Myers–Perry Rotating Black Holes

Myers–Perry black holes are the asymptotically flat vacuum generalization of Kerr to $D\geq4$. Their decisive new feature is not merely a changed power of radius: spatial rotations occur in several mutually orthogonal two-planes, so a generic solution carries $N=\lfloor(D-1)/2\rfloor$ independent angular momenta. The radial horizon equation then shows a dimension-dependent split—some spin sectors end at extremality, whereas a singly spinning hole in $D\geq6$ admits arbitrarily large rotation parameter and enters an ultraspinning regime.

Helpful background. Exact Solutions of Einstein’s Equations supplies the full metric family and coordinate domains; Global Structure supplies null horizons and ergoregions; the Einstein equation fixes the vacuum equation; Black-Hole Perturbation Theory supplies the stability analysis beyond this page; and Black-Hole Geometry and Mechanics supplies the charge and Killing-generator framework.

Rotation planes and conserved charges

Write

$$D=2N+1+\epsilon,$$

where $\epsilon=0$ for odd $D$ and $\epsilon=1$ for even $D$. Each orthogonal rotation plane has an azimuth $\phi_i$ and a length parameter $a_i$. Define

$$\Pi(r)=\prod_{i=1}^{N}(r^2+a_i^2).$$

With the asymptotic time and azimuths normalized conventionally, the metric parameter $\mu$ and rotation parameters determine the ADM charges by

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

Thus $\mu$ has dimension $L^{D-3}$ and $a_i$ has dimension $L$. Reversing one azimuth reverses the corresponding $a_i$ and $J_i$ without changing the intrinsic geometry. The exact metric, charge normalization, and dimensional cases are derived in Myers and Perry 1986, §§2–4, pp. 304–328 and organized in Emparan and Reall 2008, §§3.1 and 4.2.

Horizon equation and generator

Candidate Killing horizons occur at positive roots of

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

At the largest regular root $r_+$, the horizon generator and angular velocities are

$$\chi=\partial_t+\sum_{i=1}^{N}\Omega_i\partial_{\phi_i}, \qquad \Omega_i=\frac{a_i}{r_+^2+a_i^2}.$$

The horizon area has the compact form

$$\mathcal A_H=\Omega_{D-2}\,\mu r_+,$$

where the horizon equation has already been used. Setting every $a_i$ to zero gives $r_+^{D-3}=\mu$ and recovers the Tangherlini area.

When at least one spin is nonzero, the stationary-limit surface generally differs from the horizon. In direction-cosine coordinates, $g_{tt}$ depends on latitude as well as radius, and the ergoregion is the domain where the asymptotically timelike field $\partial_t$ becomes spacelike. The horizon is instead where the co-rotating combination $\chi$ is null. When every $a_i=0$, the two definitions remain distinct but their loci coincide and no ergoregion remains. Confusing the surfaces on a rotating solution would erase the very region in which negative Killing energy and rotational extraction are possible.

For variations within the regular vacuum family,

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

and dimensional homogeneity gives

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

These identities are strong normalization checks, but they do not by themselves decide dynamical stability.

Why the spin bound depends on dimension

In four dimensions there is one spin and

$$r^2+a^2-\mu r=0.$$

A positive horizon exists only when $\lvert a\rvert\leq\mu/2$. Equality is extremal and gives the familiar Kerr double root.

In five dimensions there are two independent spins, say $a$ and $b$:

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

Regular horizon existence requires $\mu\geq(\lvert a\rvert+\lvert b\rvert)^2$. Equality gives a regular extremal horizon only when both spins are nonzero. If $b=0$, then $r_+^2=\mu-a^2$; the limiting equality collapses the area and is singular rather than a regular extremal black hole.

For a singly spinning solution in arbitrary dimension, the nonzero horizon radius obeys

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

When $D\geq6$, the left side is continuous and strictly increasing from zero to infinity for positive $r_+$. It therefore has exactly one positive root for every finite $a$. There is no algebraic Kerr bound in this sector. At large $a$ and fixed mass,

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

so the horizon becomes thin transverse to the rotation plane. This is the ultraspinning limit. It resembles a black membrane locally and motivates a Gregory–Laflamme-type instability analysis, but “admits a horizon” and “is stable” are logically separate claims.

Limits and interpretation

Three checks prevent dimension-specific statements from being overgeneralized. The $D=4$ member must reproduce Kerr with $\mu=2G_4M$ and $J=Ma$. The zero-spin member must reproduce Tangherlini with the same $\mu$. Finally, the number of spins must be the rank of the spatial rotation group, not the number of angular coordinates one happens to introduce.

Myers–Perry horizons have topology $S^{D-2}$ throughout their regular parameter domain. That does not mean asymptotically flat higher-dimensional horizons must be spherical. In five dimensions, balanced rings provide a second topology and can share the same asymptotic charges with a Myers–Perry hole.

Check your understanding

Show directly that the singly spinning horizon radius decreases with positive $a$ but cannot reach zero at finite $a$ when $D\geq6$.

Solution

Let

$$h(r,a)=r^{D-5}(r^2+a^2).$$

For $r\gt0$ and $D\geq6$,

$$\frac{\partial h}{\partial r} =(D-3)r^{D-4}+(D-5)a^2r^{D-6}\gt0.$$

Because $h(0,a)=0$ and $h$ diverges as $r$ grows, $h=\mu\gt0$ has one positive root. Implicit differentiation gives

$$\frac{dr_+}{da} =-\frac{2ar_+^{D-5}}{\partial h/\partial r}\lt0$$

for $a\gt0$. The root shrinks monotonically, but $h(0,a)$ remains zero for every finite $a$, so the positive root cannot disappear at finite spin.

The next page replaces spherical topology by a balanced $S^1\times S^2$ horizon and makes charge nonuniqueness explicit: Black Rings and Horizon Nonuniqueness.

References

• Emparan, Roberto, and Harvey S. Reall, “Black Holes in Higher Dimensions,” Living Reviews in Relativity 11, 6 (2008). DOI. Open PDF.
• Myers, Robert C., and Malcolm J. Perry, “Black Holes in Higher Dimensional Space-Times,” Annals of Physics 172, 304–347 (1986). DOI.