---
title: 'Myers-Perry Black Hole: Rotating Spacetimes'
url: https://www.emergentmind.com/topics/myers-perry-black-hole
type: topic
---

# Myers-Perry Black Hole: Rotating Spacetimes

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 $\Delta(r) = (r^2 + a^2)(r^2 + b^2) - \mu r^2 = 0$ [1111.1903, 1708.09576].

## 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 [1407.0834]. 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 [2509.13050, 1412.5416].

## 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 [2505.04800]. 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 [1407.0834]. 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 [1303.2077]. The system undergoes second-order phase transitions akin to Van der Waals fluids, with critical exponents $\alpha=0$, $\beta=1/2$, $\gamma=1$, $\delta=3$ [1306.4516].

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 [1012.4805]. Large-$D$ effective theory provides complete maps of ultraspinning instabilities and bifurcating stationary branches, which are inherited by almost-equal-spin phases [2307.00848].

## 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 [1308.5630]. 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 [2303.16036].

## 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 [2509.13050, 1412.5416]. 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.

---

**References:**  
- General high-dimensional solution and multipole expansion: [1111.1903], [2505.04800]  
- Shadow and photon region: [1407.0834]  
- Extreme limit and phase transitions: [1303.2077], [1306.4516]  
- Geometrothermodynamics: [1303.2077]  
- Instabilities and near-horizon perturbations: [1012.4805], [2307.00848]  
- Quantum corrections: [1308.5630]  
- Love numbers and $\mathrm{SL}(2,\mathbb{R})$ symmetry: [2303.16036]  
- Distorted black holes: [2509.13050], [1412.5416]  
- Particle acceleration and horizon structure: [1708.09576]  
- Thermodynamic and dynamical properties: [1302.5079], [1304.0836], [1405.5678], [1809.10457]

Source: https://www.emergentmind.com/topics/myers-perry-black-hole