Spacetime expansion - time dimension expansion

[exander]

TL;DR Summary

Is time dimension expanding?

I am trying to find some sources about time dimension expansion during spacetime expansion.

Why do We assume that only spacial dimensions expand? I only found a lot of weird arguments.

Btw, temporal dimension expansion does not mean that second was shorter in the past. Similarly to spacial dimension expansion, second is still second as meter is still meter. What it would mean is that two instances in time looks further apart the more time has passed, similarly as two places in space are further apart the more time has passed.

For example, consider two observers A and B and no spacial expansion.
Observer A watches two supernovae, one in galaxy 25 000 light years far away and the other one in galaxy 100 000 light years far away.
Observer B watches the same two supernovae, but is 100 000 further away, so one in galaxy 125 000 light years far away and the other one in galaxy 200 000 light years far away.
If A perceives 10 years between this two supernova, then B perceives more than 10 years.
This effect seems to influence observations similarly to spacial expansion.

Any ideas, sources about this?

 

[Hill]

TL;DR Summary: Is time dimension expanding?

Why do We assume that only spacial dimensions expand?

Because of the FLWR metric:

The temporal coefficient is constant, but the spatial one depends on time.

 

[phinds]

Any ideas, sources about this?

Interesting idea, but not one that has any basis in reality. Time does not "expand" the way the universe does (or in any other way that could realistically be described as similar) and your example does not really show anything like "expansion" of time, just that things can take different amounts of time. If I travel from DC to Boston by a direct route, it takes me a different amount of time than if I do it going by way of Chicago. I do both at 60 mph --- does that mean my time "expanded" when I went by way of Chicago? No, it just means it took longer.

 

[Ibix]

Why do We assume that only spacial dimensions expand?

We don't. You can perfectly well rescale everything so that your time coordinate has a variable relationship with the proper time of a comoving observer. It doesn't change anything, just moves bits of the maths around.

The point is that the "space is expanding" description is one that is useful to observers who are at rest with respect to the average of matter around them. They are the only ones who see a pure "homogeneous space is expanding" model around them. So it makes sense to use their clocks as the notion of time and stuff all the "expansion" into the spatial slices. You aren't obliged to do this, but if you choose not to all you do is make the already-complicated maths yet more complicated.

 

[PeterDonis]

Why do We assume that only spacial dimensions expand?

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.

 

[Orodruin]

Because of the FLWR metric:
View attachment 339634
The temporal coefficient is constant, but the spatial one depends on time.

There is an additional point to this. You could very well write down the metric
$$
ds^2 = - b(t)^2 dt^2 + a(t)^2 d\Sigma^2
$$
as there is nothing forbidding this. However, in this situation you can always change the time coordinate such that $b(t) = 1$.

 

[Ibix]

Observer A watches two supernovae, one in galaxy 25 000 light years far away and the other one in galaxy 100 000 light years far away.
Observer B watches the same two supernovae, but is 100 000 further away, so one in galaxy 125 000 light years far away and the other one in galaxy 200 000 light years far away.
If A perceives 10 years between this two supernova, then B perceives more than 10 years.
This effect seems to influence observations similarly to spacial expansion.

So, just to address this point directly, this result is exactly what you get in an FLRW universe with comoving observers. You are free to attribute the difference in times between sightings to expanding space, expanding time, or a mixture of both, and there are other ways of splitting spacetime into time and space if you wish. It's all a matter of choice. "Space expands" is just one choice that drops easily out of the Friedman equations and is easy to put into words, and that's the one that's in the popular consciousness. It's not an assumption that's really there in the underlying model.

 

[Bosko]

There is an additional point to this. You could very well write down the metric
$$
ds^2 = - b(t)^2 dt^2 + a(t)^2 d\Sigma^2
$$
as there is nothing forbidding this. However, in this situation you can always change the time coordinate such that $b(t) = 1$.

Can we tell more about $b(t)$ in the past related to $a(t)$ ?
I'm just wondering if it's possible to draw some conclusions about time in the past.
For example, one of the things I'm thinking about is the null geodesic (ray of light) which comes from the distant past.
Thank you

 

[Ibix]

Can we tell more about $b(t)$ in the past related to $a(t)$ ?

First note that @Orodruin's $t$ and $a$ are not the same as the ones in Hill's standard FLRW formula. You are free to define one of Orodruin's $a$, $b$ or the relationship between the two $t$s.

If you define $b$, then by comparing the first terms in the line elements we can write $\sqrt{b(t_O)}dt_{O}=dt_{FLRW}$, where $t_O$ is Orodruin's $t$ coordinate and $t_{FLRW}$ is the usual FLRW coordinate. Solve that for a relationship between the two $t$ coordinates and plug them into the FLRW $a$ (the functional form of which is constrained by observation) to get Orodruin's $a$.

If you set $b(t)=1$ you recover the usual FLRW coordinates. If you set it to anything else you get what you might call expanding or contracting time, or something less amenable to snappy description.

 

[Orodruin]

Can we tell more about $b(t)$ in the past related to $a(t)$ ?
I'm just wondering if it's possible to draw some conclusions about time in the past.
For example, one of the things I'm thinking about is the null geodesic (ray of light) which comes from the distant past.
Thank you

The entire point was that $b(t)$ by itself is unphysical. Only the relation to $a(t)$ is relevant as indicated by the coordinate transformation that puts b = 1.

 

[exander]

Thank you for the responses. What I assume from most of the responses is that it that this just an interpretation, and We can attribute the expansion to expansion of space, time or both arbitrarily, without any change at all.

For me, it is logical to assume that if We think that all spacial dimensions expand at the same rate, that temporal dimension expands as well with the same rate - whole spacetime is expanding. It doesn't really change anything, but the interpretation would be different, wouldn't it? Because if temporal dimension is expanding as well, it would mean that space as is not expanding as fast as we assume, because part of the assumed expansion is actually expansion of time. Would under such conditions be expansion accelerating? And would there be a real way to distinguish what is actually expanding?

 

[PeterDonis]

We can attribute the expansion to expansion of space, time or both arbitrarily, without any change at all.

We can do these things, as long as we recognize that all of these interpretations are limited and should not be taken too literally. If you want to make actual predictions, you need to use math, not try to reason in ordinary language based on an interpretation.

would there be a real way to distinguish what is actually expanding?

I already told you in post #5 what is actually expanding, and it isn't any of the things you say. This is an example of interpretations being limited: all of them are focusing your attention on wrong things. The right thing to focus your attention on is the actual objects in the actual universe, their actual observed behavior, and how well our best current model accounts for that behavior.

 

[Ibix]

Would under such conditions be expansion accelerating?

Yes. You can't make physical phenomena go away by a coordinate change.

And would there be a real way to distinguish what is actually expanding?

Formally it's the FLRW congruence that has a positive expansion scalar. Whether you choose to interpret that as "expanding space" or "expanding time" is a matter of choice; neither is more "actually" than the other.

 

[PeterDonis]

Formally it's the FLRW congruence that has a positive expansion scalar. Whether you choose to interpret that as "expanding space" or "expanding time" is a matter of choice; neither is more "actually" than the other.

Or you could choose neither, and not talk about "space expanding" or "time expanding" at all, and just talk about the actual objects.

 

[exander]

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?

 

[Hill]

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?

Einstein field equation of General Relativity under assumption of the homogeneous and isotropic Universe.

 

[Nugatory]

But something actually has to change for this to happen?

You and I are standing at the North Pole and facing in slightly different directions. We both start walking south in the directions that we are facing, so following slightly different lines of longitude. We find that by the time we reach the equator the space between us has grown to many kilometers.

What has “actually changed”?

 

[Vanadium 50]

This is more of an effect. What is the cause

300 years ago people might say "An object in motion remains in motion? That sounds like an effect. But what is its cause?"

 

[Ibix]

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?

The expansion scalar of a congruence measures the rate of change of the volume between eight nearby objects forming the corners of a cube. But it does it without referencing a global notion of space or time. You don't need a notion of expanding space or time at all, that's the point.

You can imagine a similar measure in Newtonian physics. If you look at fragments from a bomb blast in zero g, the volume defined by any nearby group of fragments increases. The only "cause" of it is whatever initially set the fragments flying and Newton's first law, that stuff doesn't just change direction for no reason.

The only differences in the cosmological case are that the worldlines don't back-trace to a single location and there are distinct classes of solution depending on the ratio between the fragments' velocity and their average density.

 

[PeterDonis]

We can say that what is expanding are worldliness of comoving observers. But something actually has to change for this to happen?

What changes is the proper distance between the worldlines (and there is an invariant, i.e., coordinate-independent meaning to that statement, which, when unpacked, leads you back to the expansion scalar that has already been referenced). There is no something else that has to change for that to happen. That is the change.

 

[Dale]

it is logical to assume that if We think that all spacial dimensions expand at the same rate, that temporal dimension expands as well with the same rate - whole spacetime is expanding

It sounds like you are suggesting $a(t)=b(t)$ which is contradicted by experiment.

 

[pervect]

There is an additional point to this. You could very well write down the metric
$$
ds^2 = - b(t)^2 dt^2 + a(t)^2 d\Sigma^2
$$
as there is nothing forbidding this. However, in this situation you can always change the time coordinate such that $b(t) = 1$.

I believe this correct observation implies that the whole idea of "expanding space" isn't fundamental, it's really just a remark about a particular choice of coordinates.

Then the answer to the question would be that it's a convention that we don't talk about time expanding.

I would also say that textbooks do not emphasize this point, and I don't think I've seen many papers published about it. See for instance "Expanding confusion", by Davis and Lineweaver, https://arxiv.org/abs/astro-ph/0310808. (Google reports a respectable 348 citations).

Interestingly, the authors here say "we cite numerous misleading, or easily misinterpreted, statements in the literature" when making their points. But they never get around to saying that the whole argument is basically about coordinate choices. Additionally, they make remarks such as

Since the expansion of the universe is the basis of the big bang model, these misunderstandings are fundamental.

Perhaps I am missing something, as one of my first reactions when I have an insight which doesn't seem to be echoed in the literature is to wonder if they are correct. But I don't see any flaws in my arguments. Also, I may simply be unaware of papers with my viewpoint. Additionally, the idea doesn't directly affect experimental results, but is more a question of interpretation, and how we teach the subject, so it may not attract much interest.

 

[Orodruin]

I believe this correct observation implies that the whole idea of "expanding space" isn't fundamental, it's really just a remark about a particular choice of coordinates.

A lot of our typical statements and interpretations in cosmology are based on applying the standard FLRW form of the metric. I wrote an Insight on this several years ago: https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

 

[PeterDonis]

Additionally, they make remarks such as

"The expansion of the universe" can refer simply to the positive expansion scalar of the congruence of comoving worldlines, which is an invariant. So phrasing it that way does not necessarily invite the misleading interpretations that "expansion of space" does.

 

[Vanadium 50]

Btw, temporal dimension expansion does not mean that second was shorter in the past.

I am still trying to see if this makes logical sense.

You set up a clock ticking once per second, and record both time time and the number of ticks. At time A, you say 10 seconds and 10 clicks pass. At a later time, you say that 20 seconds passed between A and B, but it was still 10 ticks and that the clocks ran at the same rate.

How is this possible?

 

[Hill]

AFAIU, what is so special about the "comoving, synchronous" coordinates in which the metric acquires $g_{tt}=1$ is that its time coordinate is the time on the clock of a comoving observer. IOW, if we are a comoving observer then after one hour on our clock the entire universe is one hour older.

Here is illustration from the MTW's Gravitation:

 

[PeterDonis]

IOW, if we are a comoving observer then after one hour on our clock the entire universe is one hour older.

You have to be very careful making statements like this. The elapsed time on the one comoving observer's clock is an invariant. But the elapsed time for "the entire universe", on its face, is not; it depends on your coordinate choice. To make an invariant statement along these lines requires more care.

Here is how I would do it: if we are a comoving observer, and we consider two events on our worldline separated by one hour, then we can define two homogeneous and isotropic spacelike 3-surfaces containing those two events and call those "the entire universe" at those two times by our clock--and then we will find that the elapsed time between those two hypersurfaces on any comoving worldline will be the same as ours, one hour.

What makes this an invariant statement is that we specified that each spacelike 3-surface was homogeneous and isotropic. That is an invariant geometric specification that picks out a unique spacelike 3-surface for any given event on a comoving worldline. (It is also the specification that underlies the construction of standard FRW coordinates, but one still has to distinguish the coordinates themselves from the underlying invariant geometry.)

 

[Hill]

You have to be very careful making statements like this. The elapsed time on the one comoving observer's clock is an invariant. But the elapsed time for "the entire universe", on its face, is not; it depends on your coordinate choice. To make an invariant statement along these lines requires more care.

Here is how I would do it: if we are a comoving observer, and we consider two events on our worldline separated by one hour, then we can define two homogeneous and isotropic spacelike 3-surfaces containing those two events and call those "the entire universe" at those two times by our clock--and then we will find that the elapsed time between those two hypersurfaces on any comoving worldline will be the same as ours, one hour.

What makes this an invariant statement is that we specified that each spacelike 3-surface was homogeneous and isotropic. That is an invariant geometric specification that picks out a unique spacelike 3-surface for any given event on a comoving worldline. (It is also the specification that underlies the construction of standard FRW coordinates, but one still has to distinguish the coordinates themselves from the underlying invariant geometry.)

Yes, absolutely. I've only expressed the "bottom line". For completeness, here is the "construction in §27.4" which the figure refers to:

 

[pervect]

I would ask the following questions.

Consider the Milne universe, which can be thought of as a special case of the FLRW universe with a(t) = kt.

(See post #2 in this thread, i.e. the metric in https://www.physicsforums.com/threa...time-dimension-expansion.1059648/post-7054986 for the definition of a(t). The Milne universe is alsomentioned by Orodruin as a "pathological example" in his insight article).

Is the Milne universe expanding?

Now consider Minkowskii space, where a(t) = 1. Is Minkowskii space expanding?

While asking these questions probably gives a strong hint as to my point of view, I'll go ahead and give my answers and my interpretations.

My answers to these would be that the Milne universe is expanding (since a(t) increases with time), and Minkowskii space is not expanding (because a(t) is constant).

The fact that the Milne universe is diffeomorphic to the Minkowskii universe , implying that they are basically the same universe with different coordinate labels, doesn't bother me, because I regard "expansion" as being a statement about coordinates - which I don't regard as "fundamental".

As far as congruences go - while I'd agree that some congruences expand and others don't, I don't regard statements about congruences as deriving from the space-times they are in, rather they are statements and properties about the congruence, not statements about the space-time.

 

[PeterDonis]

Is the Milne universe expanding?

The congruence of "comoving" worldlines in the Milne universe has a positive expansion scalar, so with the usual interpretation of "universe expanding", yes.

Is Minkowskii space expanding?

No, because there is no valid geodesic congruence covering all of Minkowski spacetime that has a positive expansion scalar.

The Milne universe only covers a particular patch of Minkowski spacetime (the interior of the future light cone of a chosen "origin" event), and the Milne congruence is only a valid congruence on that patch (at the chosen "origin" event, which itself is not in the Milne patch, all of the Milne congruence worldlines intersect). That is why the Milne congruence can have a positive expansion scalar without contradicting the statement about Minkowski spacetime that I made above--because it doesn't cover all of Minkowski spacetime.

My answers to these would be that the Milne universe is expanding (since a(t) increases with time), and Minkowskii space is not expanding (because a(t) is constant).

While these answers are correct with the appropriate definitions of a(t) in each case, I think a better approach is to look at congruences as I did above, since the definition of a(t) is coordinate dependent, while congruences and their expansion scalars are invariant, and the existence or nonexistence of congruences with particular properties in particular spacetimes or patches of spacetimes is also invariant.

The fact that the Milne universe is diffeomorphic to the Minkowskii universe

Is not a fact. The correct fact is that the Milne universe is identical to (which implies diffeomorphic with) a particular patch of Minkowski spacetime, as stated above.

implying that they are basically the same universe with different coordinate labels

Not really, since the regions of spacetime they occupy are not identical. See above.

I regard "expansion" as being a statement about coordinates - which I don't regard as "fundamental".

But given that there is an invariant concept of "expansion" (the expansion scalar of a congruence of worldlines), it seems to me to be a better approach to use that, since it allows "expansion" to be a matter of actual physics instead of coordinate choice.

I don't regard statements about congruences as deriving from the space-times they are in

I don't see why not. The properties of congruences are determined by the geometry of the spacetime, as are the existence and non-existence of congruences with particular properties.

 

[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.