Myers–Perry Black Holes

Myers–Perry black holes generalize Kerr to asymptotically flat vacuum gravity in higher dimensions. Their defining novelty is the rank of the spatial rotation group: a generic $D$-dimensional member has $\lfloor(D-1)/2\rfloor$ independent angular momenta. Horizon existence and extremality then depend on dimension and on which spin planes are populated, so there is no single higher-dimensional Kerr bound.

Myers–Perry rotating geometry

Property	Value
Object class	Exact solution family
Theory	$D$-dimensional vacuum general relativity
Dimensions	$D\ge5$ here, with Kerr recovered at $D=4$
Parameters	mass parameter $\mu$ and independent rotation lengths $a_i$
Physical status	Exact higher-dimensional vacuum solution; no direct astrophysical identification implied

Write

$$D=2N+1+\epsilon, \qquad N=\left\lfloor\frac{D-1}{2}\right\rfloor,$$

where $\epsilon=0$ in odd $D$ and $\epsilon=1$ in even $D$. With direction cosines $\mu_i$, azimuths $\phi_i$, and

$$\sum_{i=1}^{N}\mu_i^2+\epsilon\alpha^2=1,$$

define

$$\Pi(r)=\prod_{i=1}^{N}(r^2+a_i^2), \qquad F=1-\sum_{i=1}^{N}\frac{a_i^2\mu_i^2}{r^2+a_i^2},$$

one exact form is

$$\begin{aligned} ds^2={}&-dt^2+\frac{\mu r^{2-\epsilon}}{\Pi F} \left(dt-\sum_i a_i\mu_i^2d\phi_i\right)^2 +\frac{\Pi F}{\Pi-\mu r^{2-\epsilon}}dr^2\\ &+\sum_i(r^2+a_i^2)(d\mu_i^2+\mu_i^2d\phi_i^2) +\epsilon r^2d\alpha^2 . \end{aligned}$$

The original solution and charge normalization were derived by Myers and Perry 1986, printed pp. 304–347.

Charges, horizons, and regularity

With asymptotic time and $2\pi$ azimuths,

$$M=\frac{(D-2)\Omega_{D-2}}{16\pi G_D}\mu, \qquad J_i=\frac{2}{D-2}Ma_i.$$

Thus $\mu$ has dimension $L^{D-3}$ and each $a_i$ has dimension $L$. Candidate horizons are positive roots of

$$\Pi(r)-\mu r^{2-\epsilon}=0.$$

At the largest regular root,

$$\chi=\partial_t+\sum_i\Omega_i\partial_{\phi_i}, \qquad \Omega_i=\frac{a_i}{r_+^2+a_i^2}, \qquad A_H=\Omega_{D-2}\mu r_+.$$

Horizon-penetrating coordinates remove the radial coordinate pole at a simple root. The rotation axes where a direction cosine vanishes are regular only with the correct azimuthal periods. Genuine singular sets are diagnosed by $\Pi F=0$, invariant curvature, and accessibility from the chosen domain; the four-dimensional ring picture cannot simply be copied into every dimension.

Dimension-dependent extremality

In five dimensions with two spins $a,b$,

$$(r^2+a^2)(r^2+b^2)-\mu r^2=0,$$

and regular horizon existence requires $\mu\ge(\lvert a\rvert+\lvert b\rvert)^2$. Equality is a regular extremal horizon only when both spins are nonzero. In the singly spinning case, equality collapses the area and exposes the singularity rather than producing a regular extremal endpoint.

For one nonzero spin in general dimension,

$$r_+^{D-5}(r_+^2+a^2)=\mu.$$

When $D\ge6$, the left side increases from zero to infinity, so a positive root exists for every finite $a$. There is no algebraic superextremal sector at large single spin. Instead,

$$r_+\sim\left(\frac{\mu}{a^2}\right)^{1/(D-5)}$$

at fixed mass, producing an ultraspinning horizon thin transverse to the rotation plane. Multiple-spin sectors can retain extremal boundaries. “Superextremal Myers–Perry” therefore has meaning only after dimension and spin sector are specified.

Mechanics, limits, and dynamics

The vacuum family obeys

$$dM=\frac{\kappa}{8\pi G_D}\,dA_H+\sum_i\Omega_i\,dJ_i,$$

and the Smarr relation

$$(D-3)M=(D-2) \left(\frac{\kappa A_H}{8\pi G_D}+\sum_i\Omega_iJ_i\right).$$

These are charge-normalization checks, not stability criteria.

Zero-spin limit. Taking every $a_i\to0$ gives the Schwarzschild–Tangherlini relation $r_+^{D-3}=\mu$ and a round $S^{D-2}$ horizon.

Four-dimensional limit. Setting $D=4$ leaves one spin, $\mu=2G_4M$, and the Kerr horizon equation. This checks both the mass and angular-momentum factors.

Ultraspinning limit. In singly spinning $D\ge6$, $a/r_+\to\infty$ yields a locally membrane-like patch. The limit motivates, but does not prove, a Gregory–Laflamme-type instability.

Extremal limits. In spin sectors admitting a regular double root, $\kappa\to0$ at finite area and a near-horizon scaling retains a throat. A collapsed zero-area endpoint is not the same limit.

Ultraspinning solutions can develop nonaxisymmetric or axisymmetric instabilities even though their horizons exist exactly. The higher-dimensional phase structure and the separation between algebraic bounds and perturbative stability are reviewed by Emparan and Reall 2008, §§ 4.2.1 and 4.4 and demonstrated in numerical perturbation studies such as Dias et al. 2009, complete article, printed pp. 111701-1–111701-5.

Ergoregions, topology, and interpretation

When at least one spin is nonzero, the stationary-limit surface is generally not the event horizon. The asymptotic time field $\partial_t$ becomes spacelike where its norm changes sign, while the co-rotating combination $\chi$ is null at $r_+$. The intervening ergoregion permits negative asymptotic Killing energy and rotational extraction. Its latitude dependence involves the direction cosines, so a single “ergosphere radius” is generally not available. In the all-zero-spin Tangherlini limit, the definitions remain distinct but the two loci coincide and the ergoregion disappears.

Regular Myers–Perry horizons have topology $S^{D-2}$, even when the intrinsic geometry becomes strongly flattened in an ultraspinning plane. That topology is a property of this family, not a higher-dimensional uniqueness theorem. Five-dimensional rings demonstrate that a regular asymptotically flat vacuum horizon can instead have $S^1\times S^2$ topology and share the same asymptotic mass and angular momentum with a spherical hole.

Parameter counting is also global. Independent $J_i$ correspond to mutually orthogonal rotation two-planes, not to every azimuth introduced in a coordinate chart. Reversing one azimuth reverses the associated $(a_i,J_i)$ without changing scalar invariants. Comparing solutions therefore requires the ordered set of charge magnitudes together with orientation conventions, dimension, and asymptotic frame.

No direct astrophysical inference follows from these exact higher-dimensional geometries. Compactification could change the asymptotics and excite Kaluza–Klein fields, while a genuinely noncompact extra dimension would alter long-distance gravity. Any phenomenological use must specify that additional model rather than treating the vacuum metric alone as an observed object.

Connections to the detailed theory

The complete metric, coordinate constraints, and dimension-by-dimension roots are presented in Myers-Perry Black Holes. Charges, ergoregions, mechanics, and ultraspinning interpretation are organized in Myers–Perry Rotating Black Holes. Neither exact treatment makes the family an observed population or classifies every higher-dimensional vacuum horizon.

References

• Dias, Óscar J. C., Pau Figueras, Ricardo Monteiro, Jorge E. Santos, and Roberto Emparan. “Instability and New Phases of Higher-Dimensional Rotating Black Holes.” Physical Review D 80 (2009): 111701. DOI.
• Emparan, Roberto, and Harvey S. Reall. “Black Holes in Higher Dimensions.” Living Reviews in Relativity 11 (2008): 6. DOI.
• Myers, Robert C., and Malcolm J. Perry. “Black Holes in Higher Dimensional Space-Times.” Annals of Physics 172 (1986): 304–347. DOI.