# How does the expansion of the Universe affect Black Holes?

As know our Universe is expanding, which I have always visualised as some stretching of the metric of space-time, which "flattens" it. In the same time. our theoretical perception of Black Holes usually includes some kind of singularity (or at least some extreme curvature), bounded by an event horizon. Clearly, if my naive understanding for the expansion of the Universe is correct, then it would be a natural instrument that may potentially dissolve Black Holes, which I've never heard of before => it's probably wrong.

Thus, I'd like to ask for a clarification what the Expansion of the Universe actually does and how it affects Black Holes (and the curvature of space-time, in general)?

• Related: physics.stackexchange.com/q/2110/2451 and links therein. – Qmechanic Oct 10 '15 at 17:31
• This is indeed somehow related with my question, although it deals mainly with matter. Black holes are not necessarily "matter"-related, rather than particular solutions of the Einstein equations (=> regions in space-time) with a singularity. – Newbie Oct 11 '15 at 13:52

There isn't an analytic solution for a universe expanding exactly like ours, but a black hole in a universe with just a cosmological constant (no matter) is described by the de Sitter - Schwarzschild metric. As our universe expands and the matter density decreases this will become an increasingly good approximation.

The de Sitter - Schwarschild metric is:

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + d\Omega^2$$

where the function $f(r)$ is:

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

with $M$ the mass of the black hole and $\Lambda$ the cosmological constant. The de Sitter - Schwarzschild metric has two horizons given by the two roots of $f(r) = 0$. The horizon at large $r$ is the cosmological horizon associated with a de Sitter universe, and the horizon at small $r$ is the black hole horizon. The bit of the spacetime in between the two horizons is where we live.

Solving for the position of the inner horizon is messy because we end up with a cubic equation, however it should be obvious that a non-zero value for $\Lambda$ means the black hole horizon moves outwards. In effect the expansion of the universe pulls the horizon outwards and increases the size of the black hole.

• Very interesting. Good to see that my GR still needs work ;) – Kyle Kanos Oct 9 '15 at 10:34