Musings on the infinite

[steuard]

A few weeks ago, my friend and former colleague Sean Carroll was a guest on the Colbert Report to promote his book about the nature of time. Toward the end of the interview, they discussed the idea of the "multiverse", which Sean uses to refer to the (possibly) infinite number of "universe-sized" regions within the vast web of space and time where we live. The notion is that if we could somehow travel far enough (faster than light) to regions many times more distant than our telescopes can see, we could find countless independent "universes" that can never talk to each other at all. Some of them would be much like our own but others could be very different, maybe even with different laws of physics. Steven Colbert seemed quite interested:

Colbert: Am I in these other universes?

Carroll: There will be people very much like you.

Colbert: In these other universes, is it possible that my show's on at 11 and John Stewart is at 11:30?

Carroll: Maybe more often!

It's a cute exchange, and it's a variant on the old idea that "in an infinitely big universe, everything that could possibly happen must happen somewhere."

Trouble is, I don't know that I buy that argument, for rather subtle reasons. However we define them, the number of "independent universe-sized regions" of space and time is countably infinite: we could in principle come up with some way of labeling each one by an integer. But many sets (like the real numbers) are uncountably infinite: no matter how you try to label each real number by an integer, you'll miss the vast majority of them. The real numbers are just a much bigger infinity than the integers are. Going on, the set of all possible curves in space is a yet larger infinity. (Assuming space and time are continuous! If they turn out to be discrete, then the set of curves has the same infinite size as the real numbers.)

The thing is, the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers: if anything, it's a still larger infinity. So no matter how large our multiverse may be, it's mathematically impossible for every possible history to occur somewhere. Does that mean that our Steven Colbert (on at 11:30) is the only one? Quite possibly so. I'm not convinced that the multiverse idea opens up as many possibilities as people sometimes think.

Comments

[spoonless.livejournal.com]

I think must be a lot of assumptions you're making here that I'm missing.

However we define them, the number of "independent universe-sized regions" of space and time is countably infinite: we could in principle come up with some way of labeling each one by an integer.

Can you elaborate on that? There are a lot of different kinds of multiverses in physics, with all sorts of rules... is this in Sean's particular set of ideas? or some theorem for eternal inflation in general? Or something based on your own reasoning?


The thing is, the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers: if anything, it's a still larger infinity.

I think I disagree with this. As long as you believe the holographic principle (and for the most part, I do), there are only a finite number of quantum bits in our Hubble volume anyway. So I think(*) that means the number of all possible configurations for a Hubble volume of our size is finite, not infinite. If you allow for any possible size, then it's infinite, but still only countably infinite since you can pick the size by picking the number of pixels that make up the holographic screen corresponding to our Hubble horizon.

(*) I guess the part I'm glossing over here is what about the actual "parameters" of the theory, eg things like the cosmological constant? Well, I'm not sure I understand that part very well but if pure string theory is really parameter-free and each parameter is really determined by degrees of freedom that are frozen within a particular region of the multiverse, then maybe it doesn't change what I'm saying. Or, maybe I'm just looking at this in totally the wrong way... could be.

[marphod.livejournal.com]

I've got to agree with this. How do you come to the assumption "that the number of the number of "independent universe-sized regions" of space and time is countably infinite? That's a HUGE assumption.

I've never seen such posited as an axiom for multiverse physics. In fact, most theories of multiverse physics I have seen make an assumption that, in the end, is very much the opposite.

[madmanatw.livejournal.com]

I'm only into this as a hobby, but it is my suspicion that "multiverses" as used by the Many Worlds interpretation of QM and "multiverses" as used by M-theory, branes colliding, and theories involving gravity "leaking" are pretty completely different. The MW version is an interesting, self consistent way of dealing with QM weirdness, but I don't think that the theories try to, well, _put_ those "alternate universes" anywhere. Whereas with branes, etc you have "universe sized regions" (well, universes) all embedded in a higher dimensional space (or something). The number of universes in a many worlds branching system may be a different cardinality of infinity than the number of independent universe sized regions... though I admit I don't know why number for the latter would need to be countable. Seems plausible, but, ya know, I'm not the one with the PhD in this stuff. :)

[steuard.livejournal.com]

I've alluded to this elsewhere in the thread, but I'm a bit annoyed by the sheer number of wildly different ways that people use the term "multiverse". On the one end you have Sean's usage here, which to me feels too "small" for the word: causally disconnected regions of our own space-time manifold. On the other end you have people who use "multiverse" to refer to the whole tree of branching quantum histories, which feels entirely too large to me (or, well, like a misuse of the word somehow).

If I had my druthers, I think I'd at least want the different "universes" in the "multiverse" to be geometrically disconnected in some way. I'm really not sure which of the proposals out there are consistent with that: brane-world scenarios probably are, for example. But as far as I can tell, Sean was discussing one very specific and limited case, whatever we call it, and I'm pretty sure that case must be countable.

[madmanatw.livejournal.com]

The one I've seen used for that is that two things/points/whatever are in the same "universe" if their source is the same local big bang (or functional equivalent). We pulled it out in a recent conversation I was in elsewhere where confusion between regions of our universe outside our own light cone (spacially separated universe sized regions, perhaps even) and separate "nearby" universes needed to be cleared up. (We were discussing the Dark Flow and possibility of gravity "leaking".)

It unfortunately doesn't work as well for the Many Worlds 'multiverse' usage, since you could argue that's all sourced in the same Big Bang, so for now I'm just going to hand wave them.

[steuard.livejournal.com]

two things/points/whatever are in the same "universe" if their source is the same local big bang

Well, by that definition I think much of what Sean is talking about really would be a multiverse. Perhaps that is a fair definition, or at least as fair as my "disconnected spacetime manifold" suggestion. (Heck, my suggestion may be too strong to be interesting, depending on how you interpret it.)

There was a paper I saw a while back (I don't recall enough to track it down easily) that attempted to define all the ways in which "multiverse" is used. My memory is that it identified a good half dozen distinct senses of the term.

[steuard.livejournal.com]

the number of "independent universe-sized regions" of space and time is countably infinite

Can you elaborate on that? There are a lot of different kinds of multiverses in physics

As I understand the context of Sean's conversation with Colbert, he was using "multiverse" to refer more or less to "causally disconnected regions of the full spacetime manifold". So he's essentially discussing different regions of space, not any of the more complicated (and likely less countable) versions of what "multiverse" might mean. (I feel like that term gets overused.)

the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers

As long as you believe the holographic principle (and for the most part, I do), there are only a finite number of quantum bits in our Hubble volume anyway.

This is a very fair point. I tried to nod toward related concepts a bit with my comment "assuming space and time are continuous": I think you may get the same reduction of degrees of freedom from uncountable to countable whether you take a holographic approach or just discretize space into Planck-length-sized cells (as I understand it, the big difference is whether the number of qbits scales like area or volume). I'm not even sure that you need all of holography to make the area-law argument: I seem to recall that there's a proof based just on quantum fields in GR that black holes have the highest possible entropy density.

I think I'd need to know more about quantum information theory to be sure whether this really does break my argument, though. (Hmm, the more I think of it, the more I suspect it does.) Perhaps the real lesson from all this is that our (my?) intuitive notion of smooth paths and histories is just wildly inaccurate as a guide to true underlying physics. (Amazingly inaccurate.) I've known that for a while, but I clearly still haven't gotten my brain fully around it.