Gravitational Collapse with de Sitter Asymptotics

A positive cosmological constant changes more than the force balance of collapse. It replaces asymptotically flat null infinity by spacelike conformal infinity, introduces cosmological horizons, and supplies a new length scale. Homogeneous dust gives an exact laboratory in which the interior evolution, exterior geometry, horizon structure, and limiting cases can all be checked without confusing a coordinate horizon with a collapse endpoint.

Helpful background. Black-Hole Geometry and Mechanics supplies the horizon distinctions used below; Black-Hole Perturbation Theory supplies the exterior-decay language; Formulations of General Relativity supplies constraint and matching methods; Exact Solutions of Einstein’s Equations supplies the Schwarzschild–de Sitter family; Causality and Global Structure supplies conformal boundaries; and Computational Relativity supplies the diagnostics needed beyond spherical symmetry.

Einstein–dust evolution with positive Λ

In four dimensions the field equation is

$$G_{ab}+\Lambda g_{ab}=8\pi G T_{ab}, \qquad \Lambda\gt 0.$$

The associated curvature length is $L_{\mathrm{dS}}=\sqrt{3/\Lambda}$. For pressureless homogeneous matter, take the interior metric and stress tensor to be

$$ds^2=-d\tau^2+a(\tau)^2 \left(\frac{dX^2}{1-kX^2}+X^2d\Omega^2\right), \qquad T_{ab}=\rho u_a u_b.$$

Here $X$ is dimensionless, $a$ has dimensions of length, $k\in\{-1,0,1\}$, and the displayed radial chart has $1-kX^2\gt0$. The overdot denotes dust proper time $\tau$. Energy conservation gives $\rho a^3=\mu$, with constant $\mu$. The Friedmann and acceleration equations are

$$\left(\frac{\dot a}{a}\right)^2 =\frac{8\pi G\rho}{3}-\frac{k}{a^2}+\frac{\Lambda}{3}, \qquad \frac{\ddot a}{a}=-\frac{4\pi G\rho}{3}+\frac{\Lambda}{3}.$$

The signs already reveal the competition. Dust attraction grows as $a^{-3}$ during contraction, whereas the positive-$\Lambda$ term is constant in the acceleration equation. A sufficiently developed contraction therefore becomes locally dust dominated, but the initial velocity, curvature $k$, density, and $\Lambda$ decide whether the solution ever enters that regime. The exact branch analysis contains collapse, bounce, and an unstable static threshold; it does not imply that generic matter has the same phase diagram. These branches and their assumptions are worked out in Marković and Shapiro 2000, §§III–V, pp. 3–6.

Matching to Schwarzschild–de Sitter

Let the dust surface be $X=X_b$, so its areal radius is $R(\tau)=a(\tau)X_b$. Spherical matching gives a Schwarzschild–de Sitter exterior,

$$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2, \qquad f(r)=1-\frac{2GM}{r}-\frac{\Lambda r^2}{3},$$

with constant mass

$$M=\frac{4\pi}{3}\rho R^3 =\frac{4\pi}{3}\mu X_b^3.$$

This $M$ is the spherical gravitational mass of the dust, excluding the cosmological contribution. For $k\ne0$ it is not simply density times the proper interior volume. Using $\dot R=X_b\dot a$ in the Friedmann equation gives the surface equation

$$\dot R^2 =\frac{2GM}{R}+\frac{\Lambda R^2}{3}-kX_b^2, \qquad \dot R^2+f(R)=1-kX_b^2.$$

This identity is an independent junction check: the mass inferred from the interior density is exactly the mass in the exterior lapse. The complete spherical construction, including its junction conditions and horizon roots, appears in Marković and Shapiro 2000, §II, pp. 1–3.

When

$$0\lt 9G^2M^2\Lambda\lt 1,$$

$f$ has two positive simple zeros. The smaller is the black-hole Killing horizon $r_b$ and the larger is the cosmological Killing horizon $r_c$; $t$ is timelike only in the static region $r_b\lt r\lt r_c$. At equality the roots coalesce. A nondegenerate Nariai spacetime is obtained by an additional near-horizon coordinate scaling, not simply by keeping this static chart at the double root. A collapsing surface crossing $r_b$ is not crossing the cosmological horizon, and neither crossing alone locates the event horizon on earlier interior slices. The event horizon is global.

The surface acceleration gives a second useful threshold:

$$\ddot R=-\frac{GM}{R^2}+\frac{\Lambda R}{3}, \qquad R_{\rm bal}=\left(\frac{3GM}{\Lambda}\right)^{1/3}.$$

A shell momentarily at rest accelerates inward below $R_{\rm bal}$ and outward above it. At exact balance, the derivative of the right-hand side is $\Lambda\gt0$, so a displacement grows away from the equilibrium rather than being restored. The balance radius describes the force competition, while $r_b$ and $r_c$ describe null horizons; they answer different questions.

Causal endpoint and exterior redshift

An asymptotically flat collapse endpoint is often described using future null infinity $\mathscr I^+$. That language cannot simply be copied to de Sitter asymptotics: conformal future infinity is spacelike, while a static observer has a cosmological horizon bounding the region that can communicate with the observer. The observer-relative character of this cosmological event horizon is developed in Gibbons and Hawking 1977, pp. 2738–2741. Black-hole and cosmological horizons must consequently be specified relative to the chosen asymptotic completion and observer domain.

This boundary geometry also changes dispersal. Outgoing disturbances leave a static patch through its cosmological horizon rather than propagating along an infinite asymptotically flat exterior to null infinity. In a rigorous but narrower vacuum result, sufficiently small high-regularity perturbations of slowly rotating Kerr–de Sitter data converge exponentially to a nearby Kerr–de Sitter metric Hintz and Vasy 2018, Theorem 1.1, pp. 4–5. This rapid exterior decay does not erase an interior singularity or prove that arbitrary matter collapse settles to Schwarzschild–de Sitter. Such a stability statement requires a declared matter model, data norm, gauge, and asymptotic region.

The dust model proves a narrower result: for the exact homogeneous ball, once its parameters select the collapsing branch and the exterior lies below the Nariai bound, the matched exterior contains distinct black-hole and cosmological horizons. Pressure, shocks, rotation, gravitational radiation, and nonspherical horizon dynamics lie outside that model.

Limiting checks

Three limits expose most sign errors.

• With $\Lambda\to0$, $f\to1-2GM/r$ and the surface equation reduces to Oppenheimer–Snyder collapse.
• With $M\to0$, the black-hole root disappears and $r_c\to\sqrt{3/\Lambda}$, the de Sitter cosmological radius.
• At $9G^2M^2\Lambda\to1$, the two positive roots coalesce; treating them as well-separated horizons near this limit is invalid.

These tests also separate scale from topology. On scales $R\ll L_{\mathrm{dS}}$, the term $\Lambda R^2/3$ is locally small, yet the global future boundary remains de Sitter-like.

Exercise: derive the Nariai bound

Show that a double positive root of $f(r)=1-2GM/r-\Lambda r^2/3$ occurs precisely when $9G^2M^2\Lambda=1$.

Solution

A double root satisfies $f(r_N)=0=f'(r_N)$. From

$$f'(r)=\frac{2GM}{r^2}-\frac{2\Lambda r}{3},$$

one obtains $GM=\Lambda r_N^3/3$. Substitution into $f(r_N)=0$ gives $1-\Lambda r_N^2=0$, hence $r_N=1/\sqrt{\Lambda}$ and $GM=1/(3\sqrt{\Lambda})$. Squaring the latter relation yields $9G^2M^2\Lambda=1$. Below this value the cubic has distinct black-hole and cosmological roots; at the value they coalesce.

References

• Gibbons, G. W., and Stephen W. Hawking. “Cosmological Event Horizons, Thermodynamics, and Particle Creation.” Physical Review D 15 (1977): 2738–2751. doi:10.1103/PhysRevD.15.2738.
• Hintz, Peter, and András Vasy. “The Global Non-linear Stability of the Kerr–de Sitter Family of Black Holes.” Acta Mathematica 220 (2018): 1–206. doi:10.4310/ACTA.2018.v220.n1.a1. Open PDF.
• Marković, Draža, and Stuart L. Shapiro. “Gravitational Collapse with a Cosmological Constant.” Physical Review D 61 (2000): 084029. doi:10.1103/PhysRevD.61.084029. Open PDF.