This is a question that’s bugged me for a while.

First, here’s why I think this is a reasonable question to ask.

Suppose you cram a mass M into a spherical volume of radius R such that R is less than the Schwarzschild radius, i.e., R \leq 2GM/c^2. Then it’s a pretty well understood consequence of general relativity that the mass will collapse to form a black hole.

Current estimates of the mass of the (observable) Universe vary quite a bit. This webpage seems pretty representative, though, giving a value for the mass of 3 x 10^52 kg.

This gives a value for the corresponding Schwarzschild radius of about 6 billion light years.

The radius of the observable Universe is, of course, quite a bit bigger than this. But the Universe is also expanding, and at some point in the past its radius was quite a bit less than six billion light years.

If that was the case, why didn’t it collapse to form a singularity? In short, how come we’re still here?

Any cosmologists out there who can enlighten me?

**Update:** In comments, Dave Bacon points to an enlightening essay from John Baez, explaining some of what’s going on.

My interpretation of the essay is that the standard lore I learned as an undergraduate (namely, that if you take a mass M and compress it into a smaller radius than the Schwarzschild radius then a black hole must inevitably form) is wrong, and that the FRW cosmology provides a counterexample.

This begs the question of *when*, exactly, a black hole can be guaranteed to form.