Myers-Perry Black Holes

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Myers–Perry metrics generalize Kerr to asymptotically flat vacuum gravity in $D\ge4$ . The rotation group of the $(D-1)$ spatial dimensions has rank $N=\lfloor(D-1)/2\rfloor$ , so a generic black hole carries $N$ independent angular momenta. Its horizon is determined by a single radial polynomial, but the existence of an extremal spin bound depends sharply on dimension and on how many rotation parameters are nonzero.

Required background. The Einstein Equation supplies the higher-dimensional vacuum equation.

Helpful background. The Riemann Curvature Tensor supplies singularity checks; Exact Solutions of Einstein’s Equations places the family; Global Structure develops horizon extensions; and Killing Fields and Isometry Groups supplies commuting rotation planes.

Independent rotation planes

Write

D=2N+1+\epsilon,

where $\epsilon=0$ for odd $D$ and $\epsilon=1$ for even $D$ . Introduce direction cosines $\mu_i$ , azimuths $\phi_i$ , and, in even dimension, an extra coordinate $\alpha$ , with

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

Each pair $(\mu_i,\phi_i)$ describes one orthogonal rotation two-plane and has its own parameter $a_i$ . Define

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

A general Myers–Perry line element is

\begin{aligned} ds^2={}&-dt^2 +\frac{\mu r^{2-\epsilon}}{\Pi F} \left(dt-\sum_{i=1}^Na_i\mu_i^2d\phi_i\right)^2 +\frac{\Pi F}{\Pi-\mu r^{2-\epsilon}}dr^2\\ &+\sum_{i=1}^N(r^2+a_i^2) (d\mu_i^2+\mu_i^2d\phi_i^2) +\epsilon r^2d\alpha^2. \end{aligned}

Changing every $a_i$ sign is equivalent to reversing the corresponding azimuthal orientation. Direct substitution gives $R_{ab}=0$ . The general metric, conserved-charge normalization, and dimension-dependent horizon regimes are derived in Emparan and Reall 2008, §§ 3.1 and 4.2.

The ADM charges are

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

Thus $\mu$ is a length $^{D-3}$ metric parameter and each $a_i$ is a length. These formulas also recover Kerr in $D=4$ , where $\mu=2G_4M$ and $J=Ma$ .

Horizons and angular velocities

The horizon radii are positive roots of

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

At the outer root $r_+$ , the Killing generator is

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

The zero-rotation limit gives the Tangherlini relation $r_+^{D-3}=\mu$ . In $D=4$ , the radial equation is $r^2+a^2-\mu r=0$ , so a horizon exists only for $\lvert a\rvert\le\mu/2$ .

In $D=5$ there are two spins $a,b$ and

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

A regular outer horizon requires $\mu\ge(\lvert a\rvert+\lvert b\rvert)^2$ . Equality is a regular extremal horizon only when $ab\ne0$ . If one spin vanishes, $r_+^2=\mu-a^2$ and regularity requires the strict bound $\mu>a^2$ ; equality sends $r_+$ and the horizon area to zero and exposes the ring singularity. Thus the singly spinning five-dimensional family has a finite bound but no regular extremal endpoint.

For a singly spinning solution in general dimension, the nonzero root obeys

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

When $D\ge6$ , the left side rises continuously from zero to infinity, so a positive root exists for arbitrarily large $a$ at fixed $\mu$ . There is no Kerr-like upper bound in this singly spinning sector. In the ultraspinning regime $a\gg r_+$ , the horizon extends along the rotation plane and is thin in the transverse directions, like a flattened membrane. This existence statement does not establish dynamical stability; ultraspinning and Gregory–Laflamme-type instabilities are separate perturbation problems.

Singularities, axes, and limiting sectors

The angular coordinates degenerate where direction cosines vanish, but those are normally regular rotation axes once each $\phi_i$ has period $2\pi$ . Genuine singular behavior is diagnosed by $\Pi F=0$ together with curvature invariants and the accessible domain. In four dimensions with nonzero spin, where $\mu_1=\sin\theta$ and $\alpha=\cos\theta$ , this recovers the Kerr ring at $r=0$ , $\alpha=0$ (equivalently $\mu_1=1$ ). When that spin vanishes, the locus collapses to the Schwarzschild center rather than remaining a ring. In higher dimensions the singular set changes with the number of vanishing spin parameters and should not be inferred from the four-dimensional picture alone.

Several limits provide independent checks:

$a_i\to0$ for every $i$ gives Schwarzschild–Tangherlini with the same $\mu$ ;
retaining one $a_i$ and setting the others to zero gives the singly spinning metric and its radial equation above;
$D=4$ gives Kerr after identifying the one direction cosine with $\sin\theta$ ;
the asymptotic $g_{t\phi_i}$ coefficients reproduce $J_i=2Ma_i/(D-2)$ .

The metric has spherical horizon topology $S^{D-2}$ in its regular black-hole parameter range. The existence of black rings in $D=5$ shows that this topology is not forced by asymptotic flatness and stationarity in higher dimensions.

Common pitfalls

Importing the Kerr bound into every dimension. The singly spinning horizon equation has a positive root for all $a$ when $D\ge6$ . Multiple nonzero spins can impose different extremality restrictions.

Counting coordinate azimuths instead of rotation rank. Independent angular momenta correspond to mutually orthogonal two-planes; their number is $\lfloor(D-1)/2\rfloor$ .

Exercises

Prove directly that the singly spinning horizon radius decreases but never vanishes at finite $a$ for $D\ge6$ .

Solution

Define

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

For $r>0$ and $D\ge6$ ,

\frac{\partial h}{\partial r} =(D-3)r^{D-4}+(D-5)a^2r^{D-6}>0.

Also $h(0,a)=0$ and $h\to\infty$ as $r\to\infty$ , so $h=\mu>0$ has exactly one positive root for every finite $a$ . Implicit differentiation gives

\frac{dr_+}{da} =-\frac{2ar_+^{D-5}}{\partial h/\partial r}<0

for $a>0$ . The root shrinks continuously and approaches zero only as $a\to\infty$ .

References

Roberto Emparan and Harvey S. Reall, “Black Holes in Higher Dimensions,” Living Reviews in Relativity 11, 6 (2008). DOI. Open PDF.