Isn't space expansion logically required?

[Finny]

Gerinski:
"GR still assumes that objects exist locally and objectively in that spacetime."

As Peter Donis explained, "GR" says no such thing. But quantum field theory does address such 'local objectivity".

On Einstein's curved spacetime there is no preferred vacuum state. A problem arises when you want to make statements about 'objects' [particles] which are globally valid, or when you change the reference frame as you do in the Unruh effect: Coincident observers, one inertial and one accelerating, do not in general agree on particle counts.

A problem with the particle concept is that one cannot attribute to it a permanent existence. It only exists at the moment it is detected. Our quantum models suggest in then reverts to its normal field state.

 

[Jimster41]

No, because I don't understand what this even means. I think you need to take a step back and think about what my statement "spacetime does not expand; it just is" really means.

This one is definitely making me think, and it's taking awhile.

You would or would not say that space-time has "shape"?

I am having a hard time thinking about a changing metric and an effect on CMB in a space-time volume without spatial bound (and hence without shape or any real sense of "volume" at all?). This seems most problematic in the reverse direction (time-wise). Thinking about an infinite volume in which density of mass and energy just "varies" in such a specific way - toward super high density (rather than just some set of similar differences) is counter-intuitive to the point of being non-physical isn't it?

Also, when talking about the idea of "sufficiently large" torus being indistinguishable from flat and infinite, and then the conversation about "measuring zero" you are referring to the fact that a torus at some point must show directional asymmetry in curvature, but that the value could be locally so small that no entity on (in) it could detect that asymmetry?

 

[PeterDonis]

You would or would not say that space-time has "shape"?

"Shape" is a vague term. Spacetime has geometry, which is described by the metric.

I am having a hard time thinking about a changing metric and an effect on CMB in a space-time volume without spatial bound (and hence without shape or any real sense of "volume" at all?).

A manifold without boundary can still have a geometry. Think of an infinite 2-surface with bumps and pits in it, as compared to a perfectly flat Euclidean plane. Both are 2-surfaces without boundary, but they have different geometries, described by different metrics.

Thinking about an infinite volume in which density of mass and energy just "varies" in such a specific way - toward super high density (rather than just some set of similar differences) is counter-intuitive

Possibly, depending on your intuition.

to the point of being non-physical isn't it?

No. GR is a perfectly consistent and well-confirmed physical theory, and includes solutions which are spatially infinite. So your intuition is simply telling you something incorrect in this instance.

when talking about the idea of "sufficiently large" torus being indistinguishable from flat and infinite, and then the conversation about "measuring zero" you are referring to the fact that a torus at some point must show directional asymmetry in curvature

No. The term "torus" here is being used to describe a topology, not a geometry. You can have a manifold with a 3-torus topology that is flat, i.e., zero curvature. For a 2-dimensional analogue, think of a flat square with opposite sides identified, like the screens of old video games such as Asteroids, where if your spaceship went off the right end of the screen, it reappeared at the left end. Such a manifold is perfectly consistent mathematically, and the 3-dimensional version is a perfectly possible spacelike slice in an appropriate solution of the Einstein Field Equation.

 

[Chalnoth]

A direct measurement would, yes. But there may be indirect ways of testing the finite vs. infinite question that do not depend on pinning down a measurement to exactly zero. Bear in mind that I'm not saying this will ever happen; I'm just saying that we can't rule it out.

A more meaningful way to look at it, IMO, is to start with your comment about the particle horizon that comes with a nonzero cosmological constant. One can infer from that that the "portion of the universe that matters", so to speak, for physics at our location is finite. From that standpoint, the question of whether the universe as a whole is spatially finite or infinite doesn't matter, practically speaking.

However, this thread, at least as I understand it, is about whether expansion is logically required. Answering a question like that requires considering all logically possible models, not just models that are practically useful.

I don't think so. But in practical terms, it doesn't matter, because no such proof is on the horizon. As of right now, we can't say the universe is infinite. And we definitely can't say that option is preferred.

 

[Jimster41]

"Shape" is a vague term. Spacetime has geometry, which is described by the metric.
A manifold without boundary can still have a geometry. Think of an infinite 2-surface with bumps and pits in it, as compared to a perfectly flat Euclidean plane. Both are 2-surfaces without boundary, but they have different geometries, described by different metrics.

I realize I have been picturing this, but without really being able to interpret it
https://www.physicsforums.com/attachments/davisdiagramoriginal2-jpg.55869/

It is really a diagram of a light cone. But I was confused by it when I saw it because I was trying to imagine what the light cone of the CMB would be on it, and my conclusion was that it should have been (at least at one time) the entire universe. Then at some point (moment of last scattering, the moment inflation kicked off inflation?) did the geometry of spacetime change so fast that it got left inside some horizon?

A manifold without boundary can still have a geometry. Think of an infinite 2-surface with bumps and pits in it, as compared to a perfectly flat Euclidean plane. Both are 2-surfaces without boundary, but they have different geometries, described by different metrics.

Is the idea that is is changing over time everywhere (at infinity) or are we only comfortable with the conclusion that is is changing locally, rather suggesting that it is a feature of some submanifold.

No. The term "torus" here is being used to describe a topology, not a geometry. You can have a manifold with a 3-torus topology that is flat, i.e., zero curvature. For a 2-dimensional analogue, think of a flat square with opposite sides identified, like the screens of old video games such as Asteroids, where if your spaceship went off the right end of the screen, it reappeared at the left end. Such a manifold is perfectly consistent mathematically, and the 3-dimensional version is a perfectly possible spacelike slice in an appropriate solution of the Einstein Field Equation.

I've heard that analogy before and it always sort of made me crazy. I can't see how that one step to the right can be different, i.e. result in a reversal of relative location with respect to origin. In other words that right hand edge of the screen has to process the an incoming object (from the left) quite a lot differently than an edge drawn just to the left of it. So it doesn't seem smooth (which seems like a natural requirement) to me - where as if you roll that sheet up, it makes sense. But now there is curvature.

 

[Finny]

I am having a hard time thinking about a changing metric and an effect on CMB in a space-time volume without spatial bound (and hence without shape or any real sense of "volume" at all?).

Don't worry about the outer extremities of the cosmos...the possible lack of a spatial boundary.

The CMB you can observe now originated from much closer in...from the surface of last scattering. Any signals from far beyond are causally disconnected and have no affect on you.

As has been described, our models suggest the 'boundary' was infinite even when the CMB originated at about 380,000 years of cosmic age and signals from the outer extremities beyond havn't affected us yet...nor does it appear ever will.

 

[Jimster41]

Don't worry about the outer extremities of the cosmos...the possible lack of a spatial boundary.

The CMB you can observe now originated from much closer in...from the surface of last scattering. Any signals from far beyond are causally disconnected and have no affect on you.

As has been described, our models suggest the 'boundary' was infinite even when the CMB originated at about 380,000 years of cosmic age and signals from the outer extremities beyond havn't affected us yet...nor does it appear ever will.

I do appreciate the words of comfort because this stuff worries me, but it's because I don't feel like I understand it...

So the CMB was, is, also infinite?
And the universe was infinite when the "metric" was tiny, compared to what it is now. This just seems contradictory at a point...
But I gather that's the difference between topology and geometry. The early universe was (possibly, or assumed to be) topologically infinite or unbounded (as it is now) but geometrically infinitesimal?

 

[PeterDonis]

I was trying to imagine what the light cone of the CMB would be on it, and my conclusion was that it should have been (at least at one time) the entire universe.

The CMB is everywhere in the universe. It doesn't really have a "light cone" because the CMB is radiation, i.e., it moves on null worldlines, not timelike ones. Only timelike objects are usually thought of as having past and future light cones. On diagrams like the one you linked to, the CMB is best thought of as a family of lines filling the entire diagram; on the conformal diagram (the bottom one), they will be 45 degree lines.

at some point (moment of last scattering, the moment inflation kicked off inflation?) did the geometry of spacetime change so fast that it got left inside some horizon?

The geometry of spacetime does not change; spacetime is a 4-dimensional geometric object that just is. It describes the entire history of the universe.

The geometry of space can be thought of as changing with time, but thinking of it this way requires you to choose a particular splitting of spacetime into space and time. There is no unque way to do this, so you have to be careful drawing conclusions from this kind of thinking.

Is the idea that is is changing over time everywhere (at infinity)

See above.

I can't see how that one step to the right can be different, i.e. result in a reversal of relative location with respect to origin.

It doesn't. In this scenario, any given point is both to the left and to the right of the origin: there are spatial paths in both directions that connect the two points. It's no different than the fact that you can go from, say, New York to Calcutta by plane in either of two directions, so Calcutta is both east and west of New York.

 

[Finny]

So the CMB was, is, also infinite?

Not as commonly used in discussions, but yes it is everywhere. We usually, I think, talk of the CMB as what WE observe. Most we can never observe, like most of the universe itself. If you have seen explanations of the early universe as 'the size of a dense pea', that refers to the observable universe at that time. [Although things were utterly opaque back then and we could not actually 'see' anything.] We can now 'see' much further as the 'visible' universe has expanded.

A very distant observer also sees CMB surrounding them, but it may not be any of the CMBR region we see. Or it could overlap. They may well be causually disconnected from us, if far enough away, so we may never be able to share anything in common.

 

[Jimster41]

he CMB is everywhere in the universe. It doesn't really have a "light cone" because the CMB is radiation, i.e., it moves on null worldlines, not timelike ones. Only timelike objects are usually thought of as having past and future light cones. On diagrams like the one you linked to, the CMB is best thought of as a family of lines filling the entire diagram; on the conformal diagram (the bottom one), they will be 45 degree lines.

think I get the difference between a light cone and a null line. The null line of a particle wave whatever of radiation moves at 45deg on a space-time diagram. No time like object can move outside the cone bounded or swept out by that line, because it would have to move faster than light to do so. Light-like objects only every occupy those lines. I've read this but it's more clear now. Hopefully that's correct.

The geometry of spacetime does not change; spacetime is a 4-dimensional geometric object that just is. It describes the entire history of the universe.

The geometry of space can be thought of as changing with time, but thinking of it this way requires you to choose a particular splitting of spacetime into space and time. There is no unique way to do this, so you have to be careful drawing conclusions from this kind of thinking.

I think I understand the conundrum that is posed by Relativity's lack of a preferred Frame of Reference, and therefore an infinite set of choices about how to split space time. I think that part of this discussion has centered on the semantics of whether or not a thing that must be 4d or 2d but with one dimension as time is a single thing that is not "split" or a thing that is split into distinct parts. The fact that space and time are qualitatively different as dimensions (their relative signatures, and maybe their roles in geometry and topology?), but that there is no prohibition w/respect to their simultaneous but different assignment to the vector space, definitely makes it confusing/troubling

Also, and I think this is the key to unlocking the problem I have had with understanding this. Is it then correct to say that in the early universe, the geometry of space was infinitesimal, but the topology of space was infinite?

It doesn't. In this scenario, any given point is both to the left and to the right of the origin: there are spatial paths in both directions that connect the two points. It's no different than the fact that you can go from, say, New York to Calcutta by plane in either of two directions, so Calcutta is both east and west of New York.

I'd like to say that clears it up but I know that the only reason I can do that is because the geometry of the planet supports that topology (if that is saying it right). I thought the answer might be something like what you say, that the line representing the right and left side of the screen is just a visual artifact that the math doesn't need, but something seems suspicious about the idea that it is not located anywhere specifically (in which case the surface would not be smooth). I keep trying to walk it from the right side of the screen toward the origin (of some Pac Man in the middle of the video game). At some point said pac man will be looking frontwards at his backside? This seems like a troubling topology.

 

[Finny]

The geometry of spacetime does not change; spacetime is a 4-dimensional geometric object that just is. It describes the entire history of the universe.

Does this apply to the inflationary period? pre inflationary??

 

[PeterDonis]

Hopefully that's correct.

Basically, yes. The only caveat is that null lines only appear as 45 degree lines on certain kinds of spacetime diagrams, the ones that are "conformally equivalent" to the usual SR spacetime diagram of Minkowski spacetime. For example, in the 3 diagrams of the universe that you linked to, the bottom one meets this criterion, but the others do not (as you can see from the fact that the "light cone" drawn on those diagrams is not marked out by 45 degree lines).

Is it then correct to say that in the early universe, the geometry of space was infinitesimal, but the topology of space was infinite?

The geometry is not "infinitesimal"; "infinitesimal" describes a small piece of spacetime, not its geometry as a whole.

The topology of space is not "infinite"; it's "unbounded" (or, more pedantically, a "manifold without boundary") and "non-compact" ("compact" is the precise term in topology for what we would think of as a "finite" region in ordinary language).

part of this discussion has centered on the semantics of whether or not a thing that must be 4d or 2d but with one dimension as time is a single thing that is not "split" or a thing that is split into distinct parts.

One dimension being timelike does not "split" anything; it's just a fact about the geometry of the spacetime.

the only reason I can do that is because the geometry of the planet supports that topology (if that is saying it right)

Not really. Topology is prior to geometry. The correct statement is simply that the topology of the manifold allows it.

the line representing the right and left side of the screen is just a visual artifact that the math doesn't need, but something seems suspicious about the idea that it is not located anywhere specifically (in which case the surface would not be smooth)

Why not?

 

[PeterDonis]

Does this apply to the inflationary period? pre inflationary??

It applies to the entire history of the universe except (possibly) for a period at the very beginning when the density was of the order of the Planck density and a spacetime description was not possible. I say "possibly" because we don't know for sure that there was such a period.

 

[Jimster41]

Basically, yes. The only caveat is that null lines only appear as 45 degree lines on certain kinds of spacetime diagrams, the ones that are "conformally equivalent" to the usual SR spacetime diagram of Minkowski spacetime. For example, in the 3 diagrams of the universe that you linked to, the bottom one meets this criterion, but the others do not (as you can see from the fact that the "light cone" drawn on those diagrams is not marked out by 45 degree lines).
The geometry is not "infinitesimal"; "infinitesimal" describes a small piece of spacetime, not its geometry as a whole.

The topology of space is not "infinite"; it's "unbounded" (or, more pedantically, a "manifold without boundary") and "non-compact" ("compact" is the precise term in topology for what we would think of as a "finite" region in ordinary language).
One dimension being timelike does not "split" anything; it's just a fact about the geometry of the spacetime.
Not really. Topology is prior to geometry. The correct statement is simply that the topology of the manifold allows it.
Why not?

So the metric was infinitesimal but the topology was "without boundary". Or is that still wrong?

Why not smooth? I was thinking that something different had to define that boundary exactly because it can't be arbitrarily close to the origin like any other line. But I can imagine there are mathematical/geometric proofs around this question, that show it can be different in that way wouldn't necessarily mean "not smooth", ways that rationalize it's location with respect to the origin some other way. In other words I can imagine that someone has proved that a topology can allow for that kind of connection (flying around flat-land one way on a plane). In fact I can imagine that someone could think up topological spaces of all kinds and prove formal characteristics for lots of interesting reasons. But then my question becomes (since there is always another question) is such a topology possible in the real geometry of spacetime?

Whether or not a geometric object like a triangle is "split" into two + one sides, does seem goofy, unless it's a right triangle. I would have used a tetrahedron as an example but I don't know what a "right tetrahedron" is called, or even if there is one - the point only being that there are geometric objects with internal asymmetry. Is spacetime not one of those? To be clear though, I don't have a need for it to be "split" I thought your post #46 sounded good. In fact I think I've been proposing all along that it's most interesting viewed as a single strangely flexible, but also constrained "geometric object".

I Learned a lot here Peter, by the way.

 

[PeterDonis]

So the metric was infinitesimal but the topology was "without boundary". Or is that still wrong?

The "metric is infinitesimal" part is. The metric doesn't have a size. Particular regions of spacetime have a size; a small enough region around a particular point could be considered "infinitesimal". When in doubt, try thinking of a simpler example like a Euclidean plane or the surface of a 2-sphere. You wouldn't describe the geometry or the metric of such surfaces as "infinitesimal"; you would use that word to describe the size of a very small piece of the surface.

I was thinking that something different had to define that boundary exactly because it can't be arbitrarily close to the origin like any other line.

The "boundary" of the square in the 2-dimensional "flat torus" analogy I gave (the Asteroids video game screen) is just an arbitrary boundary, like the prime meridian or the international date line on the surface of the Earth. It has no physical significance; it's just a convenience for human use in description. The origin is likewise an arbitrary choice.

What I think you may be trying to get at here is that those two arbitrary choices are not independent, just as the choices of the prime meridian and the international date line on the Earth (or more precisely the 180 degree meridian) are not independent. Once we designate some particular meridian on the Earth as "0 degrees", which meridian is "180 degrees" is automatically chosen (it's the one opposite the "0 degrees" meridian). Similarly, once we choose a particular point on the flat 2-torus as the "origin", which particular square is the "boundary of the video game screen" is automatically chosen: it's the set of points that are at the maximum possible distance from the origin.

there are geometric objects with internal asymmetry. Is spacetime not one of those?

No. The asymmetry of a right triangle, as opposed to an equilateral triangle, has to do with the group of transformations that leave the triangle invariant being smaller for the right triangle than for the equilateral triangle. The difference between timelike and spacelike in spacetime is a different kind of difference, so to speak; you can have a spacetime with the maximal possible set of transformations leaving its metric invariant (for example, flat Minkowski spacetime) that still has timelike, spacelike, and null vectors in it.