is computed from the warped/twisted near-horizon data; the Frolov–Thorne temperature $T_\varphi$ arises from the matching of angular velocities.
• Applying the Cardy formula with these parameters reproduces the Bekenstein–Hawking entropy even in the presence of acceleration, rotation, charge, and external magnetic (Melvin) fields.

Topologically, the spindle structure and quantized conical deficits established in the uplift of the metric to $D=11$ supergravity (e.g., on SE$_7$) guarantee that the lower-dimensional conical singularities are globally regular in the higher-dimensional solution (Ferrero et al., 2020). The entropy can be matched to topologically twisted partition functions of the dual ${\cal N}=2$ 3d field theories compactified on the spindle.

Recent work has also applied generalized topology to black hole thermodynamics, defining topological numbers of black hole phase space, showing how acceleration and cosmological constant shift these numbers and tracing Hawking–Page-type and Van der Waals-type transitions to jumps in these invariants (Liu et al., 18 Sep 2024).

6. Particle and Disk Dynamics, Observational Signatures

The combined effects of acceleration and rotation are manifest in:

• The geodesic structure, circular orbits, and stability. Both ISCO (innermost stable circular orbit) and OSCO (outermost) radii increase with acceleration, while required angular momentum decreases. Co-rotating (prograde) and counter-rotating (retrograde) orbits display distinct characteristics, with acceleration always pushing ISCO outward and reducing luminosity from the disk (Ashoorioon et al., 14 Aug 2024).
• Precession is always in the direction of black hole rotation, regardless of spin sign, and is slightly greater for counter-rotating orbits.
• In magnetized environments, counter-rotating (retrograde) orbits in non-axisymmetric (oblique) magnetospheres possess larger escape zones and achieve higher Lorentz factors than prograde orbits, a result with implications for jet launching and particle acceleration in AGN and microquasars (Kopacek et al., 2021).
• Radiative flux and thermal spectra from accretion disks are always suppressed by acceleration; for fixed spin, accelerating black holes are dimmer and radiate at lower peak energies than their non-accelerating Kerr counterparts.

These dynamical properties offer concrete, potentially observable signatures distinguishing accelerating, rotating black holes from standard Kerr(s) in astrophysical systems.

7. Extensions, Generalizations, and Open Problems

Accelerating black holes with rotation have been generalized to:

• Include charge, cosmological constant, NUT parameter, and dilaton fields (Einstein–Maxwell–Dilaton and low-energy string scenarios) (Siahaan, 2018, Raza et al., 18 Mar 2025).
• Higher-codimension embeddings (string/M-theory truncations on Sasaki–Einstein and Kähler–Einstein manifolds), where spindle-topology and quantization of parameters ensure smoothness (Ferrero et al., 2020).
• Einstein–Gauss–Bonnet and Lovelock gravities in higher dimensions, generating analytic solutions with intrinsic “gravitational hairs” that cannot be gauged away by a coordinate transformation, with intricate causal and thermodynamical structures (Anabalon et al., 6 Apr 2024).

Ongoing topics include the full classification of horizon topologies, nontrivial signatures of acceleration for black hole mergers and gravitational waves, the interaction with ambient field structures (e.g., cosmic strings and magnetic fields), and the microscopic counting of entropy in the dual CFT/string context.

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This synthesis encapsulates the current understanding and significance of accelerating black holes with rotation, covering their geometric construction, dynamical and thermal properties, quantum corrections, holographic duals, topological traits, and observable consequences, as established in the literature over the past two decades.