Schwarzschild-Tangherlini Spacetimes

The Schwarzschild–Tangherlini solution is the unique local spherically symmetric vacuum geometry in $D\ge4$ under the usual nondegeneracy assumptions. Its potential falls as $r^{-(D-3)}$, so the horizon radius scales as $M^{1/(D-3)}$ rather than linearly with mass. Keeping the unit-sphere area and $G_D$ explicit fixes the conversion between the metric length parameter and ADM mass.

Required background. The Einstein Equation supplies the $D$-dimensional vacuum field equation.

Helpful background. The Riemann Curvature Tensor supplies the curvature check; Exact Solutions of Einstein’s Equations places the family; Global Structure develops horizon extensions; and Killing Fields and Isometry Groups supplies static spherical symmetry.

The higher-dimensional spherical vacuum metric

Let $d\Omega_{D-2}^2$ be the unit round metric and take areal radius $r$. A static ansatz can be written

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

For $R_{ab}=0$, a radial Einstein equation reduces to

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

Hence

$$f(r)=1-\frac{\mu}{r^{D-3}} =1-\left(\frac{r_0}{r}\right)^{D-3}, \qquad r_0^{D-3}=\mu.$$

The remaining independent field equation is then satisfied, providing a useful check of the reduction. Tangherlini first gave this higher-dimensional vacuum solution and its charged analogue in Tangherlini 1963, pp. 636–651.

The unit-sphere area

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

enters the ADM normalization:

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

This follows either from the higher-dimensional ADM integral or by matching

$$g_{tt}=-(1+2\Phi+\cdots), \qquad \Phi=-\frac{8\pi G_DM}{(D-2)\Omega_{D-2}r^{D-3}}$$

to the weak-field Gauss law. Calling $\mu$ the mass without this factor obscures dimension and convention dependence.

Horizon and curvature

For $M>0$, $f$ has a simple positive root at $r=r_0$. The static chart covers $r>r_0$. An ingoing coordinate

$$v=t+\int\frac{dr}{f(r)}$$

changes the radial-time sector to $-f\,dv^2+2\,dv\,dr$, which is regular at the simple root. The horizon has topology $S^{D-2}$ and area

$$\mathcal A_H=\Omega_{D-2}r_0^{D-2}.$$

With the asymptotic normalization of $t$, its surface gravity is

$$\kappa=\frac12f'(r_0)=\frac{D-3}{2r_0}.$$

The horizon is not a curvature singularity. In vacuum the Ricci tensor vanishes, while

$$R_{abcd}R^{abcd} =(D-1)(D-2)^2(D-3) \frac{\mu^2}{r^{2(D-1)}}.$$

This is finite at $r=r_0$ and diverges at $r=0$. For $D=4$, $\mu=2G_4M$ and the expression becomes $48G_4^2M^2/r^6$, recovering Schwarzschild exactly.

Dimensional scaling

Solving the mass relation for the horizon radius gives

$$r_0=\left[ \frac{16\pi G_DM}{(D-2)\Omega_{D-2}} \right]^{1/(D-3)}.$$

Thus a fixed multiplicative change in $M$ produces a smaller fractional change in $r_0$ as $D$ increases. The far-field curvature also decays more rapidly: the Kretschmann scalar falls as $r^{-2(D-1)}$. Neither statement means gravity is uniformly “weaker” in higher dimensions, because $G_D$ has dimension $L^{D-2}$ and comparisons require a fixed physical scale.

The construction fails at $D=3$: the exponent $D-3$ vanishes and the asymptotically flat Newtonian Green function is logarithmic. Three-dimensional black holes instead require, for example, the negative cosmological constant and quotient structure of BTZ. At $D=2$, Einstein’s tensor vanishes identically. These lower-dimensional cases are not obtained by naively substituting $D$ into the Tangherlini formula.

For $M<0$, $f$ has no positive horizon and $r=0$ is naked. The exact metric still solves vacuum Einstein equations away from the origin, but it is not a regular black hole. This distinction between an exact local solution and an admissible black-hole domain applies across all dimensions.

The same solution also supplies a clean dimensional Smarr check. Using the asymptotic time normalization, define $T=\kappa/(2\pi)$ and $S=\mathcal A_H/(4G_D)$. The formulas above give

$$TS=\frac{D-3}{D-2}M,$$

or $(D-3)M=(D-2)TS$. This follows from the different scaling weights of mass and area and independently tests the factors of $D-2$, $\Omega_{D-2}$, and $G_D$. It is recorded here as a geometric consistency relation; thermodynamic interpretation and the first law are developed with the necessary mechanics elsewhere.

For the charged Tangherlini analogue, the radial function contains two mass- and charge-dependent inverse powers and can have inner and outer horizons. Its normalization depends on the Maxwell action convention. One should therefore not obtain it by replacing $2M/r$ with a guessed higher-dimensional expression; derive the charge coefficient from the stated Einstein–Maxwell equations and a flux integral.

Common pitfalls

Dropping $\Omega_{D-2}$. The metric parameter has dimension $L^{D-3}$; the physical ADM mass includes both $G_D$ and the unit-sphere area.

Treating $D$ as a continuous substitution at special dimensions. The solution assumes $D\ge4$. The field equations and Green functions change qualitatively at $D=3$ and $D=2$.

Exercises

Use the mass–radius relation to express the horizon area as a power of $M$.

Solution

Substitute the expression for $r_0$ into $\mathcal A_H=\Omega_{D-2}r_0^{D-2}$:

$$\mathcal A_H =\Omega_{D-2} \left[ \frac{16\pi G_DM}{(D-2)\Omega_{D-2}} \right]^{\frac{D-2}{D-3}}.$$

Hence $\mathcal A_H\propto M^{(D-2)/(D-3)}$. For $D=4$ the exponent is $2$; for $D=5$ it is $3/2$.

References

• Frank R. Tangherlini, “Schwarzschild Field in $n$ Dimensions and the Dimensionality of Space Problem,” Il Nuovo Cimento 27, 636–651 (1963). DOI.