Bubble universes and infinite in extent

[laymanB]

I was watching a conference on YouTube about cosmology and one of the speakers was Anthony Aguirre. He was giving a lecture on different cosmological models that are infinite in time as well as space. Here is a link to one of his papers and one of the models he was presenting. There is not much need to read the whole paper, I just wanted to have something to reference.

https://arxiv.org/pdf/gr-qc/0301042.pdf

My question involves what I seem to hear a lot from models about the multiverse and eternal inflation. Which is that each nucleation event produces a bubble universe which itself is spatially infinite, there are an infinite number of these bubbles universes, and they are embedded in some kind of inflating background spacetime, which itself is spatially infinite.

Now my confusing lies in how can a certain bubble universe have infinite volume and have collisions with other bubbles while all being embedded in a spatially infinite background spacetime? Are these cosmologists changing the meaning of spatially infinite to be used in different senses in the same sentence, paper, or lecture?

 

[andrewkirk]

I haven't seen the video or paper you refer to, but here's an explanation of how infinite spaces can collide with one another.

Put simply, they have to be contained in a higher dimensional space. So if the bubble universes are four-dimensional, they must be contained in a higher dimensional space, at least five dimensions but possibly many more.

To see how this works, consider a 3D space that has a lot of infinite 1D universes in it. A 1D space is a line, so what we are imagining here are a bunch of infinitely long strings in this 3D space. If the strings wobble about, they can bang into one another.

Similarly you can imagine a bunch of infinite rubber sheets in our 3D space, that can wobble about and bang into one another. This doesn't work quite as well as the strings, because, since sheets are 2D there is only one 'spare' dimension in the 3D space so the sheets are forced to sit effectively parallel to one another otherwise they'd have to pass through one another like ghosts. In contrast, the strings can head off in all different directions without blocking one another.

If the space that contains the universes has several more dimensions than the 4D universes, there is plenty of scope for them to be arranged at all different angles without having to pass through one another.

 

[laymanB]

Put simply, they have to be contained in a higher dimensional space.

That makes more sense. Thanks.

 

[kurros]

But is that what actually happens in eternal inflation models? I am no expert in them but I don't seem to remember being told about extra dimensions in that context before. What you describe sounds more like some brane-worlds scenario, which is a whole other thing.

 

[laymanB]

But is that what actually happens in eternal inflation models? I am no expert in them but I don't seem to remember being told about extra dimensions in that context before. What you describe sounds more like some brane-worlds scenario, which is a whole other thing.

I agree that when reading about eternal inflation models, one is not usually told explicitly that they require additional spatial dimensions. But I don't see how you can have a logically consistent eternal inflation model without extra spatial dimensions, unless each bubble universe is finite in extent. If even one bubble universe is claimed to be spatially infinite, then it seems to me that there is no where else for other bubbles to reside without higher dimensions. An infinite universe to me means that it goes on forever in each spatial dimension. I.e. there is "no where else".

 

[JMz]

I agree that when reading about eternal inflation models, one is not usually told explicitly that they require additional spatial dimensions. But I don't see how you can have a logically consistent eternal inflation model without extra spatial dimensions, unless each bubble universe is finite in extent. If even one bubble universe is claimed to be spatially infinite, then it seems to me that there is no where else for other bubbles to reside without higher dimensions. An infinite universe to me means that it goes on forever in each spatial dimension. I.e. there is "no where else".

I do not recall being told of needing extra spatial dimensions for this. The explanation for non-interaction that I have heard is that these "other universes" are simply too far away: unimaginably distant, and increasingly so, because the space between them is constantly inflating so fast that their separation grows, even as they themselves may grow.

 

[Bandersnatch]

I agree that when reading about eternal inflation models, one is not usually told explicitly that they require additional spatial dimensions. But I don't see how you can have a logically consistent eternal inflation model without extra spatial dimensions, unless each bubble universe is finite in extent. If even one bubble universe is claimed to be spatially infinite, then it seems to me that there is no where else for other bubbles to reside without higher dimensions. An infinite universe to me means that it goes on forever in each spatial dimension. I.e. there is "no where else".

There was a similar discussion recently in this thread:
https://www.physicsforums.com/threa...ppen-everywhere-in-the-universe.943564/page-4
I think it only became somewhat clarified towards the end, at least for me (and not by much).
What I gathered from that exchange is that in general, when you have a spacetime manifold, the extent of its spatial slices depends on how you slice it ('foliate'). I.e., whether space is infinite or not is not uniquely determined.
If there's matter in the universe, like in ours, then this fact leads to one type of slicing to be 'natural'. This natural foliation produces slices of infinite spatial extent.
But when there's no matter, only dark energy, like during the inflationary epoch (an exponentially expanding 'de Sitter universe'), then there is no preferred foliation, and the concept of spatial infiniteness is just an arbitrary choice. The inflationary bubble universes are neither uniquely infinite nor finite. It's only when the inflation field decays, the bubble 'becomes' infinite.
At least that's my hackneyed naive understanding of what's going on. It kinda looks like one really needs to take a course in GR to wrap one's head around it.

 

[JMz]

That makes some sense to me, but in that case, what's different about the class of foliations with infinite, or with finite, spatial extent? Global topology? E.g., are the finite ones all toroidal? Or metric, e.g., the finite ones must have non-constant spatial curvature? (I'm guessing they aren't all non-orientable!)

 

[laymanB]

@Bandersnatch thanks for that thread. I read the last page of posts and those where the kinds of questions I was having and the usual helpful posts by @PeterDonis .

So I guess my next question would be: Can you compare spacelike slicing between different spacetimes? In other words, can we take a certain foliation for our FRW spacetime and compare it to the same foliation of the inflating deSitter spacetime?

Second question: Is there something more "physical" about a specific foliation on FRW spacetime versus others? In other words, can we take a spacelike slice along some hyperbola that goes asymptotically to infinity, and if so, does this tell us anything physical about the geometry?

Third question: Do our observations that point to an open FRW universe have implications for an inflating deSitter spacetime? Isn't deSitter spacetime modeled on a 4-dimensional manifold like ours?

I am clearly out of my current level of knowledge, so be gentle .

 

[laymanB]

So this is the YouTube video that got me thinking about this thread topic. If you watch from 22:00 to around 30:00 you will get the gist of the argument from Anthony Aguirre.



So essentially he is saying you can have bubble universes that are finite or infinite depending on how you slice them and these bubbles can have collisions with each other. Makes perfect sense to me now.

George Ellis disagrees with Mr. Aguirre in this series of lectures and likes to appeal to a quote by mathematician David Hilbert:
"The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to."

 

[PeterDonis]

Can you compare spacelike slicing between different spacetimes?

In general, no.

can we take a certain foliation for our FRW spacetime and compare it to the same foliation of the inflating deSitter spacetime?

Inflating de Sitter spacetime is an FRW spacetime. In fact, there are spacelike slicings on de Sitter spacetime that correspond to closed, flat, and open FRW spacetimes. (Each of these slicings covers a different portion of the entire de Sitter spacetime manifold.)

 

[PeterDonis]

what's different about the class of foliations with infinite, or with finite, spatial extent?

The 3-geometry of the spacelike slices, and the portion of the overall 4-d spacetime that the foliation covers.

Global topology?

Of the spacelike slices? Yes. The closed slicing on de Sitter spacetime has spacelike slices with topology $S^3$ (the 3-sphere). The flat and open slicings have spacelike slices with topology $R^3$.

Or metric, e.g., the finite ones must have non-constant spatial curvature?

Not in de Sitter spacetime. All three of the slicings I named have constant curvature. Of course, you can always construct a slicing that has non-constant spatial curvature, but it won't be one of the three I named.

 

[PeterDonis]

Is there something more "physical" about a specific foliation on FRW spacetime versus others?

A foliation corresponds to a family of observers--the observers who stay at constant spatial coordinates in the coordinate chart constructed from the foliation. Some foliations correspond to families of observers with particular special properties; for example, the standard foliations on FRW spacetime correspond to "comoving" observers--observers who always see the universe as homogeneous and isotropic. But there is no requirement that a foliation correspond to a family of observers with any special properties.

can we take a spacelike slice along some hyperbola that goes asymptotically to infinity

I don't know what you mean by this.

Do our observations that point to an open FRW universe

What observations are these? Please give references.

 

[laymanB]

Inflating de Sitter spacetime is an FRW spacetime.

That helps, thanks.

I don't know what you mean by this.

Me neither. But what I was trying to convey was what Aguirre is referencing from about 24:00 to 25:00 in the above video. Where he is showing a number of spacelike surface hyperbolas that can be chosen to represent the Minkowski spacetime as infinite within a future light cone of some point.

He seems to be saying (starting at 22:50) that even in a closed, k=1, FRW model you can chose a spacelike slicing that makes that closed $S^{3}$ spatially infinite, if you chose the proper slicing. I don't understand that.

What observations are these? Please give references.

Sorry, I meant flat, not open.

Does our choice of foliation have an effect on how we measure time? Could we choose slicing that makes the Big Bang happen yesterday?

 

[PeterDonis]

he is showing a number of spacelike surface hyperbolas that can be chosen to represent the Minkowski spacetime as infinite within a future light cone of some point.

Those are hyperbolas of constant proper time from the origin; that is, if you draw straight timelike lines from the origin in all possible timelike directions (i.e., within the future light cone), the points where each timelike line intersects a given hyperbola all have the same proper time (arc length) along the timelike line. (It's the hyperbolic analogue of lines radiating out from the center of a circle in all directions, and all intersecting the circle at the same proper length from the origin.) Yes, each hyperbola is infinitely long.

Does our choice of foliation have an effect on how we measure time?

Coordinate time, yes, since choosing a foliation is choosing a coordinate chart.

Proper time, no; proper time along a given timelike curve is an invariant, independent of your choice of coordinates.

Could we choose slicing that makes the Big Bang happen yesterday?

No, since that is proper time, not coordinate time.

 

[laymanB]

Those are hyperbolas of constant proper time from the origin; that is, if you draw straight timelike lines from the origin in all possible timelike directions (i.e., within the future light cone), the points where each timelike line intersects a given hyperbola all have the same proper time (arc length) along the timelike line. (It's the hyperbolic analogue of lines radiating out from the center of a circle in all directions, and all intersecting the circle at the same proper length from the origin.) Yes, each hyperbola is infinitely long.

Okay, that makes more sense. So, if you are on a null line on the edge of the cone, you do not ever cross one of these hyperbolas of constant proper time because they are asymptotic to the null lines, therefore the null line has no proper time? And the straight line in the center taking the "shortest" path to the hyperbola is the longest proper time because we are talking about Minkowski spacetime?

Does a spacelike surface mean that each point on that surface is causally disconnected from each other?

I still don't understand the part (22:50 in video) about the $S^3$, k=+1, FRW spacetime to be sliced in such a way to be spatially infinite?

 

[PeterDonis]

if you are on a null line on the edge of the cone, you do not ever cross one of these hyperbolas of constant proper time because they are asymptotic to the null lines

Yes.

therefore the null line has no proper time?

Not "therefore", because the null line having zero proper time is not a logical consequence of the hyberbolas being asymptotic to it. But the two statements are certainly connected.

the straight line in the center taking the "shortest" path to the hyperbola is the longest proper time because we are talking about Minkowski spacetime?

No. All straight lines from the origin to the hyperbola have the same length (proper time) between those two points (origin and point of intersection with the hyperbola) because we are talking about Minkowski spacetime. Just as all straight lines from the center of a circle to the circle have the same length in Euclidean space.

Does a spacelike surface mean that each point on that surface is causally disconnected from each other?

Yes.

 

[laymanB]

No. All straight lines from the origin to the hyperbola have the same length (proper time) between those two points (origin and point of intersection with the hyperbola) because we are talking about Minkowski spacetime. Just as all straight lines from the center of a circle to the circle have the same length in Euclidean space.

Yeah sorry, I realize you said that in your previous post.

Thanks for all the insight and help.

 

[nikkkom]

So this is the YouTube video that got me thinking about this thread topic. If you watch from 22:00 to around 30:00 you will get the gist of the argument from Anthony Aguirre.



So essentially he is saying you can have bubble universes that are finite or infinite depending on how you slice them and these bubbles can have collisions with each other. Makes perfect sense to me now.

Excellent video, thanks!

However, I think I need to reevaluate what I'd call "infinite bubble". Just the fact that there exist a space-like foliation of it with all slices infinite, does not sound like a sufficient reason to call it that: even in a flat Minkowski spacetime (no expansion/inflation), a volume of space enclosed by a sphere expanding at the speed of light would then by this reasoning called "infinite" as well, which does not match what people usually call "infinite": the sphere is finite at any fixed time!

 

[laymanB]

Excellent video, thanks!

However, I think I need to reevaluate what I'd call "infinite bubble". Just the fact that there exist a space-like foliation of it with all slices infinite, does not sound like a sufficient reason to call it that: even in a flat Minkowski spacetime (no expansion/inflation), a volume of space enclosed by a sphere expanding at the speed of light would then by this reasoning called "infinite" as well, which does not match what people usually call "infinite": the sphere is finite at any fixed time!

Unless I'm not thinking about it correctly, I agree. It seems to me that the relative aspect of relativity, where you can choose arbitrary coordinate charts and foliations and none are more correct or privileged seems to lend itself to equivocation when communicating a theory. That may not be a default of the theory, just a translation issue between mathematics and vernacular language. It would seem to me that is what is occurring in these eternal inflation models is the sense of the word infinite is being changed in the course of discussion. To me, it seems analogous to saying that thinking about the universe as temporally finite is the wrong way to think about it because we can slice spacetime in such a way as to show that the Big Bang happened yesterday, or that time does not flow at all. Now, I'm not sure that one has the liberty to foliate in such a manner in GR or if that even makes sense. I still have much to learn.

 

[PeterDonis]

the sphere is finite at any fixed time

The sphere is finite at any fixed time in standard Minkowski coordinates. But in FRW coordinates on the interior of the future light cone of the origin, each of the hyperbolas is a spacelike surface of constant coordinate time, and the "sphere" you speak of (the light cone itself) is "at infinity", outside the region covered by the coordinates. This choice of FRW coordinates on a subregion of Minkowski spacetime is called the "Milne universe" (after the astronomer who proposed it as a model--the model is not an accurate model of our actual universe, but as a mathematical model it's perfectly consistent). Timelike observers who all start at the origin and go in all possible future timelike directions from the origin are "comoving" observers in this universe, and since each of them has the same proper time to their intersection with a given hyperbola, the hyperbolas, as above, are surfaces of constant time for these observers. In other words, as above, this is an FRW universe with open, infinite spacelike slices of constant time (it is in fact the limiting case of such a universe as the matter density goes to zero).

 

[PeterDonis]

I'm not sure that one has the liberty to foliate in such a manner in GR

One doesn't. See my post #15.

 

[laymanB]

One doesn't. See my post #15.

Man I have a short memory! Thanks.

Edit: So let's forget about measuring time since the Big Bang, being this is an invariant proper time. The question that I'm trying to pose is that if we don't use invariant quantities and call any spacetime spatially "infinite" based on some arbitrary foliation, could we also not construct a theory based on an arbitrary coordinate time where time does something "strange" like stand still or run backwards?

 

[JMz]

Man I have a short memory! Thanks.

Edit: So let's forget about measuring time since the Big Bang, being this is an invariant proper time. The question that I'm trying to pose is that if we don't use invariant quantities and call any spacetime spatially "infinite" based on some arbitrary foliation, could we also not construct a theory based on an arbitrary coordinate time where time does something "strange" like stand still or run backwards?

The usual requirement for any coordinate system is that it be smooth, and smoothly invertible, as a map to the Reals. Standing still (constancy) or going backward violate invertibility.

 

[laymanB]

So if our observable universe is $E^3$ according to our best observations, we conclude if it is simply connected that it is spatially infinite? Based on what foliation is this definition of spatially infinite made?

 

[PeterDonis]

if we don't use invariant quantities and call any spacetime spatially "infinite" based on some arbitrary foliation, could we also not construct a theory based on an arbitrary coordinate time where time does something "strange" like stand still or run backwards?

The short answer is no. See below for a slightly longer answer. A longer answer than that would go beyond the "B" level of this thread.

The usual requirement for any coordinate system is that it be smooth, and smoothly invertible, as a map to the Reals. Standing still (constancy) or going backward violate invertibility.

It's not quite that simple. You can construct smooth coordinate charts that have no timelike coordinate at all; you can also construct smooth coordinate charts in which such a coordinate stays the same or goes backwards along a timelike curve.

What you can't do is construct a coordinate chart based on a foliation of the spacetime by spacelike 3-surfaces, in which the time coordinate stands still or goes backwards. Any foliation by spacelike 3-surfaces induces a coordinate chart in which the surfaces are smoothly, invertibly labeled by a timelike coordinate, which can then be combined with a set of 3 spacelike coordinates labeling points on the 3-surfaces to construct a valid coordinate chart.

 

[laymanB]

See below for a slightly longer answer. A longer answer than that would go beyond the "B" level of this thread.

Thanks. I am trying to learn the math so one of these days I can start posting at "I".

I am not trying to dispute the majority consensus among the cosmological community that the best fit data is for a FRW spacetime that is flat and infinite, given a simply connected, trivial topology. What I am trying to do in this thread is understand what type of slicing that definition of infinity is based on. And then to see if there are any logical (mathematical) contradictions against other bubbles universes existing within this 3+1 dimensional spacetime without needing additional dimensions.

 

[JMz]

The short answer is no. See below for a slightly longer answer. A longer answer than that would go beyond the "B" level of this thread.

It's not quite that simple. You can construct smooth coordinate charts that have no timelike coordinate at all; you can also construct smooth coordinate charts in which such a coordinate stays the same or goes backwards along a timelike curve.

What you can't do is construct a coordinate chart based on a foliation of the spacetime by spacelike 3-surfaces, in which the time coordinate stands still or goes backwards. Any foliation by spacelike 3-surfaces induces a coordinate chart in which the surfaces are smoothly, invertibly labeled by a timelike coordinate, which can then be combined with a set of 3 spacelike coordinates labeling points on the 3-surfaces to construct a valid coordinate chart.

Quite right. I took the constraint of spacelike 3-surfaces as being implied in the question.

 

[PeterDonis]

What I am trying to do in this thread is understand what type of slicing that definition of infinity is based on.

The standard FRW slicing: each spacelike slice of constant time is a slice on which every comoving observer has the same proper time since the Big Bang.

to see if there are any logical (mathematical) contradictions against other bubbles universes existing within this 3+1 dimensional spacetime

It depends on what you mean by "within" and "this 3+1 dimensional spacetime". FRW coordinates cover the entire patch of spacetime that you describe as "flat and infinite". That is, they cover an infinite series of spacelike 3-surfaces, indexed by the time coordinate $t$, each of which is flat, infinite Euclidean 3-space. If the question is whether there are other "bubble universes" within that patch of spacetime, the answer is no.

However, there is a different question you could ask: could that patch of spacetime, the one that FRW coordinates cover the entirety of, itself be a "bubble" contained within a "larger" 3+1 dimensional spacetime, which would also contain other patches that are not covered by the FRW coordinates on the patch of spacetime we call our "universe"? AFAIK the answer to that question is yes--at least, that such a model is mathematically consistent. And such a model, if consistent, would be an example of our universe being a "bubble" in a larger 3+1 manifold without requiring any higher dimensions.