Isn't space expansion logically required?

[Jimster41]

I presume that the OP meant that the state space of the universe is increasing. That is the set of possible configurations everything in universe could potentially be in is increasing. That seems like a reasonable idea.

That was one of the first questions I asked when I got here. Whether expansion was the cause of entropy (not precisely the same question but close). I was told it was wrong to think so, but even now, as you say, it seems like a reasonable idea.

One alternative is that the number of possible configurations, the "Liouville space" was set "on day one" or was "always set", and the actual state has been moving through that space over time in the direction of increasing probability and entropy . That idea has always seemed more awkward to me.

I think the problem with the idea that the phase space of the universe is growing, is that it implies that the universe isn't "The Universe". What could be "feeding" that growth, cause by definition it ain't the universe feeding itself.

The problem with the "was always" is that it seems almost equally illogical in terms of the "antinomy of cosmogenesis", and it leaves the driver of change un-addressed, and the idea of expansion (arguably the most important discovery of science) as a possible source of change itself, somehow - ignored

As I understand it is possible to suggest what we experience as space-time (which is expanding) is not necessarily synonymous with "The Universe". From there both notions seem a lot easier to think about! Or rather space-time seems easier to think about, as fed by something, or "set " by something. "The Universe" still defies thought at some point. IMHO

 

[PeterDonis]

Doesn't GR show that really geometry is the fundamental thing

Yes, but it's spacetime geometry. Spacetime does not expand; it just is. The spacetime geometry of our universe happens to admit a slicing into "space" and "time" such that the scale factor of "space" increases with "time"; but there's no requirement that you have to view it using such a slicing, or even view it as split into "space" and "time" at all.

the spatial extension of spacetime

Which depends on how you slice up spacetime into "space" and "time". You don't have to do that. We do it because of the limitations of our cognitive faculties, not because of any limitation inherent in the physics. But even given that, as Chalnoth said, there is no requirement that spacetime must admit a slicing such that the scale

 

[Jimster41]

Yes, but it's spacetime geometry. Spacetime does not expand; it just is. The spacetime geometry of our universe happens to admit a slicing into "space" and "time" such that the scale factor of "space" increases with "time"; but there's no requirement that you have to view it using such a slicing, or even view it as split into "space" and "time" at all.
Which depends on how you slice up spacetime into "space" and "time". You don't have to do that. We do it because of the limitations of our cognitive faculties, not because of any limitation inherent in the physics. But even given that, as Chalnoth said, there is no requirement that spacetime must admit a slicing such that the scale

Is there an example of such an alternative relational model? I thought GR was arguably a fundamental observation. And that even if an alien had different words for things like "distance" and "getting old" things for which she would have to have words, once we got those figured out we would recognize that she had identified the same operational mechanics of the geometry we share. For instance would the signature difference between the temporal and spatial dimensions (or some identifiable dual of it) at least be an expected shared constraint? Or could she have ignored that altogether? Likewise on the scale factor, could any meaningful model ignore the observation of cosmological red-shift, or CMB distribution, and the question of orientation of energy density gradient, the second law of thermodynamics, and QM superposition. These would have to have duals, and consistent interactions, in any relational theory of physics and space-time?

It seems like we are discussing the difficulty of simultaneously knowing your description of something is idiosyncratic to your own experience, while also having some real faith that it refers to something that exists, and has real symmetry with respect to that description. Absent the latter we seem like an odd gathering of busy-body solipsists.

I'm not suggesting that abstract things like geometry (strings) are most real by the way (I am not a Platonist), rather that we may at present be otherwise entirely sightless w/respect to what the physical thing they describe really is, but that something that looks like we think it does, is really there.

 

[Finny]

My own favorite description of space and time:
"Time is what keeps everything from happening at once; Space is what keeps everything from happening to me."

Space is more than just geometry, space is where fields reside.

Thats most likely a simplistic view. A place to start, perhaps, but probably not to conclude your thinking about space...distance...spacetime...time...Following are some contradictory views by some well known physicsts... some things to consider...And if you read between the lines, issues arising between GR and QFT...

Carlo Rovelli:
"Special relativity weakens the notion of absolute time; general relativity weakens it further. Relativity shows time is not constant...and varies between observers due to relative speed and or differences in gravitational potential. This means space-time is a dynamical field...we learn from GR that spacetime is a dynamical field and we learn from QM that all dynamical fields are quantized..."

[The first part probably reflects the change in time in varying gravitational potentials [GR] and between observers in relative motion.]

"...Conventional QFT relies ….on the existence of a non–dynamical background spacetime metric..[but]…with GR we have understood that there is no such non–dynamical background spacetime metric in nature….
http://arxiv.org/abs/gr-qc/0604045
Unfinished revolution

[Yet in the wonderful Wheeler-Dewitt quantum mechanical equation there is no time variable!]

Lee Smolin
Abstract: There are a number of arguments in the philosophical, physical and cosmological literatures for the thesis that time is not fundamental to the description of nature. According to this view, time should be only an approximate notion which emerges from a more fundamental, timeless description only in certain limiting approximations. ... The view that time is real and not emergent is, I will argue, supported by considerations arising from all these issues It leads finally to a need for a notion of law in cosmology which replaces the freedom to choose initial conditions with a notion of laws evolving in time. The arguments presented here have been developed in collaboration with Roberto Mangabeira Unger .
-- http://pirsa.org/08100049/


"Forget time"Authors: Carlo Rovelli
(Submitted on 23 Mar 2009 (v1), last revised 27 Mar 2009 (this version, v3))
Abstract: Following a line of research that I have developed for several years, I argue that the best strategy for understanding quantum gravity is to build a picture of the physical world where the notion of time plays no role. I summarize here this point of view, explaining why I think that in a fundamental description of nature we must "forget time", and how this can be done in the classical and in the quantum theory. The idea is to develop a formalism that treats dependent and independent variables on the same footing. In short, I propose to interpret mechanics as a theory of relations between variables, rather than the theory of the evolution of variables in time. http://arxiv.org/abs/0903.3832

 

[PeterDonis]

Is there an example of such an alternative relational model?

I don't understand what you mean by this. I wasn't talking about any "alternative" to GR; I was talking about GR.

I thought GR was arguably a fundamental observation.

I don't understand what you mean by this either. GR is a theory, not an observation. Tidal gravity is a fundamental observation, but identifying tidal gravity with spacetime curvature, which is what GR does, is not; it's a theory, which can only be judged by the accuracy of its predictions. No observation will ever tell you directly that spacetime curvature exists; you have to adopt a theory that tells you what observations indicate spacetime curvature (as GR says that observations of tidal gravity indicate spacetime curvature).

would the signature difference between the temporal and spatial dimensions (or some identifiable dual of it) at least be an expected shared constraint?

Since we measure "temporal dimensions" with clocks, and "spatial dimensions" with rulers, they would seem to be fundamentally different things, physically, so we would expect any valid theory to have to include that difference.

on the scale factor, could any meaningful model ignore the observation of cosmological red-shift, or CMB distribution, and the question of orientation of energy density gradient, the second law of thermodynamics, and QM superposition. These would have to have duals, and consistent interactions, in any relational theory of physics and space-time?

Most of these are observations, but you've mixed in some interpretations too. For example, "cosmological red shift" presupposes that the red shift we observe in light from distant galaxies is of cosmological origin--i.e., that it's due to the expansion of the universe. That's a theoretical conclusion, not an observation (but the red shifts themselves are observations). Similarly, the second law of thermodynamics is not really an observation: entropy is not something we directly observe, it's a theoretical construct.

Also, I don't understand what you mean by "orientation of energy density gradient".

 

[Jimster41]

By "alternative relational theory" I was trying to imagine an alternative description of reality, one that didn't view it as "split into slice into space and time". As I think you say is natural, or needed, later. The fact that GR admits any set of slices, seems subtle but as I argued earlier I don't think it implies "un-reality" rather the opposite.

Your second statement, the question of difference between observation, theory and knowing is exactly what was interesting about the discussion.

I'm paraphrasing Polanyi from a long time ago here - At some point we "indwell" in knowledge that at first can only be explicit, starting with raw observation, from that theory, sensitivity and a map, then at some point, through repetition, it becomes "tacit", unconsidered, sensory, and we are poised at a new height, for new raw observation. Like the observation that my car goes left when I turn the wheel left, and the theory that my steering wheel is mechanically connected to my tires, and the front end of my car... I'm never thinking of that when driving. I'm thinking about this d@$#% forum.

I can imagine at some distant future point a sentient creature or entity tied to a gravitational wave sensing apparatus, looking out at an asteroid field "seeing" the beautiful contours of raw spacetime distortion. To that being your statement that no observation will tell you that spacetime exists seems solipsistic - how are her instruments different from our eyes or fingers?

Clearly this is off into the philosophy of science... which certainly can be a rat hole.

 

[PeterDonis]

By "alternative relational theory" I was trying to imagine an alternative description of reality, one that didn't view it as "split into slice into space and time".

In GR, you don't have to view spacetime as split into space and time. That's a convenience for us humans, not a necessary part of the theory. GR is perfectly capable of describing all the physics without ever splitting spacetime into space and time.

 

[Gerinski]

Spacetime does not expand; it just is.

Don't you agree that the extension of the spatial dimensions of spacetime do expand (get larger) as the extension of the time dimension increases?

 

[Gerinski]

Carlo Rovelli:
"Special relativity weakens the notion of absolute time; general relativity weakens it further. Relativity shows time is not constant...and varies between observers due to relative speed and or differences in gravitational potential. This means space-time is a dynamical field...we learn from GR that spacetime is a dynamical field and we learn from QM that all dynamical fields are quantized..."

"...Conventional QFT relies ….on the existence of a non–dynamical background spacetime metric..[but]…with GR we have understood that there is no such non–dynamical background spacetime metric in nature….

This reminds me of this reasoning:"Newton believed that physical objects and phenomena have a local and objective existence in a 'canvas' of absolute space and time. To make it more clear, let's split this in 2 statements:

1. things exist locally and objectively
2. space and time are absolute

Einstein with his General Relativity (GR) showed that space and time are not absolute, they are flexible and subjective, different observers will perceive them differently, but GR still assumes that objects exist locally and objectively in that spacetime.

Quantum Theory (QT) on the other hand showed that existence is neither local nor objective, it's all a bunch of probability waves, non-locality was confirmed, observation and information is what defines the properties of objects and phenomena (for example path information yes/no defines the outcome of the famous double slit experiment), but QT still assumes a physical, objectively existing spacetime background.

So you see, GR showed that postulate 2 of Newton was false but kept postulate 1, while QT showed that postulate 1 was false but kept postulate 2.

So the ultimate theory, if it exists, needs to be one which gets rid of both postulates at the same time. Existence is neither local nor objective, and spacetime is not an objective physical entity. Everything is the outcome of information processing, and spacetime is only a set of relationships governing how information events can relate to each other and which appearance they must take when perceived by a consciousness. Physical existence is a sort of illusion, although we of course must take it as very real for ourselves, but it is not material in the sense that we assume, its 'material appearance' is only a consequence of the rules governing how the information behaves as perceived by any consciousness able to process it."Debate is welcome :-)

 

[PeterDonis]

Don't you agree that the extension of the spatial dimensions of spacetime do expand (get larger) as the extension of the time dimension increases?

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.

 

[PeterDonis]

Einstein with his General Relativity (GR) showed that space and time are not absolute, they are flexible and subjective, different observers will perceive them differently

Be careful; the way you are putting this implies a contradiction with this:

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

How can objects exist "locally and objectively" in something that is "flexible and subjective"? "Space" and "time" may be "flexible and subjective", but that just means you should not be looking at "space" and "time" separately; you should be looking at spacetime. Spacetime is not "flexible and subjective" in GR; it's objective and invariant. What it isn't, in GR, is independent of the matter and energy content of the universe. In Newtonian physics, space and time are each, separately, constant and independent of anything else. In SR, "space" and "time" can no longer be considered separately, but spacetime is still constant and independent of anything else. GR makes spacetime, as Rovelli says, "a dynamical field", i.e., its geometry is now dependent on the matter and energy content of the universe. But that geometry is still objective and invariant; different observers do not measure different spacetime geometries.

The "flexible and subjective" part is that there is no unique way to split up spacetime into "space" and "time". But that split is itself not a necessary part of GR; it's just a convenience for us humans, because we find it difficult to let go of our intuitive concept of "space" and "time" as separate things. You don't need to make that split to model any physics or compute any observables. So if you are trying to understand the fundamentals, the best thing I can recommend is to simply throw away the whole idea of splitting spacetime into "space" and "time", as well as anything else that is flexible and subjective. Einstein commented that the name "relativity" was a misnomer; his theory should have been called the "theory of invariants", because the whole point was supposed to be to emphasize that all of the physics is contained in things that are not "flexible and subjective", but objective and invariant.

 

[Finny]

The OP:

At the event of 'me now', there is more time extension since the Big Bang than there was at the event when the solar system formed, and even more than at the event when the first galaxies formed. I don't care whether also the future 'already exists'. The extension of the time dimension is larger 'in my now' than in my past, and smaller now than in the future.

It's not a bad question. But of course one needs to define 'larger' and smaller' time...We can agree space and time seem different, whatever they are. And keep in mind everyone's 'reality' is local. Yours is not the same as something many light years distant; nor it is the same as someone who is caually disconnected...out of the reach of light.

In our universe it seems like despite vastly different gravitational and spatial backgrounds [from moments after the big bang to a large ,cold, dead universe at the end] with slowly evolving entropy and informational conditions, as far as I can tell local time plods along at a steady pace. I don't know of any theory that requires 'an extension of the time dimension'.

You can also consider anti particles...don't they move backward in time in our models? I don't think we should describe that as a 'smaller' time dimension.

 

[rootone]

This reminds me of this reasoning:
Existence is neither local nor objective, and spacetime is not an objective physical entity. Everything is the outcome of information processing, and spacetime is only a set of relationships governing how information events can relate to each other and which appearance they must take when perceived by a consciousness. Physical existence is a sort of illusion, although we of course must take it as very real for ourselves, but it is not material in the sense that we assume, its 'material appearance' is only a consequence of the rules governing how the information behaves as perceived by any consciousness able to process it."Debate is welcome :-)

It sounds a lot like the 'We are a simulation' proposal.
For me that doesn't work because a simulation has to simulate something, so is that 'something' the real something?, or is it another simulation, ad infinitum.
... not to mention what is the ACTUAL physical thing which does the simulating?

 

[Gerinski]

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.

I can guess that you take the view of block time, but even so, that block spacetime would not be like a constantly thick slice bread as it is sometimes depicted in popular science books, it would be more like a cone bread, getting larger in its space dimensions as it gets larger in its time dimension.
If that's not the case, kindly enlighten me.

TX

 

[PeterDonis]

You can also consider anti particles...don't they move backward in time in our models?

The short answer is "no". A longer answer really belongs in the quantum physics forum.

 

[PeterDonis]

I can guess that you take the view of block time

As a model, that's how relativity views spacetime, yes: as a 4-dimensional geometric object that just exists, and does not change. But that's a model; it should not be taken as making metaphysical claims about what "reality" is like.

that block spacetime would not be like a constantly thick slice bread as it is sometimes depicted in popular science books, it would be more like a cone bread, getting larger in its space dimensions as it gets larger in its time dimension.

Remember that our best current model says that the universe is spatially infinite. You can't really view a spatially infinite model as "getting larger in its space dimensions" in the way you describe.

For a closed universe model, where the spatial topology is that of a 3-sphere, you can think of it as something like a loaf of bread that thins to a point at each end and is thickest in the middle, yes. But describing that as "being larger in its space dimensions" in the middle presupposes a particular split of spacetime into space and time. See my comments on that in earlier posts. And, as I just noted, this model, as best we can tell, does not describe our actual universe.

 

[guywithdoubts]

It is perfectly possible for space to either expand or contract. Doing neither is only possible in perfectly empty space with no cosmological constant.

Now, is the latter only possible because there is no way to measure distances (since there are no contents in this hypothetical universe) and therefore we cannot talk about geometry that well, or perhaps at all? Or is, then, geometry independent of these measurements? If so, wouldn't that mean space, as geometry, is "something", in that it does not need other elements for it to be a framework of them?

 

[Chalnoth]

Remember that our best current model says that the universe is spatially infinite.

There's no reason to take that seriously. "Close to flat" is a long way away from being perfectly flat. And even a perfectly-flat universe can be finite (e.g. a toroidal universe).

The main problem here is that measuring the local spatial curvature says literally nothing about the overall topology.

Now, is the latter only possible because there is no way to measure distances (since there are no contents in this hypothetical universe) and therefore we cannot talk about geometry that well, or perhaps at all? Or is, then, geometry independent of these measurements? If so, wouldn't that mean space, as geometry, is "something", in that it does not need other elements for it to be a framework of them?

No, it's just because an empty universe is a flat space-time.

 

[PeterDonis]

measuring the local spatial curvature says literally nothing about the overall topology.

This is true, but there are other ways of getting data on the overall topology. For example, if the universe were spatially a flat 3-torus, we would expect to see multiple images of identical objects in widely different directions, which, AFAIK, we don't. One could argue that there hasn't been enough time yet for light to "circumnavigate" the universe in this way, but that still means a model with non-trivial spatial topology has more explaining to do.

 

[Gerinski]

Remember that our best current model says that the universe is spatially infinite. You can't really view a spatially infinite model as "getting larger in its space dimensions" in the way you describe.

I guess you mean it's spatially infinite towards the future, not that it is spatially infinite at its 13 billion years age or that it was also spatially infinite when it was 1 million years old.
I was taking about spatial finiteness at certain time extension magnitudes (certain ages).

 

[Chalnoth]

This is true, but there are other ways of getting data on the overall topology. For example, if the universe were spatially a flat 3-torus, we would expect to see multiple images of identical objects in widely different directions, which, AFAIK, we don't. One could argue that there hasn't been enough time yet for light to "circumnavigate" the universe in this way, but that still means a model with non-trivial spatial topology has more explaining to do.

Only if the universe wrapped back on itself before reaching the particle horizon. There may be clever ways of pushing the distance out further. But we can never push that distance infinitely-far. And because of the cosmological constant, observers within our universe will never be able to see parts of the universe that are currently beyond their horizon.

 

[PeterDonis]

I guess you mean it's spatially infinite towards the future, not that it is spatially infinite at its 13 billion years age or that it was also spatially infinite when it was 1 million years old.

No. The spatially infinite model is spatially infinite, period. Anyway, you are once again implicitly assuming a split of spacetime into "space" and "time". Stop doing that; it's only going to continue causing you confusion.

 

[PeterDonis]

we can never push that distance infinitely-far

Meaning, we can never distinguish a spatial 3-torus with a sufficiently large "size" from a spatially infinite universe? I can't think of any way of doing so. But I would be hesitant to make a flat statement that it's impossible. The two models are still different, and in principle some way could be found to test for that difference experimentally.

 

[Chalnoth]

Meaning, we can never distinguish a spatial 3-torus with a sufficiently large "size" from a spatially infinite universe? I can't think of any way of doing so. But I would be hesitant to make a flat statement that it's impossible. The two models are still different, and in principle some way could be found to test for that difference experimentally.

Measuring an infinite universe would require measuring exactly zero on some measurable quantity or other. As measurement error will always be nonzero, that is impossible.

There are certainly methods to place a lower bound on the size of the universe outside of our particle horizon (though those always require certain assumptions about the nature of the universe beyond that horizon). But it's not possible to push that lower bound out to infinity.

 

[PeterDonis]

Measuring an infinite universe would require measuring exactly zero on some measurable quantity or other.

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.

it's not possible to push that lower bound out to infinity.

True. But ruling out a spatially infinite universe requires establishing an upper bound, not a lower bound.

 

[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.