Tangherlini Black Holes

Tangherlini’s solution is the static, spherically symmetric vacuum black hole in spacetime dimension $D\geq4$. Its metric looks deceptively like Schwarzschild’s, but the gravitational potential falls as $r^{-(D-3)}$, the unit-sphere area enters the mass normalization, and the horizon radius scales as $M^{1/(D-3)}$. Deriving these factors once provides the clean reference point for rotating holes, rings, strings, and higher-dimensional mechanics throughout this chapter.

Helpful background. Exact Solutions of Einstein’s Equations supplies the distinction between a metric chart and the complete solution family; Global Structure supplies null horizons and regular extensions; the Einstein equation fixes the vacuum field equation; Black-Hole Perturbation Theory supplies the later stability interface; and Black-Hole Geometry and Mechanics supplies the horizon-generator and charge conventions used here.

The spherical vacuum solution

Let $d\Omega_{D-2}^2$ be the metric on a unit round $(D-2)$-sphere and let $r$ be its areal radius. A static spherical line element may be put in the form

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{D-2}^2.$$

In vacuum, one radial Einstein equation is

$$\frac{d}{dr}\left[r^{D-3}(1-f)\right]=0.$$

Consequently

$$f(r)=1-\frac{\mu}{r^{D-3}}, \qquad \mu=r_H^{D-3}.$$

The remaining independent equation is then satisfied. This is the higher-dimensional Schwarzschild solution first obtained in Tangherlini 1963, pp. 636–651. Its positive constant $\mu$ has dimension $L^{D-3}$; it is not itself the ADM mass.

The area of the unit sphere is

$$\Omega_{D-2}=\frac{2\pi^{(D-1)/2}}{\Gamma((D-1)/2)}.$$

Matching the asymptotic field to the $D$-dimensional ADM charge gives

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

This normalization also follows from Gauss’s law: the flux is spread across a sphere whose area is $\Omega_{D-2}r^{D-2}$, so the potential falls as $r^{-(D-3)}$. The complete normalization and its relation to the higher-dimensional solution families are reviewed in Emparan and Reall 2008, §3.2.

Horizon geometry and mechanics

For $M\gt0$, $f$ has one simple positive root,

$$r_H=\left[\frac{16\pi G_D M}{(D-2)\Omega_{D-2}}\right]^{1/(D-3)}.$$

The static coordinates become singular there, but the geometry does not. Defining an ingoing coordinate by $dv=dt+dr/f$ changes the radial-time sector to

$$-f\,dv^2+2\,dv\,dr,$$

which is regular at $r=r_H$. The future horizon is generated by the asymptotically unit-normalized Killing field $\xi=\partial_t$. A spatial cross-section is a round $S^{D-2}$ with

$$\mathcal A_H=\Omega_{D-2}r_H^{D-2}, \qquad \kappa=\frac12 f'(r_H)=\frac{D-3}{2r_H}.$$

The horizon is therefore nondegenerate for every finite positive mass. Its intrinsic scalar curvature is $(D-2)(D-3)/r_H^2$, while the spacetime Kretschmann scalar is finite at $r_H$ and diverges at $r=0$. The zero of $f$ is a coordinate horizon; $r=0$ is the curvature singularity.

The area and surface gravity pass two independent mechanical checks. First, differentiating the mass-radius relation gives

$$dM=\frac{\kappa}{8\pi G_D}\,d\mathcal A_H.$$

Second, homogeneity under $r_H\mapsto\lambda r_H$ gives the vacuum Smarr relation

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

The coefficients differ because $M$ scales as length$^{D-3}$ whereas area scales as length$^{D-2}$. These relations concern the asymptotically flat, uncharged family with fixed $G_D$; cosmological pressure, charge, angular momentum, and compactification lengths add conjugate terms.

What dimension changes

At fixed $G_DM$, the horizon-radius exponent decreases as dimension rises. The area therefore scales as

$$\mathcal A_H\propto (G_DM)^{(D-2)/(D-3)},$$

and the surface gravity scales as $(G_DM)^{-1/(D-3)}$. These are dimensional comparisons, not a claim that gravity is uniformly weaker at large $D$: $G_D$ itself has dimension $L^{D-2}$, so comparing different dimensions requires a specified common length or compactification scale.

The formula does not extend continuously to $D=3$. There the exponent $D-3$ vanishes, asymptotically flat vacuum gravity has no Tangherlini black hole, and black-hole solutions such as BTZ require different asymptotics. Likewise, $M\lt0$ leaves no positive root and exposes the singularity. Solving the field equation away from $r=0$ is not enough to establish a regular black-hole spacetime.

The solution is also a useful lesson about what “the same mass” means across dimensions. An ADM mass is defined relative to a chosen asymptotic geometry and Newton constant. A Tangherlini hole is localized in every spatial direction and has spherical sections at infinity. A uniform black string in the same total spacetime dimension instead has a compact or translational direction, a different falloff power, and a gravitational tension. Its mass cannot be inserted into the formula above without first reducing to the appropriate transverse problem.

Finally, surface gravity depends on the normalization of the generator. Asymptotic flatness fixes $\partial_t$ to unit norm at infinity, so the value quoted here is unambiguous. In a finite cavity, a cosmological spacetime, or a compactification without the same asymptotic time normalization, rescaling the Killing field rescales $\kappa$. Horizon radius and area remain intrinsic, whereas a numerical surface gravity must always be read together with its normalization condition.

Schwarzschild as a normalization check

Set $D=4$. Since $\Omega_2=4\pi$,

$$\mu=\frac{16\pi G_4M}{2(4\pi)}=2G_4M, \qquad r_H=2G_4M.$$

Then $\mathcal A_H=16\pi G_4^2M^2$ and $\kappa=1/(4G_4M)$, exactly the Schwarzschild values. This limit simultaneously checks the power of $r$, the factor $D-2$, the unit-sphere area, and the normalization of the Killing time.

Check your understanding

A five-dimensional Tangherlini black hole has mass $M$ and $\Omega_3=2\pi^2$. Find $r_H$, $\mathcal A_H$, and $\kappa$, and verify the Smarr relation.

Solution

For $D=5$,

$$r_H^2=\frac{16\pi G_5M}{3(2\pi^2)} =\frac{8G_5M}{3\pi}.$$

Hence

$$\mathcal A_H=2\pi^2r_H^3, \qquad \kappa=\frac1{r_H}.$$

The right-hand side of the Smarr formula is

$$3\frac{\kappa\mathcal A_H}{8\pi G_5} =3\frac{2\pi^2r_H^2}{8\pi G_5} =\frac{3\pi r_H^2}{4G_5} =2M,$$

which equals $(D-3)M=2M$.

The next page adds independent rotation planes and shows why a Kerr-like spin bound survives in some dimensions but not others: Myers–Perry Rotating Black Holes.

References

• Emparan, Roberto, and Harvey S. Reall, “Black Holes in Higher Dimensions,” Living Reviews in Relativity 11, 6 (2008). DOI. Open PDF.
• Tangherlini, Frank R., “Schwarzschild Field in $n$ Dimensions and the Dimensionality of Space Problem,” Il Nuovo Cimento 27, 636–651 (1963). DOI.