Myers-Perry Black Hole: Rotating Spacetimes

Updated 2 December 2025

Myers-Perry black holes are higher-dimensional, rotating, asymptotically flat solutions with up to floor((D-1)/2) independent angular momentum parameters.
They exhibit complex horizon geometries, multipole expansions, and distinctive scattering signatures that extend beyond the four-dimensional Kerr paradigm.
These solutions provide a testbed for exploring gravitational thermodynamics, perturbative instabilities, and quantum corrections in high-energy and string theory contexts.

The Myers-Perry (MP) black hole is the unique asymptotically flat, stationary, rotating vacuum black hole solution of the $D$ -dimensional Einstein equations. Generalizing the four-dimensional Kerr solution, MP black holes possess up to $N = \lfloor (D-1)/2 \rfloor$ independent angular momentum parameters, with richer horizon and multipole structure than their four-dimensional analogues. The five-dimensional case ( $D=5$ ) is particularly well-studied, offering tractable metrics with significant phenomenology in gravity, high-energy, and string-theory contexts.

1. Metric Structure and Global Properties

In $D$ spacetime dimensions, the MP metric with $N$ independent spins in Boyer-Lindquist-type coordinates is

$ds^2 = -dt^2 + \sum_{i=1}^N (r^2+a_i^2)\left(d\mu_i^2 + \mu_i^2 d\phi_i^2\right) + \epsilon\, r^2 d\alpha^2 + \frac{\mu\,r^\epsilon}{\Pi\,F}\left(dt+\sum_{i=1}^N a_i \mu_i^2 d\phi_i\right)^2 + \frac{\Pi F}{\Pi - \mu r^\epsilon} dr^2\,,$

where

$\Pi(r) = \prod_{i=1}^N (r^2+a_i^2), \quad F(r,\mu_i) = 1 - \sum_{i=1}^{N} \frac{a_i^2 \mu_i^2}{r^2+a_i^2}, \quad \epsilon = D-2N-1, \quad \sum_{i=1}^N \mu_i^2 + \epsilon \alpha^2 = 1.$

For $D=5$ ( $N=2$ ), the black hole can have up to two independent rotation parameters $a$ and $b$ . The mass and angular momenta are

$M = \frac{3\pi}{8 G_5} \mu, \quad J_{i} = \frac{2}{3} M a_i.$

The event horizon sits at the largest positive real solution of $\Delta(r) = (r^2 + a^2)(r^2 + b^2) - \mu r^2 = 0$ (Myers, 2011, An et al., 2017).

2. Horizon Geometry, Extremality, and Topology

The topology of the event horizon remains $S^{D-2}$ . Non-extremal MP black holes have two horizons in $D=5$ , which coincide in the extremal limit defined by $(1 - a^2 - b^2)^2 = 4a^2b^2$ . For $a=b$ (equal spins), the extremality is reached at $r_0^2 = 4a^2$ , beyond which the horizon disappears and a naked singularity forms (Papnoi et al., 2014). The surface gravity and angular velocities are, for each rotation plane,

$\kappa = \frac{\Delta'(r_+)}{2(r_+^2 + a^2)}, \quad \Omega_i = \frac{a_i}{r_+^2 + a_i^2}.$

The sphere $S^{D-2}$ may be distorted if the black hole is subject to external fields; however, the topology remains unchanged even under axisymmetric distortions (Abdolrahimi et al., 16 Sep 2025, Abdolrahimi et al., 2014).

3. Multipole Structure and Scattering Signatures

MP black holes exhibit a multipole expansion richer than the Kerr case. In $D\ge5$ dimensions, the metric's large- $r$ asymptotics include mass, current, and genuine stress multipoles: $g_{00} \sim 1 - \sum_\ell \left(\text{mass multipoles}\right) - \sum_\ell \left(\text{stress multipoles}\right),$ with the stress multipoles absent in four-dimensional solutions (Bianchi et al., 7 May 2025). For $D=5$ and equal spins, all higher mass multipoles vanish, while current dipole and stress quadrupole survive:

If $a_1=a_2$ , the gravitational field's distinctiveness arises from a nonvanishing stress quadrupole in addition to current-type multipoles.
These stress multipoles induce corrections in scattering eikonal phases and appear at subleading orders in high-energy scattering, making their detection a characteristic probe of higher-dimensional black hole spacetimes.

4. Geodesic Structure, Shadows, and Photon Regions

The separability of the Hamilton-Jacobi equation allows for analytical treatment of null geodesics and the construction of "shadow" boundaries, crucial for potential observational signatures:

For $D=5$ , $a=b$ , the boundary of the black hole shadow is described by explicit parametric equations for the celestial coordinates $(\alpha,\beta)$ in terms of conserved impact parameters $(\xi_1, \xi_2, \eta)$ : $\alpha = -\left(\frac{\xi_1}{\sin\theta_0} + \frac{\xi_2}{\cos\theta_0}\right), \quad \beta = \pm \sqrt{\eta - \xi_1^2 \cot^2\theta_0 - \xi_2^2 \tan^2\theta_0 + a^2}$ The shadow forms a deformed disk whose size $R_s$ decreases and distortion $\delta_s$ increases monotonically with spin (Papnoi et al., 2014). For $a$ exceeding extremality, only a partial arc remains, corresponding to orbits that can escape from the near-singularity region.

5. Thermodynamics, Instabilities, and Phase Structure

MP black hole thermodynamics exhibit nontrivial structure in $D>4$ :

The temperature and entropy, as functions of horizon area and spin(s), are

$S = \frac{A}{4G}, \quad T = \frac{\kappa}{2\pi}$

where $A$ is the horizon area.

Geometrothermodynamics (GTD) reveals singularities in the thermodynamic curvature at points coinciding with phase transitions, matching the divergence loci of specific heats and susceptibilities (Bravetti et al., 2013). The system undergoes second-order phase transitions akin to Van der Waals fluids, with critical exponents $\alpha=0$ , $\beta=1/2$ , $\gamma=1$ , $\delta=3$ (Poshteh et al., 2013).

Classically, $D=5$ MP black holes are stable to linear perturbations, except in higher dimensions ( $D\ge 7$ ), where extremal cohomogeneity-1 solutions exhibit near-horizon instabilities linked to the violation of the Breitenlohner-Freedman bound for certain gravitational harmonics (Durkee et al., 2010). Large- $D$ effective theory provides complete maps of ultraspinning instabilities and bifurcating stationary branches, which are inherited by almost-equal-spin phases (Suzuki et al., 2023).

6. Quantum Corrections, Love Numbers, and Advanced Phenomena

Quantum-gravity-inspired renormalization group improvements of the classical MP metric introduce running Newton's constant $G(r)$ , modifying the inner geometry, producing a minimal black hole mass for horizon formation, removing classical ultra-spinning solutions, softening curvature singularities, and imposing finite remnant states (Litim et al., 2013). Scalar tidal Love numbers of $D=5$ MP black holes do not generically vanish and show logarithmic running except in resonant cases tied to enhanced $\mathrm{SL}(2,\mathbb{R})$ near-zone symmetries. These conformal symmetries are geometrized as isometries of "subtracted" black hole metrics that preserve horizon structure but truncate the asymptotic region. In the extremal (AdS $_2$ ) limit, the Love symmetries match exact near-horizon isometries (Charalambous et al., 2023).

7. Black Holes in External Fields, Distorted Horizons, and Physical Implications

The family of exact solutions for a five-dimensional Myers-Perry black hole immersed in external stationary $U(1)\times U(1)$ symmetric matter encodes axisymmetric distortions via prolate spheroidal multipoles. The influence of these multipoles results in arbitrary deformations ("bumps") of the $S^3$ horizon. The ergoregion remains robust in the presence of specific classes of external distortions, illustrating physical separation between the horizon geometry and the structure of the ergosphere (Abdolrahimi et al., 16 Sep 2025, Abdolrahimi et al., 2014). The ratio $J^2/M^3$ of angular momentum squared to the mass cubed can be made arbitrarily large via appropriate distortions, in contrast to the undistorted case.