What Changes with Dimension?

Four-dimensional black-hole intuition mixes robust geometric principles with accidents of dimension. Horizons remain null hypersurfaces generated by Killing fields in stationary exact solutions, but local vacuum curvature, allowed topology, rotational bounds, charge normalization, and uniqueness all change. The comparison below isolates which claim survives, which needs new hypotheses, and which already fails among the benchmark metrics in this chapter.

Required background. The Einstein Equation supplies the dimension-dependent field equations.

Helpful background. The Riemann Curvature Tensor supplies local diagnostics; Exact Solutions of Einstein’s Equations places the families; Global Structure develops topology and causality; and Killing Fields and Isometry Groups supplies stationary and rotational symmetries.

Local curvature and radial falloff

In $D=3$, the Weyl tensor vanishes identically and vacuum with cosmological constant implies

$$R_{abcd}=\frac{2\Lambda}{(D-1)(D-2)} (g_{ac}g_{bd}-g_{ad}g_{bc}).$$

BTZ is therefore locally AdS$_3$ everywhere in its regular region; its mass and spin reside in the quotient identification and asymptotic charges. In $D\ge4$, Ricci-flatness leaves nontrivial Weyl curvature. Tangherlini and Myers–Perry have local tidal fields and central curvature singularities.

For an isolated spherical mass in $D\ge4$,

$$g_{tt}+1=O(r^{-(D-3)}).$$

For a uniform $p$-brane built from a $d=D-p$ dimensional seed, the falloff is instead $r^{-(d-3)}$ in the transverse radius. Thus radial power counts noncompact transverse dimensions, not merely the total coordinate dimension.

Horizon topology and global identifications

Four-dimensional asymptotically flat stationary vacuum black holes satisfying the standard regularity hypotheses have spherical horizon cross-sections. Higher dimensions permit more positive-Yamabe topologies and more interval structures. The exact examples here include

$$S^{D-2} \quad\text{(Tangherlini and Myers–Perry)},$$ $$S^{d-2}\times T^p \quad\text{(compact uniform branes)},$$ $$S^1\times S^2 \quad\text{(five-dimensional black ring)}.$$

Topology is global: a direct product or periodic quotient changes it without adding transverse local curvature. BTZ makes the same point in a different way, creating a black hole from constant-curvature AdS$_3$ through identification.

Restrictions still exist. Horizon topology theorems use energy conditions, asymptotics, and global assumptions; “anything is possible in higher dimensions” is false. The exact metrics show that spherical topology is not unique, not that topology is unconstrained.

Rotation and extremality

The number of independent angular momenta is

$$N=\left\lfloor\frac{D-1}{2}\right\rfloor.$$

Kerr in $D=4$ has one rotation parameter and a finite extremal bound. Myers–Perry in $D=5$ has two, and its horizon condition couples them. A singly spinning Myers–Perry black hole in $D\ge6$ has a horizon for arbitrarily large spin parameter at fixed mass parameter, entering an ultraspinning regime. Black rings add a different mechanism: spin supports an extended $S^1$, and balance fixes a relation between shape parameters.

An absence of an algebraic upper bound is not an absence of physical constraints. Ultraspinning objects can be unstable; rings can possess thin and fat branches; regularity and absence of closed timelike curves still restrict parameters. Extremality, balance, and stability are distinct concepts.

Uniqueness and charge data

In four-dimensional asymptotically flat electrovac gravity, strong uniqueness theorems can identify a stationary connected black hole by conserved charges under substantial hypotheses. In five dimensions, a Myers–Perry black hole and a black ring can have the same $(M,J)$, and two rings can share those charges. Conserved monopole charges do not encode horizon topology or interval structure.

Even charge definitions change with asymptotics:

• asymptotically flat Tangherlini and Myers–Perry metrics use $D$-dimensional ADM charges;
• BTZ charges are referenced to asymptotically AdS$_3$ boundary conditions, with global AdS at a negative mass offset;
• noncompact branes have densities, while compact branes have totals depending on their periods;
• conical or Kaluza–Klein asymptotics introduce tensions and other boundary data.

The same metric symbol $m$ therefore cannot be compared across families without its normalization formula.

What survives across dimensions

Several methods remain reliable:

1. Candidate horizons are located by invariant Killing or null-expansion conditions, then crossed in regular coordinates.
2. Curvature invariants distinguish many coordinate horizons from genuine curvature singularities, though quotients and null algebraic structure require additional data.
3. Axis regularity is tested by circumference over proper radius, not by coordinate smoothness alone.
4. Conserved charges follow from a stated asymptotic structure and Killing normalization.
5. Parameter limits include coordinate scalings, periodic identifications, and compactification lengths.

The broad classification, exact families, and known boundaries of higher-dimensional uniqueness are reviewed in Emparan and Reall 2008, §§ 3–8. The comparison here is restricted to Einstein gravity and the exact benchmark geometries already developed. Higher-curvature field equations can change both local solutions and entropy laws.

Common pitfalls

Treating a four-dimensional theorem as dimension-free. Check the theorem’s dimension, topology, asymptotics, analyticity, and connectedness hypotheses before applying it.

Comparing parameter symbols instead of charges. Convert every length parameter using the appropriate $G_D$, unit-sphere area, periods, and background normalization.

Exercises

Explain why a uniform black string and a five-dimensional Tangherlini black hole have different far-field powers despite satisfying the same five-dimensional vacuum equation.

Solution

A five-dimensional Tangherlini black hole is localized in four spatial dimensions, so its potential falls as $r^{-(5-3)}=r^{-2}$ and its infinity has $S^3$ sections. A five-dimensional uniform string is a four-dimensional Schwarzschild solution times a line. It is invariant along the string, so flux spreads through only three transverse spatial dimensions and its potential falls as $r^{-1}$. Its asymptotic sections include $S^2\times S^1$ for a compact string. The field equation is the same, but the boundary conditions and transverse dimensionality differ.

References

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