# Can you escape from a black hole in an expanding universe?

Suppose I have a mass $M$ crushed below the Schwarzschild limit. I'm assuming singularities are unphysical; the mass inside the black hole is a super dense and hot cloud of exotic matter, but it's still located within a finite volume. The universe then expands violently.

Normally, tightly bound systems don't increase in size when the universe expands because the forces binding them act faster than the expansion (e.g., the universe is expanding, but the size of the solar system stays the same). But suppose in this scenario the expansion is so violent this doesn't hold. What's stopping the mass $M$ from being dispersed into a volume larger than the Schwarzschild limit and dissolving the horizon? This would allow an indestructible and immortal observer to escape the black hole.

Contrary to what you might think, an expanding universe makes black holes bigger not smaller. So an expanding universe doesn't help you escape the black hole.

The simplest theoretical model for this is an isolated black hole in a de Sitter universe, which is described by the rather obviously named de Sitter-Schwarzschild metric. In a de sitter-Schwarzschild universe the event horizon radius $r$ is given by:

$$1 - \frac{2M}{r} - \Lambda r^2 = 0$$

where $\Lambda$ is the cosmological constant. If we rearrange this as:

$$r = \frac{2M}{1-\Lambda r^2}$$

This is a cubic equation, so to give any analytic solution is going to be messy. However it should be obvious that any value of $\Lambda$ greater than zero will make the denominator smaller on the right hand side and therefore it will give a larger value for $r$.

There is a way to sort of intuitively understand what is going on, though don't take this too literally. The event horizon marks a region in spacetime inside which light travelling outwards can never reach a distant observer. However in a de Sitter expanding universe that distant observer is being accelerated away from the black hole by the expansion. So it's even hard for the light to reach the observer. This is basically why expansion makes the event horizon grow outwards.

There is an exception to this, but it requires such extreme conditions that it's unlikely to be physically relevant. A de Sitter universe has a cosmological event horizon at very large distances. As you increase the cosmological constant this cosmological horizon moves inwards while the black hole horizon moves outwards. If you make the cosmological constant large enough the two horizons will meet and disappear (presumably leaving a naked singularity).

• Ok, but de Sitter space is one that has always been expanding and will always do so. What if instead of a cosmological constant I have some dynamical dark energy that turns on at some point, inflates the universe, then turns off? Dec 6 '15 at 14:32