Accelerating Kerr Black Holes

• Accelerating Kerr black holes are rotating solutions driven by cosmic strings, characterized by mass M, angular momentum a, and acceleration parameter α.
• The metric extends the classical Kerr solution by introducing an acceleration horizon, conical singularities, and modified geodesic and thermodynamic properties.
• Observational diagnostics such as shifted black hole shadows and altered energy extraction efficiencies offer practical insights for astrophysical and holographic studies.

An accelerating Kerr black hole is an exact, asymptotically non-flat solution to the Einstein equations representing a rotating black hole that is uniformly accelerated by cosmic strings or struts attached to its poles, as encoded in the so-called C-metric with rotation. This spacetime is characterized by four parameters: mass $M$, specific angular momentum $a$, and acceleration parameter $\alpha$ (or $A$), and often generalizations may include electromagnetic charges or a cosmological constant. The geometry combines the features of the classical Kerr solution—such as horizons, ergospheres, and ring singularity—with novel structures including an acceleration horizon and conical defects. Accelerating Kerr metrics form an important class within the Plebański–Demiański family and are crucial in modern theoretical research, including holographic dualities, energy extraction mechanisms, and global geometric analysis.

1. Metric Structure and Horizon Geometry

The canonical form of the accelerating Kerr metric in Boyer–Lindquist–like coordinates $(t, r, \theta, \phi)$ is

$$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Delta_r}{\rho^2}(dt - a\sin^2\theta\, d\phi)^2 + \frac{\rho^2}{\Delta_r} dr^2 + \frac{\rho^2}{\Delta_\theta} d\theta^2 + \frac{\Delta_\theta \sin^2\theta}{\rho^2}[a dt - (r^2+a^2) d\phi]^2 \right\}$$

where

$$\begin{aligned} \rho^2 &= r^2 + a^2 \cos^2\theta, \ \Omega &= 1 - \alpha r \cos\theta, \ \Delta_r &= (1 - \alpha^2 r^2)(r^2 - 2 M r + a^2), \ \Delta_\theta &= 1 + 2\alpha M \cos\theta + \alpha^2 a^2 \cos^2\theta. \end{aligned}$$

The event and Cauchy horizons are at $r_\pm = M \pm \sqrt{M^2 - a^2}$, and the acceleration horizon is located at $r = 1/\alpha$ for $\alpha > 0$, provided $a$0 (Gao, 20 Sep 2025). The domain of outer communications lies between $a$1 and $a$2. Unlike the asymptotically flat Kerr metric, this spacetime is conformally compactified due to the $a$3 factor.

Conical singularities are generally present on the axes $a$4, associated with cosmic strings or struts. Their deficit angles $a$5 are fixed by the parameter $a$6 and cannot be simultaneously removed without external fields or global identifications (Gao, 20 Sep 2025, Zhang et al., 2020).

2. Geodesics, Photon Orbits, and Black Hole Shadows

Test particle and photon motion in the accelerating Kerr geometry is governed by generalizations of the Carter equations. The conformal factor and angular deformations couple the radial and angular parts, but for small $a$7 the motion qualitatively resembles Kerr.

Photon circular orbits deviate from the equatorial plane with nonzero acceleration: the latitude $a$8 of these orbits increases with $a$9, shifting toward the south pole. Apparent black hole shadows exhibit a maximum radius and distortion that move to observer inclinations with $\alpha$0, determined by the magnitude of acceleration (Zhang et al., 2020). This demonstrates that the shadow's size and shape no longer attain their extreme values at the equatorial observation but at shifted latitudes, making shadow asymmetries possible observational diagnostics of acceleration-induced cosmic-string structures.

3. Black Hole Thermodynamics and Conserved Charges

Thermodynamic properties are modified by the acceleration parameter. Key quantities at the event horizon $\alpha$1 (with $\alpha$2) are:

• Angular velocity: $\alpha$3,
• Surface gravity: $\alpha$4,
• Horizon area: $\alpha$5,
• Temperature: $\alpha$6,
• Entropy: $\alpha$7 (Gao, 20 Sep 2025, Astorino, 2016).

No globally well-defined ADM mass exists due to non-flat infinity, but a consistent mass formula can be constructed via the first law and the Christodoulou–Ruffini scheme by relating $\alpha$8 to horizon properties and conical normalization parameters. The Smarr formula and first law are satisfied with conical-regularization terms for the string tension (Astorino, 2016, Astorino, 2016).

4. Energy Extraction and Magnetohydrodynamics

Accelerating Kerr black holes permit generalized energy-extraction processes. In the Penrose process, the maximal extractable rotational energy is

$\alpha$9

which decreases with increasing initial acceleration (Zeng et al., 4 Jan 2026).

Multiple iterations of the Penrose process in the accelerating Kerr background demonstrate that extraction efficiency $A$0 can surpass the $A$1 bound seen in non-accelerating Kerr when the decay radius is small. The bulk of the decrease in extractable energy appears as true extracted energy, not increased irreducible mass. For sufficiently large initial acceleration, the extractable energy rapidly collapses to nearly zero.

Magnetic reconnection processes—crucial for astrophysical jet models—are also modified: in the plunging region ($A$2), efficiency is enhanced near extremality but can be reduced away from extremality. Acceleration generally shifts permissible extraction regions outward and can raise extraction efficiencies above the Kerr baseline, but also reduces the region size for circular orbits (Wang et al., 16 Aug 2025).

5. Symmetries, Holography, and Dual Conformal Field Theories

Accelerating Kerr black holes preserve hidden conformal symmetry in the near-horizon, low-frequency regime for scalar perturbations (Siahaan, 2018). The wave equation exhibits an $A$3 structure, allowing the application of the Kerr/CFT correspondence. For both extremal and non-extremal, slowly accelerating cases, the Cardy formula with central charge $A$4 and temperatures $A$5 precisely reproduces the Bekenstein–Hawking entropy, with $A$6 (Siahaan, 2018, Astorino, 2016). In the fully covariant phase space approach, the asymptotic symmetry group at the bifurcation surface realizes a warped conformal field theory (WCFT) with central extensions: a Virasoro and a Kac–Moody algebra, matching the horizon entropy via the modular invariant WCFT Cardy formula (Xu, 2023).

When acceleration is not small, direct $A$7 embedding complicates, but it is expected that more elaborate CFT duals (for example, with $A$8-deformation) may be appropriate.

6. Global Geometry, Invariants, and Physical Interpretation

The presence of acceleration introduces novel structures:

• An acceleration horizon at $A$9 acts as a causal boundary for uniformly accelerated observers, analogous to Rindler spacetime (Gao, 20 Sep 2025).
• Geometric invariants such as the Weyl scalars, Kretschmann scalar, and Euler–Poincaré density exhibit explicit $(t, r, \theta, \phi)$0-dependence (Kraniotis, 2021). The ring singularity remains at $(t, r, \theta, \phi)$1, but all invariants have additional conformal factors of $(t, r, \theta, \phi)$2.
• Conical deficits on the symmetry axes are direct measures of cosmic-string tensions, which physically induce the acceleration (Zhang et al., 2020).
• Nonvanishing global Chern–Pontryagin charge and associated Hirzebruch signature density can induce quantum anomalies, such as helicity non-conservation for photons near the black hole (Kraniotis, 2021).

Astrophysical manifestations may include shadow asymmetries, energy-extraction efficiency deviations, and possibly chiral electromagnetic signals in observations of black-hole environments.

7. Generalizations and Theoretical Extensions

Accelerating Kerr black holes are embedded within the broader class of Plebański–Demiański metrics, supporting further generalization to include electromagnetic charges, NUT parameter, cosmological constant, and extension to string-theoretic and higher-dimensional solutions. In string theory, accelerating Kerr black holes can be used as seed metrics for solution generating techniques, yielding configurations relevant for compactified heterotic models with nontrivial dilaton and B-field structure (Siahaan, 2018).

Kerr–de Sitter and Kerr–anti–de Sitter modifications admit further rich phenomenology in horizon and acceleration structure. The Bañados–Silk–West effect for particle acceleration generalizes to these settings, with the possibility for unbounded center-of-mass energies in appropriate extremal or non-extremal contexts depending on $(t, r, \theta, \phi)$3 (Zhang et al., 2018).

Accelerating Kerr spacetimes therefore constitute a fundamental testing ground for the intersection of general relativity, quantum field theory in curved spacetime, and gravitational holography.