Lower- and Higher-Dimensional Spacetimes

Changing spacetime dimension changes more than numerical coefficients. In three dimensions vacuum curvature is locally fixed by the cosmological constant, yet global identifications produce the BTZ black hole. Above four dimensions gravity falls off faster, rotations occur in several independent planes, horizons need not be spherical, and mass and angular momentum need not identify a unique stationary solution. This chapter uses exact metrics to separate genuinely dimension-dependent structure from features shared by all dimensions.

Helpful background. The Einstein Equation supplies the $D$-dimensional vacuum equation; The Riemann Curvature Tensor supplies local curvature tests; Exact Solutions of Einstein’s Equations places the families; Global Structure develops horizons and boundaries; and Killing Fields and Isometry Groups supplies stationary and rotational symmetries.

Dimension-dependent conventions

Let $D$ denote total spacetime dimension and use

$$G_{ab}+\Lambda g_{ab}=8\pi G_D T_{ab}.$$

The area of the unit $n$-sphere is

$$\Omega_n=\frac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)}.$$

Factors of $G_D$ and $\Omega_{D-2}$ are retained because the length parameter in a metric is not automatically its physical mass. For example, the spherical vacuum solution has

$$f(r)=1-\frac{\mu}{r^{D-3}}, \qquad M=\frac{(D-2)\Omega_{D-2}}{16\pi G_D}\mu.$$

The power $D-3$ comes from the Green function of the spatial Laplacian in $D-1$ dimensions. Four-dimensional $1/r$ behavior is therefore one member of a dimensional sequence, not a universal gravitational law.

When a direction is translationally invariant and noncompact, total mass diverges and a density per unit length or volume replaces it. When it is periodically identified, the period is part of the spacetime data. The pages below state whether charges are totals, densities, or metric parameters.

Five benchmark geometries

The BTZ Black Hole begins in $D=3$. Its Riemann tensor is locally that of anti-de Sitter space everywhere away from identifications, while a discrete quotient creates horizons and a causal singularity. It is the clearest demonstration that local curvature does not determine global black-hole structure.

Schwarzschild–Tangherlini Spacetimes generalize spherical vacuum geometry to $D\ge4$ and fix the horizon–mass normalization. Myers–Perry Black Holes add the $N=\lfloor(D-1)/2\rfloor$ independent rotation planes permitted by the rank of $SO(D-1)$. Their extremality structure changes with dimension; in sufficiently high dimension a singly spinning parameter can grow without the four-dimensional Kerr bound.

Uniform Black Strings and Branes take a Ricci-flat black hole times flat directions. The local field equation factorizes, but horizon topology and global charges change. Five-Dimensional Black Rings use rotation to support an $S^1\times S^2$ horizon. In a range of mass and angular momentum, a ring and a Myers–Perry black hole coexist, so four-dimensional-style uniqueness fails.

What Changes with Dimension? then compares curvature, asymptotics, topology, rotational bounds, and uniqueness without conflating local exact solutions with their dynamical stability. A comprehensive exact-solution survey and the higher-dimensional uniqueness boundary are given in Emparan and Reall 2008, §§ 2–7.

A common verification sequence

Every dimensional metric is checked with the same questions:

1. Which value of $D$ is assumed, and what is the dimension of $G_D$?
2. Is the metric Ricci-flat or Einstein with the stated $\Lambda$?
3. How are metric parameters converted to ADM, Abbott–Deser, or Brown–Henneaux charges?
4. Which roots are regular Killing horizons, and what is the horizon topology after global identifications?
5. Which directions are compact, which are noncompact, and are charges totals or densities?
6. Does a lower-dimensional, zero-rotation, or decompactification limit recover a known geometry with the correct normalization?

This sequence catches two frequent category errors. First, a product with a flat line does not preserve asymptotic flatness in the same dimension, even though it preserves Ricci flatness. Second, a coordinate degeneration at an axis is not the same as a quotient singularity or a curvature singularity.

What is handed off

These pages establish metrics, domains, local curvature, horizon geometry, charges, and exact limiting relations. They do not prove nonlinear stability, compute quasinormal spectra, or develop black-string and ultraspinning instabilities. The Gregory–Laflamme threshold is mentioned only to mark that an exact uniform string need not be dynamically preferred. Higher-curvature gravities, supersymmetric classifications, Kaluza–Klein reduction dynamics, and holographic applications require different field equations or additional structures.

Review the chapter

Derive the radial falloff of a weak static gravitational potential in $D$ spacetime dimensions.

Solution

There are $d=D-1$ spatial dimensions. Away from a point source, spherical symmetry gives

$$0=\nabla^2\Phi =\frac1{r^{d-1}}\frac{d}{dr} \left(r^{d-1}\frac{d\Phi}{dr}\right).$$

Thus $r^{d-1}\Phi'=C$, and for $d>2$,

$$\Phi(r)=\Phi_\infty-\frac{C}{(d-2)r^{d-2}} =\Phi_\infty-\frac{C}{(D-3)r^{D-3}}.$$

The Tangherlini metric therefore contains a mass term proportional to $r^{-(D-3)}$. For $D=3$ the integration is logarithmic, one reason asymptotically flat three-dimensional gravity behaves differently.

References

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