Can empty space be considered ‘elastic’ and expandable?

[Herbascious J]

TL;DR Summary

Is a perfect vacuum able to expand in the way that we see the cosmos expanding even without any matter content? When viewing the expansion of space according to GR, it seems that the expansion is coming from a growth of space, rather than a physical acceleration of matter away from a local spot.

The intention of this question is to get to the heart of the geometrical properties of space-time according to GR, and to focus entirely on what the theory allows, and not so much on what we actually see. I would like to consider a perfect vacuum, in a euclidian infinite void, completely devoid of matter and photons. I understand that there are quantum consideration when considering a perfect vacuum, but I am more interested in the pure mechanics of GR, and would like to avoid vacuum energy and only consider the lambda cosmological constant when further explaining the possible behaviors of space-time. Of course, I am prepared to be shown that GR must have some basic matter constituent to function, and I am also prepared to be told that a Euclidian space cannot exist for various reasons, but this is the thrust of my question. So a couple of thoughts. Can this empty void exist in theory, and if so, can I mathematically expand this space so that the metrics all begin to move away from one another at some fixed velocity and would this require energy? What would this question look like with a single electron in the infinite void, of perhaps two at inter galactic distances? Perhaps the entire void is simply an extremely rare and diffuse homogenous gas of hydrogen molecules far more diffuse than ever would be observed (just to add matter to stabilize things). I understand these molecules have their own sub speed of light random velocities and therefore, an extremely low temperature. You can replace the gas with a low level photon temperate of .0001 degrees if that helps the question. So I’m really just curious about he dynamics of this medium and if it’s expanded, is the momentum of expansion happening to the individual particles, or can this expansion be purely geometric? I apologize for the loosens of the question and if it’s irrational.

 

[Dale]

Summary: Is a perfect vacuum able to expand in the way that we see the cosmos expanding even without any matter content? When viewing the expansion of space according to GR, it seems that the expansion is coming from a growth of space, rather than a physical acceleration of matter away from a local spot.

Can this empty void exist in theory, and if so, can I mathematically expand this space so that the metrics all begin to move away from one another

What you want to look for are vacuum solutions to the EFE. I believe that the Milne spacetime has at least some of th properties you are looking for.

 

[pervect]

From a geometric point of view, the Milne metric, which describes an expanding, empty universe, represents the same physical reality (an empty vacuum with no singularities or exclude points) that the non-expanding Minkowskii metric does.

The two metrics are just different ways of describing the same geometry.

We can regard two metrics as being equivalent (belong to the same equivalence class) if there is a smooth 1:1 correspondence between the space-times described by the metric, a diffeomorphism.

Basically, if you have an empty flat space-time, it's the same geometry, one way of labelling the points gives rise to the non-expanding Minkowskii metric, antother system of labelling points results in the same space-time being described by the "expanding" Milne metric.

Wiki currently has <<this link>> on the Milne metric, which might be a good place to start.

Note that while the Schwarzschild and the Minkowskii metric are both vacuum solutions of Einstein's equtions, they are not equivalent - there is no smooth 1:1 mapping (diffeomorphism) between them that preserves the Lorentz intervals.

I would say that since the identical geometry can be described as "expanding" or "not expanding", "expansion" doesn't have any direct physical significance.

 

[Dale]

I believe that the Milne metric also does not involve the cosmological constant.

 

[PeterDonis]

I believe that the Milne metric also does not involve the cosmological constant.

That's correct; the Milne metric is just a different coordinate chart on a portion of Minkowski spacetime, which has zero cosmological constant.

 

[PeterDonis]

"expansion" doesn't have any direct physical significance.

It does, but not as a property of a spacetime. Its physical significance is as a property of a family of worldlines; "expansion" means that the expansion tensor (or scalar, depending on context) of that family of worldlines is nonzero. But you can have families of worldlines with this property in all kinds of spacetimes.

 

[PAllen]

I would say that since the identical geometry can be described as "expanding" or "not expanding", "expansion" doesn't have any direct physical significance.

I agree. I view the notion of expanding space as similar to the notion of ether. That is, an inessential addition to a theory that doesn't need it (in one case SR, in the other GR). I think the expanding space notion is more helpful than ether, but no more essential.

 

[PAllen]

It does, but not as a property of a spacetime. Its physical significance is as a property of a family of worldlines; "expansion" means that the expansion tensor (or scalar, depending on context) of that family of worldlines is nonzero. But you can have families of worldlines with this property in all kinds of spacetimes.

But expanding space, especially as typically presented, is not the same as the expansion of a congruence of world lines. I agree that the key notion for cosmologies is really that the geometry allows a global homogeneous expanding congruence. Only very special geometries allow this.

 

[PeterDonis]

expanding space, especially as typically presented, is not the same as the expansion of a congruence of world lines

Actually, it is as typically presented, because the typical presentation is in the context of an expanding FRW spacetime, with the congruence of worldlines being the worldlines of comoving observers, which are naturally picked out by the symmetry of the spacetime. The expanding spaces are just the spacelike surfaces of constant time for the comoving observers, and the positive expansion of that congruence corresponds to the expansion of the spaces (in fact, in a proper rigorous presentation the expansion of the spaces is defined as the positive expansion of the congruence).

What the typical presentation fails to do is to point out that the correspondence between the two expansions is a feature that is particular to that family of spacetimes and does not hold in general.

 

[Herbascious J]

When we speak of expansion in this way, does the expansion of a vacuum require energy input to get it going, even without matter?

Admitedly, this is a little above my head. Does anyone have any introductory reading that can help? (I hope it is appropriate to ask for recommendations on this site)

 

[PeterDonis]

When we speak of expansion in this way, does the expansion of a vaccuum...

When we speak of expansion in this way, "expansion of a vacuum" is no longer physically meaningful; it's just a particular choice of coordinates on Minkowski spacetime. So it makes no sense to ask if anything is required to get it going, because it's not something real that's happening to begin with.

 

[Herbascious J]

When you say it is no longer meangingful, is that because it is empty space? In a universe like ours, when matter is being separated, things must have some kind of momentum or energy to drive them apart? I'm trying to imagine how this applies to what we see and if we can place any significance to the idea that our cosmos is expanding the way it seems to.

 

[PeterDonis]

When you say it is no longer meangingful, is that because it is empty space?

You said "expansion of a vacuum". "Vacuum" in this context means "Minkowski spacetime". And "expansion of space" applied to Minkowski spacetime is just a particular choice of coordinates, as I and others have said. But a particular choice of coordinates doesn't have physical meaning.

In a universe like ours, when matter is being separated, things must have some kind of momentum or energy to drive them apart?

You're thinking of the matter in the universe as if it were expanding the way the products of an ordinary explosion expand away from the location of the explosion. That's not how the expansion of the universe works.

To the extent it makes sense, given how the expansion of the universe does work (i.e., given that the spacetime geometry of the universe has a particular shape), to talk about momentum and energy making the matter in the universe fly apart, that momentum and energy came from the big bang.

 

[Herbascious J]

I guess what I'm hoping to understand is if the metric expansion of space itself is somehow energy dependent in a way that is different from normal relative velocities of objects (like in an explosion). I thought by framing the question in a vacuum, and then populating the vacuum with matter and seeing how the explanations differ I could try to understand it better. Thank you for the input, I will look into and the study the concepts mentioned.

 

[PeterDonis]

what I'm hoping to understand is if the metric expansion of space itself is somehow energy dependent in a way that is different from normal relative velocities of objects (like in an explosion)

As I said, the expansion of the universe works very differently from a normal explosion. So I don't think trying to find analogies between them is helpful to understanding how the expansion of the universe works. That includes trying to figure out in what way the expansion of the universe is "energy dependent". Technically, the expansion history of the universe does depend on the initial conditions at the big bang; but "depends on the initial conditions", though it is a factor in common between the expansion of the universe and an ordinary explosion, is a very weak commonality and doesn't really tell you anything useful for a further understanding.

 

[Herbascious J]

Ok, please allow me a final attempt to bend my mind around this and I'll go. By defining metric spaces (like the ones that make vacuum appear to expand), these are just dynamic (or curved space-time like) coordinate systems, and we can just as easily apply a non-expanding or even a contracting set of coordinates to the same physical volume of space and it wouldn't make a difference to the outcome of the physics. Real objects and their behavior can be explained and modeled in a variety of these different spaces, because its all relative and they are conceptual. So when we look up and see galaxies moving apart they just seem to be following an expanding coordinate set that can be modeled using GR for convenience and we just leave it at that? It's once we plug in GR and how it handles energy/gravity that we can start predicting how things will evolve and then it's just how our chosen coordinate system will change.

 

[PeterDonis]

By defining metric spaces (like the ones that make vacuum appear to expand), these are just dynamic (or curved space-time like) coordinate systems

No, metric spaces are the underlying geometric entities that we can choose different coordinate systems to describe. But choosing a different coordinate system does not change the underlying geometry of the metric space.

we can just as easily apply a non-expanding or even a contracting set of coordinates to the same physical volume of space and it wouldn't make a difference to the outcome of the physics

Describing coordinates as "non-expanding" or "contracting" is not correct. The coordinates are just sets of numbers. To even give a meaning to "expanding" or "non-expanding" or "contracting", you have to look at the actual physics, not just the coordinates.

Real objects and their behavior can be explained and modeled in a variety of these different spaces, because its all relative and they are conceptual.

No, real physics is not relative or conceptual. The expansion of the universe, properly understood, is a real thing that is really happening, and no choice of coordinates can change that. But the expansion of the universe, properly understood, is not "expansion of space". It's the galaxies and galaxy clusters moving apart.

 

[PAllen]

Actually, it is as typically presented, because the typical presentation is in the context of an expanding FRW spacetime, with the congruence of worldlines being the worldlines of comoving observers, which are naturally picked out by the symmetry of the spacetime. The expanding spaces are just the spacelike surfaces of constant time for the comoving observers, and the positive expansion of that congruence corresponds to the expansion of the spaces (in fact, in a proper rigorous presentation the expansion of the spaces is defined as the positive expansion of the congruence).

What the typical presentation fails to do is to point out that the correspondence between the two expansions is a feature that is particular to that family of spacetimes and does not hold in general.

Many also make misleading statements about recession being due to expansion of space between galaxies that are not moving relative to space, as distinct from peculiar velocity which is motion through space. This distinction is nonsense - a 4 velocity is a 4 velocity, a geodesic is a geodesic, per the math of GR. All you can state is that a comoving oberver will see isotropy. Many also make statements about cosmological red shift being different in kind to Doppler. But, per GR in general there is just a single generalization of SR Doppler to curved spacetime (with several equivalent formulations), that necessarily has the feature that the closer emitter and receiver are, the more the math matches SR Doppler (a consequence of local flatness).

 

[pervect]

It does, but not as a property of a spacetime. Its physical significance is as a property of a family of worldlines; "expansion" means that the expansion tensor (or scalar, depending on context) of that family of worldlines is nonzero. But you can have families of worldlines with this property in all kinds of spacetimes.

I'd agree with that - when I was rereading my post I regretted not adding that point.

 

[metastable]

Many also make misleading statements about recession being due to expansion of space between galaxies that are not moving relative to space, as distinct from peculiar velocity which is motion through space. This distinction is nonsense - a 4 velocity is a 4 velocity, a geodesic is a geodesic, per the math of GR. All you can state is that a comoving oberver will see isotropy.

Since the predicted combination of values for my recessional vector and recessional velocity from the rest frame of a comoving test particle will vary at every comoving position in space, how can it still be said that “space is isotropic?”

 

[PAllen]

Since the predicted combination of values for my recessional vector and recessional velocity from the rest frame of a comoving test particle will vary at every comoving position in space, how can it still be said that “space is isotropic?”

I have no idea what you question means. However it is easy to explain isotropy.

A comoving observer will see an even distribution of matter in all directions (at large scales), and redshift observed will be purely a function of distance with no variation by angular position. Finally, the CMBR will have the same frequency in all directions.

An observer moving relative to an adjacent comoving observer will see the CMBR frequency as different in different directions, and the redshift of bodies will vary based on distance and viewing angle. This is anisotropy.

 

[metastable]

I have no idea what you question means.

I choose a single non accelerating non precessing electron in the universe as a “reference” electron, and I also have a “test” non-accelerating non-prescessing electron I can observe at my position in space relative to the reference electron, and my test electron is said to be “comoving” with the reference electron, & I observe the combination of values of distance increase over time and vector of this distance increase relative to the vector of the non-precessing electron’s magnetic moment vector is different at every point in space.

If the values for distance increase over time + vector of distance increase are unique at every separate point in spacetime, how can I still say spacetime is isotropic?

 

[PeterDonis]

my test electron is said to be “comoving” with the reference electron

You are using a different meaning of "comoving" than the standard meaning in cosmology. You are using it to mean "at rest relative to". But "comoving" observers in the universe in cosmology are not at rest relative to each other; they are moving apart, because the universe is expanding. The "comoving" observers in cosmology are the ones who always see the universe as homogeneous and isotropic. But the two electrons in your scenario are not "comoving" in the sense used in cosmology, and at most one of them can possibly see the rest of the universe as homogeneous and isotropic.

 

[metastable]

You are using a different meaning of "comoving" than the standard meaning in cosmology.

I recognize there are different meanings but the meaning I actually intended for "comoving" was the cosmology meaning... which is why I said:

my test electron is said to be “comoving” with the reference electron, & I observe the combination of values of distance increase over time and vector of this distance increase

^ie even though they are said to be "comoving," the distance is still expected to increase over time...

Still even using this definition of "comoving," where the rate of distance increase between 2 non-accelerating particles depends entirely or almost entirely on separation distance--

I said:

the combination of values of distance increase over time and vector of this distance increase relative to the vector of the non-precessing electron’s magnetic moment vector

^What I'm asking is if at every point in spacetime, if at least one of these 2 values varies, how can I still say spacetime is isotropic?

 

[Dale]

If the values for distance increase over time + vector of distance increase are unique at every separate point in spacetime, how can I still say spacetime is isotropic?

If you rotate your two electron apparatus you will get the same behavior. Therefore spacetime is isotropic. If you move your two electron apparatus to a new location you will also get the same behavior. Therefore spacetime is homogenous.

 

[metastable]

If you rotate your two electron apparatus you will get the same behavior. Therefore spacetime is isotropic.

^Dale, thank you for your answer, I wasn't confused by it until I remembered that I thought you said before I can't actually rotate the magnetic moment vector of the electron to a desired vector...

Is there a method that will cause the vector of the electron's magnetic moment to stop precessing & continuously point in a desired direction? Suppose I observe a distant galaxy and I want each measurement I perform of the electron's magnetic moment vector to show that it is substantially parallel with the vector pointing towards that galaxy from my present position within some small margin of error. What forces and methods would I rely on to orient the vector of the electron's magnetic moment in this desired vector?
https://www.physicsforums.com/threads/polarized-electron-in-a-rotating-reference-frame.971485/page-2

Not that I know of. But why not just use a classical gyroscope instead?
https://www.physicsforums.com/threads/polarized-electron-in-a-rotating-reference-frame.971485/page-2

No, if you’re thinking about an electron and quantum mechanical spin
https://www.physicsforums.com/threads/polarized-electron-in-a-rotating-reference-frame.971485/page-2

 

[PeterDonis]

I recognize there are different meanings but the meaning I actually intended for "comoving" was the cosmology meaning

If you meant "comoving" in the cosmological sense, then your claim that the test electron is comoving with the reference electron is wrong. Do you see why?

 

[Dale]

I can't actually rotate the magnetic moment vector of the electron to a desired vector

All of the same quantum mechanical issues apply regardless of how you rotate your apparatus. If you can’t do something in one orientation then you can’t do it in any other orientation either. Therefore it is isotropic.

You really need to stop mixing the QM and the cosmological questions as we have discussed.

 

[PeterDonis]

I can't actually rotate the magnetic moment vector of the electron

We're not talking about trying to rotate the electron's magnetic moment vector. We're talking about setting up the scenario with all of it rotated--i.e., the line connecting the two electrons is along a different direction. That won't make any difference to what happens in the scenario; that's what the universe being isotropic means. You can also set up the scenario anywhere in the universe you like, and it won't make any difference to what happens; that's what the universe being homogeneous means.