Exact Solutions and Families

Exact black-hole metrics are most useful when they are treated as families rather than as isolated line elements. The theory, spacetime dimension, asymptotic boundary condition, global identifications, and allowed parameter domain determine whether a formula describes a regular black hole, a limiting throat, an extended object, or a horizonless continuation. This directory organizes those choices and points to pages that state what is exact, what is only a physical model, and which calculations belong in their full canonical treatments.

Choose a family by physical data

Start with the field equations and the boundary condition, then add conserved charges and dimensionality. A metric that solves one action need not solve another, and the same local geometry can acquire different charges after a quotient or compactification. Throughout this directory, “exact” means an exact solution of the stated classical theory. It does not mean dynamically stable, formed in nature, or observationally identified.

Regime	Families	Decisive data
Four-dimensional, asymptotically flat	Schwarzschild, Kerr, Reissner–Nordström, Kerr–Newman	mass, angular momentum, electromagnetic charge
Positive cosmological constant	Schwarzschild–de Sitter, Kerr–de Sitter	black-hole and cosmological horizons, generator normalization
Negative cosmological constant	Schwarzschild–AdS, Reissner–Nordström–AdS, Kerr–AdS	conformal-boundary metric, ensemble, charge frame
Lower-dimensional gravity	BTZ	quotient data, AdS scale, global identifications
Higher-dimensional and extended horizons	black branes, Myers–Perry, black rings, black strings	worldvolume periods, independent spin planes, horizon topology, gravitational tension
Extra fields or external support	accelerating black holes, Einstein–Maxwell–dilaton black holes, scalarized and hairy black holes	conical defects, coupling constants, branch and stability data

The classic four-dimensional families are nested by controlled limits: Kerr–Newman reduces to Kerr when charge vanishes, to Reissner–Nordström when rotation vanishes, and to Schwarzschild when both vanish. Cosmological families instead change the asymptotic problem. Sending the curvature radius to infinity recovers an asymptotically flat metric locally, but at finite radius the location and causal character of infinity, the normalization of energy, and the admissible boundary data differ.

Read horizon equations as domain conditions

Most pages reduce horizon existence to roots of a radial function. A simple root is ordinarily a regular Killing horizon in a horizon-penetrating chart; a repeated root marks an extremal or other degenerate limit and generally requires a near-horizon scaling. A parameter continuation beyond the discriminant can remain an exact solution while losing its black-hole horizon. The resulting superextremal geometry must not be called a black hole merely because it shares the same metric formula.

Root counting alone is insufficient when several boundaries are present. In de Sitter space, a black-hole horizon and a cosmological horizon bound a finite static patch and have orientation-dependent signed surface gravities. In AdS, the timelike conformal boundary requires boundary conditions, and rotating charges must be referred to a declared asymptotic frame. For accelerating solutions, axes and conical deficits are as important as radial roots. For rings, balance is a global regularity condition rather than an optional aesthetic choice.

Compare limits without erasing the theory

Every page supplies at least two limiting checks. Useful limits include vanishing charge or spin, zero cosmological constant, extremality, a thin-ring or ultraspinning scaling, decompactification, and a near-horizon throat. A controlled limit keeps the physical quantities and coordinates needed for a nondegenerate geometry. Direct substitution at a double root often collapses the chart and can miss the limiting spacetime entirely.

The families also separate several notions that are frequently conflated. Exactness is a statement about field equations. Regularity concerns the metric, matter fields, axes, and identifications. Stability is a perturbative or nonlinear question. Thermodynamic preference depends on an ensemble. Astrophysical status requires formation and evidence. Schwarzschild and Kerr are extraordinarily successful idealized benchmarks; large electromagnetic charge, extra dimensions, strings, dilatons, or scalar hair require additional physical assumptions and are not thereby observed.

Comparing exact solutions

For a research calculation, record the action and normalization of all fields; dimension and topology; coordinate range and identifications; asymptotic structure; independent parameters and their allowed domain; horizon generator; and the definition of conserved charges. Then test field equations, dimensions, horizon roots, regularity, and two independent limits before importing mechanics or perturbations. This discipline is particularly important across references: the symbols $M$, $Q$, $a$, and $m$ are often metric parameters in one convention and physical charges in another.

These entries give enough equations to identify and check each spacetime. Follow their links for full derivations, maximal extensions, horizon mechanics, and perturbation theory. Broad comparisons of higher-dimensional solutions can be found in Emparan and Reall’s review; the four-dimensional landscape is organized in Griffiths and Podolský’s monograph and the reference classification by Stephani et al..

References

• Emparan, Roberto, and Harvey S. Reall. “Black Holes in Higher Dimensions.” Living Reviews in Relativity 11 (2008): 6. DOI.
• Griffiths, Jerry B., and Jiří Podolský. Exact Space-Times in Einstein’s General Relativity. Cambridge: Cambridge University Press, 2009. Publisher.
• Stephani, Hans, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt. Exact Solutions of Einstein’s Field Equations, 2nd ed. Cambridge: Cambridge University Press, 2003. Publisher.