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.

From → Physics

1. It’s been a while since I studied GR, but I think the answer lies in the fact that Einstein’s field equations are time reversible. Just as there are timelike geodesics that start outside the Schwarzschild radius, cross the event horizon and end at the singularity, there are corresponding timelike geodesics going in the opposite direction that start at the singularity and leave the black hole, which is now known as a white hole.

The fact that we don’t expect to see particles travelling along the outgoing geodesics, and thus escaping from the black hole is, I think, simply a consequence of the initial conditions – all the avaliable matter existed before the black hole formed and so can only follow ingoing geodesics. (Things are different when we account for quantum effects and Hawking radiation becomes possible.)

The Big Bang, on the other hand, could be regarded as the collapse of a black hole run in reverse. The matter is now following the outgoing geodesics, which must cross the event horizon and leave the black hole.

The difference between the collapse of a black hole and the Big Bang, comes from the different initial conditions. Matter follows ingoing geodesics in a collapsing black hole, but outgoing geodesics in the expanding Big Bang. The Big Bang is more like a white hole.

(Of course, I’m being a bit careless in my reasoning above. The Schwarzschild solution assumes a vacuum beyond some radius. The Big Bang assumes a uniform density everywhere so doesn’t have an event horizon like a black hole does that divides the Universe into ‘inside’ and ‘outside’ the black hole. I hope the gist of my argument is clear though.)

2. Thanks for that, Dave. It’s extremely informative. I obviously need to revise my thinking about black holes quite a bit.

I find it interesting that the naive (but often stated) idea “if I have more than this amount of mass inside the Schwarzschild radius then a black hole will form” is wrong. It’d be nice to know how one _should_ think about black hole formation.

3. Following up on my last comment, it seems that the resolution must lie in developing a precise understanding of the singularity theorems governing black hole formation. Obviously, FRW avoids the naive idea of when black hole formation takes place; I should better understand the conditions under which it does occur.

4. Andy: Thanks for the comment. If you look at the link Dave posted, they argue that it’s not really right to think of the FRW / big bang cosmology in terms of a white hole picture. The key point seems to be this assertion about the differences between the two:

“Outside a white hole event horizon there are world lines which can be traced back into the past indefinitely without ever meeting the white hole singularity whereas in a FRW cosmology all worldline originate at the singularity.”

5. I was hoping that my last comment, in parentheses, would cover caveats like that. Other than that, the essay seems to pretty much agree with what I said.

I think it’s worth noting that there are interior solutions of Einstein’s Equations for a collapsing star that, locally, are just the FRW solution in reverse. The difference, of course, being that with a collapsing star, there is a vacuum beyond some radius, which results in a Schwarzshild metric for the exterior solution and allows geodesics that do not begin or end on the singularity.

6. Another thing worth checking up on is the extended Schwarzshild solution. (Check here: http://en.wikipedia.org/wiki/Kruskal-Szekeres_coordinates and scroll down.) It’s discussed better and in more detail in General Relativity by Wald.

At this point, I was going to embark on an explanation of what that is and why I think it’s relevant to your question in the update, but that will take too long and I would really have to draw diagrams to do it justice. I hope just giving you a reference is enough for the time being.

7. Thanks for the followup comment, Andy. I’ll definitely check out the Kruskal-Szekeres co-ordinates at some point.

Michael, do you know what happened to the QIP 2007 website qipworkshop.org?

9. Toby: It was accidentally deleted, and has now been restored from backups.

First, I am assuming you mean white hole, the time reversal of a black hole. In that case, as the essay discusses, we would be inside the horizon, and will not be able to receive messages from the outside (the time reversal of the familiar characteristics of black holes). In other words if the horizon formed there will be no way for us to know, only observers causally disconnected from us will notice.

As for the singularity, FRW is singular so no singularity theorems are evaded, though the details of the singularity differ from the simplest black holes (but in any event near the singularity we don’t trust any of our theories).