Schwarzschild–de Sitter Black Holes

Schwarzschild–de Sitter, also called the Kottler family, is the spherical vacuum solution of four-dimensional general relativity with positive cosmological constant. A regular member contains both a black-hole horizon and a cosmological horizon, with a finite static region between them. It is exact for $R_{\mu\nu}=\Lambda g_{\mu\nu}$, but it is not an asymptotically flat model of an isolated object.

Schwarzschild–de Sitter geometry

Property	Value
Object class	Exact solution family
Theory	Four-dimensional general relativity with $\Lambda\gt0$
Parameters	$m=GM$ and de Sitter radius $\ell=\sqrt{3/\Lambda}$
Asymptotics	de Sitter-type; no timelike spatial infinity
Regular black-hole domain	$0\lt9m^2\Lambda\lt1$
Physical status	Exact $\Lambda$-vacuum solution; idealized cosmological black hole

In static coordinates,

$$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2, \qquad f(r)=1-\frac{2m}{r}-\frac{r^2}{\ell^2}.$$

The same local family follows from the branch-complete Birkhoff theorem with $\Lambda$; the exceptional constant-radius branch must be treated separately, as shown by Schleich and Witt 2010, Theorem 1 and eqs. (4)–(12). The parameter called $M$ is a metric mass parameter whose global charge interpretation depends on the de Sitter boundary prescription.

Three roots and a finite static patch

The horizon equation is

$$r^3-\ell^2r+2m\ell^2=0.$$

In the regular range it has two positive roots $r_b\lt r_c$ and one negative root $r_o=-(r_b+r_c)$. Factoring,

$$f(r)=-\frac{(r-r_b)(r-r_c)(r-r_o)}{\ell^2r},$$

shows that $f\gt0$ only for $r_b\lt r\lt r_c$. The inner positive root is the black-hole horizon and the outer one the cosmological horizon of that observer patch. Ingoing or outgoing Eddington–Finkelstein coordinates are regular at each simple root; $r=0$, not either root, is the curvature singularity.

With the single static generator $T=\partial_t$, the signed surface gravity is

$$\kappa_h=\frac12 f'(r_h) =\frac12\left(\frac1{r_h}-\frac{3r_h}{\ell^2}\right).$$

Hence $\kappa_b\gt0$ and $\kappa_c\lt0$ for the same orientation. Temperatures often use magnitudes, but erasing the sign loses boundary orientation. There is no asymptotic condition $T^2\to-1$ at a timelike infinity, so rescaling $T$ rescales both values. The general normalization dependence of horizon mechanics is explicit in Bardeen, Carter, and Hawking 1973, complete article, printed pp. 161–170.

Nariai degeneracy and mechanics

A double positive root obeys $f=f'=0$, giving

$$r_b=r_c=\frac{\ell}{\sqrt3}=\frac1{\sqrt\Lambda}, \qquad m=\frac{\ell}{3\sqrt3}=\frac1{3\sqrt\Lambda}.$$

This is the Nariai boundary $9m^2\Lambda=1$. The unrescaled static generator has zero surface gravity and the coordinate width of the static interval collapses. A simultaneous coordinate and parameter scaling retains a finite dS$_2\times S^2$ geometry; merely substituting the double-root relation into the static chart does not produce that limit.

Each horizon admits a local first-law-type relation, but the two areas are constrained by the same $(m,\Lambda)$. They cannot be varied independently while both shared parameters are fixed. Moreover, the generic magnitudes $\lvert\kappa_b\rvert$ and $\lvert\kappa_c\rvert$ differ, so a single global equilibrium temperature is unavailable without an additional special construction or ensemble. The distinction between cosmological-horizon geometry and a thermal prescription is explicit in Gibbons and Hawking 1977, §§ 3–5.

Shared-root and global diagnostics

The two positive roots are not independent horizon data. Vieta’s relations give

$$r_b^2+r_br_c+r_c^2=\ell^2, \qquad 2m\ell^2=r_br_c(r_b+r_c).$$

A proposed pair $(r_b,r_c)$ must satisfy both equations for one common $(m,\ell)$. They expose missing factors of three in $\ell^2=3/\Lambda$ and explain why a multi-horizon first law cannot vary both areas freely while holding the shared parameters fixed.

The finite static patch is not the whole spacetime. Beyond $r_c$, the areal radius is timelike and de Sitter conformal infinity is spacelike. The cosmological horizon is tied to an observer domain, whereas the black-hole horizon bounds the singular interior relative to that domain. Nor is this exact $\Lambda$-vacuum automatically a black hole embedded in a matter-filled FLRW universe: adding cosmological matter changes the field equations, matching problem, and global mass definition.

Controlled limits and parameter regimes

Schwarzschild limit. Taking $\Lambda\to0^+$ at fixed $m$ sends $r_b\to2m$ and $r_c\to\infty$. On any fixed-radius region the metric tends to Schwarzschild, although its global boundary changes discontinuously.

de Sitter limit. Taking $m\to0^+$ sends $r_c\to\ell$ and collapses $r_b$ to the origin, leaving the de Sitter static patch. The limiting origin is regular because the Schwarzschild singularity disappears with $m$.

Nariai limit. Approaching $9m^2\Lambda=1$ from below merges the two positive roots. The scaled product limit is regular even though the original static chart degenerates.

For $9m^2\Lambda\gt1$, no positive-root static interval exists. The metric still solves the local $\Lambda$-vacuum equations, but it no longer describes the regular two-horizon black-hole patch. This regime is analogous to “superextremal” only at the level of root loss; its causal geometry is not the Kerr or charged overextremal geometry.

Connections to the detailed theory

The solution models a black hole in an exactly de Sitter-type vacuum, not a local object embedded self-consistently in an evolving matter-filled FLRW universe. The local theorem and exceptional branch are developed in Birkhoff’s Theorem with a Cosmological Constant. Root ordering and normalization are treated in Schwarzschild–de Sitter Horizon Structure. The cosmological meaning of positive $\Lambda$ is separated from the static metric in Cosmological Constant and Accelerated Expansion.

References

• Bardeen, James M., Brandon Carter, and Stephen W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Gibbons, Gary W., and Stephen W. Hawking. “Cosmological Event Horizons, Thermodynamics, and Particle Creation.” Physical Review D 15 (1977): 2738–2751. DOI.
• Schleich, Kristin, and Donald M. Witt. “A Simple Proof of Birkhoff’s Theorem for Cosmological Constant.” Journal of Mathematical Physics 51 (2010): 112502. DOI.