I Comparing Black Holes & Contracting Universes

[Gerinski]

Near a black hole spacetime gets contracted, more and more as we get closer to it. At the event horizon the contraction (pull of spacetime towards the singularity) equals the speed of light, we could say that space is falling into the black hole at precisely the same rate as light trying to travel outwards can catch up. Beyond the event horizon space is falling into the singularity at a rate faster than what light can catch up.

The contracting space seems to be similar to imagining a contracting universe, space(time) falling into a singularity.

In which respects can both situations be likened and in which respects are they fundamentally different?

I can guess that one important difference is that in a black hole, there remains an external spacetime to which the black hole can be said to be relative to, while in the case of a contracting universe the whole of spacetime contracts until becoming a singularity. There is no space beyond the portion which is contracting at a rate equal to the speed of light.

I'm also interested in the question: if the universe started as a singularity, and the laws of physics seem to be time-symmetrical, does the expanding universe have anything equivalent to an event horizon? And if so, what would that be?

And conversely, would a contracting universe have anything equivalent to an event horizon and if so, what would it be? The edge of spacetime itself? or some region defined by the speed of light? (such as when we speak of our current event horizon at our present coordinates in spacetime, which I believe has not much to do with the event horizon of a black hole).

Thanks!

 

[PeterDonis]

In which respects can both situations be likened and in which respects are they fundamentally different?

They're alike in that both of them contain a family of timelike worldlines which is "contracting"--this is a more precise way of describing what is going on than just saying "spacetime is contracting", which doesn't really make sense as it stands--spacetime doesn't contract or expand, it just is. You could say "space is contracting", and many pop science articles do, but that's not really precise either. The precise way is to focus in on a particular family of timelike worldlines, which can be thought of as a family of "observers" who are all freely falling in such a way that the distance between neighboring observers is continually decreasing, until it becomes zero at the instant when all of the observers hit the singularity.

The two situations are different, however, in several key respects. One such is related to this statement of yours:

I can guess that one important difference is that in a black hole, there remains an external spacetime to which the black hole can be said to be relative to, while in the case of a contracting universe the whole of spacetime contracts until becoming a singularity.

Again, it's not really correct to say "the whole of spacetime contracts" in the case of a contracting universe. A better way of saying what you're trying to say here is that in the contracting universe, there are no timelike worldlines that do not end in the singularity. In the case of the black hole, there are. The timelike worldlines that do not end in the singularity in the black hole case are the ones that stay in "the external spacetime", i.e., they never cross the hole's horizon.

There is no space beyond the portion which is contracting at a rate equal to the speed of light.

This, however, is not true of the contracting universe case. It is perfectly possible to have a contracting universe where some parts are contracting at more than the speed of light relative to other parts. But the "speed of contraction" here is just a coordinate speed, i.e., a calculated number with no physical meaning; no observer actually measures it. (The same is true for the time-reversed case of an expanding universe--it's perfectly possible for some portions to have a coordinate "speed of expansion" greater than the speed of light relative to other portions.)

I'm also interested in the question: if the universe started as a singularity, and the laws of physics seem to be time-symmetrical, does the expanding universe have anything equivalent to an event horizon?

And conversely, would a contracting universe have anything equivalent to an event horizon

No. This is another key difference between the two cases; the contracting/expanding universe does not have an event horizon, while the black hole/white hole does (the white hole would be the analogue of the expanding universe, as the black hole is of the contracting universe).

such as when we speak of our current event horizon at our present coordinates in spacetime, which I believe has not much to do with the event horizon of a black hole

Correct, it doesn't. Wikipedia has good brief descriptions of the different types of horizons in expanding universe models:

https://en.wikipedia.org/wiki/List_of_cosmological_horizons