I Spacetime expansion - time dimension expansion

[Orodruin]

The Milne coordinates on Minkowski space are a bit special as far as they go. Unlike most cases of RW spacetimes where there exists a t such that a(t)=0, the corresponding singularity is just a coordinate singularity and the spacetime can be extended to Minkowski space even if you just start with the Milne coordinates.

The Milne model is expanding in the sense that comoving observers of the Milne coordinates get further and further away from eachother. This is a property of the set of comoving observers chosen. In a similar vein, comoving observers of the usual Minkowski coordinates are different and they are at rest relative to eachother.

The fact that there are different RW coordinates on the same spacetime is related to the fact that there are several ways of foliating Minkowski space into homogeneous and isotropic spacelike surfaces.

 

[Orodruin]

The correct fact is that the Milne universe is identical to (which implies diffeomorphic with) a particular patch of Minkowski spacetime, as stated above.

I would add that the Milne universe is not geodesically complete. If you extend it in the same fashion as you can extend the exterior Schwarzschild solution beyond the horizon, you get Minkowski space.

 

[PeroK]

I understand that We can say that what is expanding are worldliness of comoving observers. But something actually has to change for this to happen? This is more of an effect. What is the cause? I can imagine that the space is expanding, or time is expanding, both, or everything is shrinking. What other explanation could be?

There are several concepts here that can be difficult to grasp. Change usually refers to change over time. Space can change over time, but spacetime includes the time dimension, so it can't change over time.

In GR spacetime itself is subject to the laws of physics, rather than being a fixed, uniform background. In particular, space must either be expanding or contracting. It cannot be static. Albeit these are coordinate dependent statements. An analogy is a ball in a gravitational field. The ball cannot hover, but must be going up or down. This is a result of the law of gravitation. Likewise, the expanding universe is a result of the laws of physics and the stuff the universe is made of.

That is the explanation provided by theoretical physics.

Finally, the concept of a coordinate free description of spacetime and cosmological phenomena generally is also difficult to grasp. You probably need to study GR seriously before this really makes sense.

 

[PeterDonis]

space must either be expanding or contracting. It cannot be static

I think you mean that for cases like the universe as a whole, spacetime cannot be static, because there are no static solutions that satisfy the usual assumptions of homogeneity and isotropy (except for the edge case of the Einstein static universe). It is certainly not the case that there are no solutions in GR at all that are static.

An analogy is a ball in a gravitational field. The ball cannot hover, but must be going up or down.

But an object held together by its own gravity, and supported against gravity by pressure, can be static (such as an idealized non-rotating planet or star or a Schwarzschild black hole). It's just that such a solution cannot describe the universe as a whole, because it's not possible for such a solution to be homogeneous and isotropic.

(And note also that for the ball in the gravitational field, the field itself can be static--it could be the field of a static planet or star or black hole--even though the ball is not. In such a case, "space", at least for the natural notion of "space" for a static spacetime, is static; the ball falling is not due to "space" expanding or contracting, it's due to the ball not being supported against gravity.)

 

[exander]

I was thinking about it a lot, and I am not sure how to explain cosmological red shift under conditions that just distance between comoving worldliness changes. How can this produce the effect of red shift? I always assume that the space in which the wave travels gets stretched.

 

[Hill]

I was thinking about it a lot, and I am not sure how to explain cosmological red shift under conditions that just distance between comoving worldliness changes. How can this produce the effect of red shift? I always assume that the space in which the wave travels gets stretched.

I think the following excerpt from Manton, Nicholas; Mee, Nicholas. The Physical World: An Inspirational Tour of Fundamental Physics explains it nicely:

 

[PeroK]

I was thinking about it a lot, and I am not sure how to explain cosmological red shift under conditions that just distance between comoving worldliness changes.

You can explain it with mathematics. As above!

 

[exander]

I think the following excerpt from Manton, Nicholas; Mee, Nicholas. The Physical World: An Inspirational Tour of Fundamental Physics explains it nicely:

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This is interesting, but kind of vague. This looks to me more like the light wave hadn't changed, but the measure changed. Meaning everything shrunk. Can We interpret this as matter shrinking, and because of that the wave seems to have larger wave length? Is there a shrinking matter theory?

 

[PeroK]

This is interesting, but kind of vague. This looks to me more like the light wave hadn't changed, but the measure changed.

Light doesn't inherently change. All redshift or blueshift is a result of the relationship between the source and receiver: relative motion, relative gravitational potential; or, separated by expanding space. Or, a combination of all three!

 

[Hill]

This is interesting, but kind of vague. This looks to me more like the light wave hadn't changed, but the measure changed. Meaning everything shrunk. Can We interpret this as matter shrinking, and because of that the wave seems to have larger wave length? Is there a shrinking matter theory?

This is standard GR. I can't find anything shrinking in this derivation. Which step is vague?

 

[PeterDonis]

Can We interpret this as matter shrinking

No.

 

[PeterDonis]

This looks to me more like the light wave hadn't changed, but the measure changed.

No. What changes is, heuristically, the angle in spacetime between the light ray's worldline and the comoving worldlines it passes. Each successive comoving worldline is at a slightly different angle in spacetime from the light ray's worldline. And that angle in spacetime is what determines the measured energy/frequency/wavelength of the light according to the comoving observer.

 

[cianfa72]

You are incorrectly describing what expands in our universe. It is not "spatial dimensions". It is the set of worldlines of comoving observers, which are observers who always see the universe as homogeneous and isotropic. The worldlines of these observers can be used to construct a very convenient coordinate chart for describing the universe, and in this chart the expansion of the set of worldlines can be, and often is, described as "space expanding". Which is unfortunate because it misleads people into thinking that the expansion is something it's not.

Sorry, the claim about comoving observers that see the universe as homogeneous and isotropic, actually means that each spacelike hypersurface of constant comoving coordinate time has a spatial positive definite metric homogeneous and isotropic.

 

[PeterDonis]

the claim about comoving observers that see the universe as homogeneous and isotropic, actually means that each spacelike hypersurface of constant comoving coordinate time has a spatial positive definite metric homogeneous and isotropic.

Yes. But any such spacelike hypersurface is an abstraction; it's not a physical thing that we can observe to be "expanding". The actual galaxies we see are physical things that we can observe, and we can take their average motion and see that it corresponds to expansion.

 

[pervect]

No.

This is interesting, but kind of vague. This looks to me more like the light wave hadn't changed, but the measure changed. Meaning everything shrunk. Can We interpret this as matter shrinking, and because of that the wave seems to have larger wave length? Is there a shrinking matter theory?

Einstein has used the idea of "shrinking" or "expanding" rulers in his popularization, "Relativity: The Special and General theory". See for instance https://www.bartleby.com/lit-hub/re...ry/xxiv-euclidean-and-non-euclidean-continuum. Einstein, however, uses the idea mainly as an example of one way of understanding non-Euclidean geometry, not applying it specifically to an expanding universe.

By making use of the following modification of this abstract experiment, we recognise that there must also be cases in which the experiment would be unsuccessful. We shall suppose that the rods “expand” by an amount proportional to the increase of temperature. We heat the central part of the marble slab, but not the periphery, in which case two of our little rods can still be brought into coincidence at every position on the table. But our construction of squares must necessarily come into disorder during the heating, because the little rods on the central region of the table expand, whereas those on the outer part do no

But it's really mostly a philosophical issue. It's philosophical because there isn't any obvious experiment to distinguish between a growing object and a shrinking ruler. The more common philosophy is to say that standardized rulers don't shrink - that's inherent in the idea of what a ruler does.

The idea of "shrinking rulers" occurs in a few other places, I believe MTW mentions it in "Gravitation", though I don't have a specific quote, and I recall at least one other paper that talks about this idea in the context of General Relativity, though it's name and author escape me. But while the general idea has occured to people, it hasn't gained much traction. I'm not aware of anyone publishing a peer-reviewed paper that specifically suggests applies this general idea to the expanding universe. I think the idea may run into some difficulties with instantaneous action at a distance , but that's my own thought, as I said I'm not aware of any peer-reviewed papers that discuss such an idea.

This is a bit of a longer answer than "No", but in the end the result is pretty similar.

 

[Hill]

The idea of "shrinking rulers" occurs in a few other places

Thorne in his book "Black Holes & Time Warps: Einstein's Outrageous Legacy" describes and discusses the two "paradigms" in chapter 11. He concludes,

It is extremely useful, in relativity research, to have both paradigms at one’s fingertips. Some problems are solved most easily and quickly using the curved spacetime paradigm; others, using flat spacetime. Black-hole problems (for example, the discovery that a black hole has no hair) are most amenable to curved spacetime techniques; gravitational-wave problems (for example, computing the waves produced when two neutron stars orbit each other) are most amenable to flat spacetime techniques. Theoretical physicists, as they mature, gradually build up insight into which paradigm will be best for which situation, and they learn to flip their minds back and forth from one paradigm to the other, as needed. They may regard spacetime as curved on Sunday, when thinking about black holes, and as flat on Monday, when thinking about gravitational waves. This mind-flip is similar to that which one experiences when looking at a drawing by M. C. Escher, for example, Figure 11.2. Since the laws that underlie the two paradigms are mathematically equivalent, we can be sure that when the same physical situation is analyzed using both paradigms, the predictions for the results of experiments will be identically the same. We thus are free to use the paradigm that best suits us in any given situation. This freedom carries power. That is why physicists were not content with Einstein’s curved spacetime paradigm and have developed the flat spacetime paradigm as a supplement to it.

 

[PeterDonis]

Thorne in his book "Black Holes & Time Warps: Einstein's Outrageous Legacy" describes and discusses the two "paradigms" in chapter 11.

Note, though, that when Thorne says the two paradigms are "mathematically equivalent", he is glossing over an important technical issue. Locally, yes, you can use either description. But globally, the "field on flat spacetime" description might not work, because the global topology of the spacetime might not be the same as the global topology of flat spacetime, meaning that globally, it is impossible to view gravity as a field on flat spacetime. This is true of black hole spacetimes, for example.

 

[Hill]

Note, though, that when Thorne says the two paradigms are "mathematically equivalent", he is glossing over an important technical issue. Locally, yes, you can use either description. But globally, the "field on flat spacetime" description might not work, because the global topology of the spacetime might not be the same as the global topology of flat spacetime, meaning that globally, it is impossible to view gravity as a field on flat spacetime. This is true of black hole spacetimes, for example.

Where can I learn more about this limitation?

 

[PeterDonis]

Where can I learn more about this limitation?

I have not found any discussion of this in any detail in the literature, I expect because nobody actually uses the flat spacetime paradigm to analyze black holes. The main uses of the flat spacetime paradigm are in analyzing gravitational waves and weak fields.