Black Holes in de Sitter Spacetimes

Positive cosmological constant turns black-hole mechanics into a two-boundary problem. A static observer region can be bounded by a black-hole horizon and a cosmological horizon, with one solution parameter controlling both. Their signed surface gravities have opposite orientations and usually unequal magnitudes. Rotation adds a frame choice, degeneracy requires scaled limits, and dynamical evolution separates global event horizons from quasi-local marginal boundaries.

Helpful background. Exact Solutions of Einstein’s Equations supplies de Sitter coordinate families; Causality and Global Structure supplies observer patches and spacelike conformal infinity; and The Einstein Equation supplies the positive-$\Lambda$ field equations.

The two-boundary problem

Set $\ell^2=3/\Lambda$. In Schwarzschild–de Sitter,

$$f(r)=1-\frac{2GM}{r}-\frac{r^2}{\ell^2}$$

has a black-hole root $r_b$ and cosmological root $r_c$ only when $0\lt9G^2M^2\Lambda\lt1$. The finite interval $r_b\lt r\lt r_c$ is static. For $T=\partial_t$,

$$\kappa_b\gt0, \qquad \kappa_c\lt0.$$

These signs are not optional temperature labels; they record the orientations of two boundaries of one region. Their magnitudes can be compared only after using the same generator. The rotating generalization replaces $r^2f$ by

$$\Delta_r=(r^2+a^2) \left(1-\frac{r^2}{\ell^2}\right)-2mr,$$

and each simple root has its own null generator $\chi_h=T+\Omega_h\Phi$. The four-dimensional formulas follow from Gibbons, Lü, Page, and Pope 2005, §§3–5.

The chapter does not construct the global de Sitter conformal extension or a quantum thermal state. It develops the classical geometry and variational identities needed to state those later questions correctly.

Check your preparation

You are ready for the direct route if you can solve and order polynomial roots, distinguish a Killing horizon from an observer event horizon, and explain why a Killing generator can be rescaled. You should also be able to tell a metric parameter from a Hamiltonian charge and a coordinate separation from a proper distance.

If the causal distinction is uncertain, review Causality and Global Structure. If horizon generators or surface gravity are unclear, use Black-Hole Geometry and Mechanics. Exact Solutions of Einstein’s Equations is the repair route for regular coordinates and product limits. The essential habit is to state the horizon component, observer patch, generator, frame, and boundary before interpreting a number.

Six pages in working order

1. Schwarzschild–de Sitter Horizon Structure solves the cubic, orders the two positive roots, proves the mass bound, and derives the Nariai double-root condition.
2. Kerr–de Sitter Horizon Mechanics introduces the rotating radial polynomial, areas, signed surface gravities, and two azimuthal frame conventions.
3. Two-Horizon Surface Gravities and Nonequilibrium proves that uncharged equal magnitudes force degeneracy and contrasts the charged lukewarm branch.
4. Nariai and Ultracold Limits magnifies coalescing horizons, derives the triple-root parameters, and checks the product curvatures.
5. Charges and First Laws in de Sitter derives the two area signs and compares patch, horizon, finite-boundary, and conformal charges.
6. Dynamical and Quasi-Local de Sitter Horizons compares event, trapping, dynamical, isolated, and asymptotic structures through the 2026-08-12 evidence cutoff.

This order moves from one polynomial to increasingly consequential boundary choices. The final two pages rely on the signs and normalizations fixed by the first four.

Three routes through the chapter

The horizon-geometry route follows Schwarzschild–de Sitter, Kerr–de Sitter, and the coalescence page. It emphasizes roots, areas, generators, and scaled limits. The mechanics route reads the two stationary families, compares their surface gravities, and proceeds to charges and first laws. It is best for preventing sign and frame mismatches.

The dynamical-boundary route begins with Schwarzschild–de Sitter to fix the stationary coincidence of Killing, event, isolated, and marginal horizons, then jumps to the frontier page. There a Vaidya–de Sitter calculation shows why a local marginal tube and a globally reconstructed event horizon separate when the mass changes. The underlying quasi-local versus event-horizon distinction, including the required structure on each boundary, is reviewed in Ashtekar and Krishnan 2025, §§2.1–4 and §7.

Across all routes, exact stationary calculations, limiting geometries, local existence theorems, numerical horizon tracks, and open comparison questions retain distinct evidence labels.

A compact convention sheet

Quantity	Convention in this chapter	Check
de Sitter scale	$\ell^2=3/\Lambda$ with $\Lambda\gt0$	Pure de Sitter has cosmological radius $\ell$
Static mass length	$m=GM$	Small black-hole root approaches $2m$
Kerr–de Sitter parameter	$m$ is kept as a metric parameter until a charge prescription is chosen	Do not call it a unique energy
Horizon labels	$r_b$ is black-hole, $r_c$ cosmological	The static patch lies between them
Surface gravity	Signed for one declared generator	Magnitude does not replace orientation
Rotating frame	State the basis $(T,\Phi)$ and any rigid azimuthal shift	Differences of horizon angular velocities survive a common shift
Extended pressure	$P=-\Lambda/(8\pi G)$	Positive $\Lambda$ gives negative $P$

The strongest algebraic checks are the $a\to0$ reduction to Schwarzschild–de Sitter, the $\Lambda\to0$ reduction to Kerr or Reissner–Nordström, and equality of the metric parameter computed at every root. A double root must satisfy the radial equation and its first derivative; an ultracold triple root also satisfies the second derivative. A limiting coordinate chart is accepted only after its metric, curvature, and proper distance remain finite.

Review the chapter

Consider an uncharged Schwarzschild–de Sitter family at fixed $\Lambda$. First, suppose a pair of distinct horizons is claimed to have equal surface- gravity magnitudes after one common normalization. Second, increase the metric parameter. Third, replace it by a Vaidya–de Sitter function $M(v)$. Diagnose the equilibrium claim, the two area variations, and which horizon can be found from data on one slice.

Solution

For distinct uncharged roots, equality would imply $r_br_c=\ell^2/3$. The common horizon equation also gives $r_b^2+r_br_c+r_c^2=\ell^2$. Together they force $(r_c-r_b)^2=0$, contradicting distinctness. Equal magnitudes occur only at the Nariai double root; one common rescaling cannot change their ratio.

At fixed $\Lambda$,

$$\delta M =\frac{\kappa_b}{8\pi G}\delta\mathcal A_b =\frac{\kappa_c}{8\pi G}\delta\mathcal A_c.$$

Since $\kappa_b\gt0$ and $\kappa_c\lt0$, increasing $M$ increases $\mathcal A_b$ and decreases $\mathcal A_c$.

In Vaidya–de Sitter, the marginal radii are instantaneous roots of $f(v,r)=0$, so a slice can locate apparent-horizon candidates after a foliation and null orientation are chosen. The event horizon cannot be identified from that slice alone; it requires the completed future and the selected observer or asymptotic endpoint.

After this chapter, you should be able to derive the de Sitter horizon bounds, translate rotating frames, distinguish signed from absolute surface gravity, take a nontrivial coalescence limit, and write a boundary-specific first law. Continue to Black Holes in Anti-de Sitter Spacetimes to see how a timelike conformal boundary changes charges, rotation, topology, and mechanics.

References

• Ashtekar, Abhay, and Badri Krishnan. “Quasi-Local Black Hole Horizons: Recent Advances.” Living Reviews in Relativity 28 (2025): 8. DOI. Open manuscript.
• Gibbons, Gary W., Hong Lü, Don N. Page, and Christopher N. Pope. “The General Kerr–de Sitter Metrics in All Dimensions.” Journal of Geometry and Physics 53 (2005): 49–73. DOI. Open manuscript.