# What happens to time as space is expanding?

• B
• AlfSalte
Alternatives to GR have been proposed which respect this symmetry, but they usually predict the exact same dynamics in nearly all cases.This stuff gets seriously technical in any event, and I'm not sure it's yet been explained in an approachable way.f

#### AlfSalte

TL;DR Summary
When space expands, what happens to time?
I have one question I hope someone here can answer for me.

Relativity theory tells us that space and time are sort of the same thing, as a spacetime. So when space is expanding, what happens to time? I find it hard to believe that time is somehow unaffected by the expansion of space, so while space is expanding, what is happening to time? Does it expand too or contract or what?

• Delta2
Time is not affected by the expansion of space-time. Time LOCALLY (technically, "proper time") always ticks away at one second per second.

Perhaps you are confused by "time dilation" which is a co-ordinate dependent measure.

Summary:: When space expands, what happens to time?

I have one question I hope someone here can answer for me.

Relativity theory tells us that space and time are sort of the same thing, as a spacetime. So when space is expanding, what happens to time? I find it hard to believe that time is somehow unaffected by the expansion of space, so while space is expanding, what is happening to time? Does it expand too or contract or what?
Good question!

When we talk about the expansion of space, we are talking about the distance to far away objects increasing over time. And by time we mean the regular time as measured by a clock on Earth. In cosmological terms this is also called comoving time.

The universe, therefore, has a scale factor ##a(t)## that determines the distance between objects and depends on comoving time ##t##. And in this model - sometimes called the FLRW model - the time coordinate itself is not "expanding", but is the parameter determining the expansion of space:

https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric

• Delta2 and Buzz Bloom
so while space is expanding, what is happening to time?

How would you measure such a thing? How do you tell if one second last year is the same as one second now? Understanding something oten begins with figuring out how to measure it.

• nnunn
The short answer is that the expansion of space is measured with respect to time.

The slightly longer answer is that the whole space-time is described as a single 4-dimensional manifold, and that manifold is then sliced. The specific slice we pick when describing the expansion is the one where there average background radiation temperature is the same across the entire slice. Then we label each slice, and measure time across them.

The really nice thing is that it all behaves very regularly. It's so simple that we can even derive the matter-only expansion in exactly the same way using simple Newtonian gravity. Thus nothing at all weird happens with the time coordinate.

But the really weird thing is that General Relativity isn't covariant to transformations that affect all four dimensions at once. I used to think it was. But apparently it isn't! I believe the name of the transformation is the Weyl transformation. Alternatives to GR have been proposed which respect this symmetry, but they usually predict the exact same dynamics in nearly all cases.

This stuff gets seriously technical in any event, and I'm not sure it's yet been explained in an approachable way.

• • Delta2, atehundel, AlfSalte and 1 other person
The short answer is that the expansion of space is measured with respect to time.
I suppose you mean the expansion red shift.
We are measuring distant objects and their 'slowing down in time'. But it is never mentioned that way, as it is commonly for velocity and gravitational time dilation.

Time is not affected by the expansion of space-time. Time LOCALLY (technically, "proper time") always ticks away at one second per second.

Perhaps you are confused by "time dilation" which is a co-ordinate dependent measure.

But LOCALLY there is also no expansion. The expansion happens in the large scale (cosmic scale). So the question remains how time expands or contracts or what on the cosmic scale.

How would you measure such a thing? How do you tell if one second last year is the same as one second now? Understanding something oten begins with figuring out how to measure it.
Yes, that is a key I guess - how would you measure such a thing. For space we can measure the expansion through some means because we can measure distant galaxies, but we cannot measure the time for those distant galaxies - or can we? Would there be some event that takes a known time so that we can measure how time is moving in distant galaxies?

How would you measure such a thing? How do you tell if one second last year is the same as one second now? Understanding something oten begins with figuring out how to measure it.
Well you might look at a local clock and compare it to very distant ones. If you used the hydrogen spectral lines, rather than the current spectral transition that defines IS unit for the second for practical reasons, you might conclude time was speeding up!

However, the cosmological red shift is not interpreted in this was but as a change in the scale factor a(t).

Regards Andrew

I would assume that the expansion of time in General relativity is in some sense similar to the angle between space and time. It is kind of there, but you can't measure it and you can just set it to some fixed value.

• • Delta2, weirdoguy and PeroK
Well you might look at a local clock and compare it to very distant ones.

How? And how do you tell it's running slower and not just redshifted?

I would assume that the expansion of time in General relativity is in some sense similar to the angle between space and time. It is kind of there, but you can't measure it and you can just set it to some fixed value.

What "angle between space and time" are you talking about? Do you have a reference?

What "angle between space and time" are you talking about? Do you have a reference?

In General Relativity you have a metric tensor g like the Minkowski or the Schwarzschild metric. Such a metric has coordinates like for example x, y, z and t. From the metric you can for example calculate the local angle between the x and the y direction. Basically you go a step in dx and a step in dy and then you look at the triangle. That's the angle a local observer can measure. In most choices for a coordinate system this will be 90° but it doesn't have to be. In the same way you can also calculate the angle between a space direction and time. This is ultimately an arbitrary number that cannot be measured, and purely depends on your choice of coordinates.

In the same way you can also calculate the angle between a space direction and time.

This, as you note, is an angle between coordinate directions. The "time" coordinate direction is not the same thing as a "time" that is actually observed.

This is ultimately an arbitrary number that cannot be measured, and purely depends on your choice of coordinates.

Yes, which means it has no physical meaning and is therefore not what the discussion in this thread is about. The "space expansion" referred to in the OP is not dependent on a coordinate choice; it corresponds to an invariant, namely the expansion scalar of the congruence of "comoving" worldlines (the worldlines of observers who always see the universe as homogeneous and isotropic). So the OP's question is whether this has any effect on "time", and if so, what.

when space is expanding, what happens to time?

The simplest answer would be "nothing". The "space expansion" corresponds to the expansion scalar of the congruence of "comoving" worldlines, i.e., the worldlines of observers who always see the universe as homogeneous and isotropic. The corresponding quantity for "time" would be the proper time along comoving worldlines, and the expansion has no effect on that at all.

How? And how do you tell it's running slower and not just redshifted?
It a matter of interpretation if you take the shift to be due to a change in the rate of time or a change of scale factor I.e. theory dependant. How would you show it's due to expansion?

How, would you experimental tell them apart?

Regards Andrew

How, would you experimental tell them apart?

You are the one who proposed a claimed experimental procedure:

you might look at a local clock and compare it to very distant ones

The answer is that you can't. There is no way to directly compare a local clock with a distant one.

It a matter of interpretation if you take the shift to be due to a change in the rate of time or a change of scale factor I.e. theory dependant.

Exactly. Which means that, as above, you can't directly compare a local clock with a distant clock, as you were proposing to do. That is what @Vanadium 50 was pointing out to you. His question was rhetorical.

• Motore
@PeterDonis given you can't compare anything local directly with anything distant you use a theory to deduce relationships and see if the are consistent.

One such theory would be the distant clock run slow another is that the scale factor changes. You can't measure either directly. What you can measure is the red shift.

I am well aware which is the current adopted model. But humour me do you or anyone know if you could tell them apart experimentally?

Regards Andrew

you can't compare anything local directly with anything distant

Yes.

do you or anyone know if you could tell them apart experimentally?

How could you given the statement quoted at the start of this post?

How could you given the statement quoted at the start of this post?
So since they are not experimentally distingushable, a perfectly acceptable interpretation of the cosmological red shift is that time ran slower in the past than now, as opposed to the current interpretation that it is due to a change in the scale factor.

If not why not.

Regards Andrew

a perfectly acceptable interpretation of the cosmological red shift is that time ran slower in the past than now,
"Time ran slower in the past" is fairly meaningless, given the relativity of simultaneity to start with.

"Time ran slower in the past" is fairly meaningless, given the relativity of simultaneity to start with.
In what way does the relativity of simultaneity come into it? There is no need to synchronise them just compare rates which is done in SR all the time.

For example we compare the rates of pulsars all the time with Earth based clocks without trying to synchronise them.

In addition is it anymore meaningless than saying the rate of a clocks vary with relative velocity?

I know it is not the current accepted interpretation but that does not mean it is meaningless.

Regards Andrew

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In what way does the relativity of simultaneity come into it? There is no need to synchronise them just compare rates which is done in SR all the time.

For example we compare the rates of pulsars all the time with Earth based clocks without trying to synchronise them.

In addition is it anymore meaningless than saying the rate of a clocks vary with relative velocity?

I know it is not the current accepted interpretation but that does not mean it is meaningless.

Regards Andrew
If you say "time ran slower yesterday". How are you defining "yesterday"? If you use comoving time that makes that a special reference frame in which the laws of physics hold, but not in other frames.

The expansion of the universe is a physical phenomenon: it says something about the spacetime of the universe. Relative time dilation is just a coordinate effect.

If you are saying that things like the Caesium clock slow down as they age, then two physical processes would only synchronise if their components had experienced the same proper time since the Big Bang. That would makes sense as an idea, but there's no evidence for it.

In any case, saying "time ran slower yesterday" or "time ran slower in the past" is meaningless.

I think the problem with "time ran slower in the past" is that one could trap a light beam in a small perfectly reflecting box (i.e. one whose walls are bound together somehow) for a couple of billion years and then release it. It would not be redshifted, so it isn't "the past" where time runs slowly.

• D.S.Beyer
a perfectly acceptable interpretation of the cosmological red shift is that time ran slower in the past than now, as opposed to the current interpretation that it is due to a change in the scale factor.

That depends on what you mean by "time ran slower in the past than now". Are you proposing a different mathematical model? Or just a change in coordinates for the same mathematical model?

I have been assuming thus far that you meant the latter, in which case, as has already been said, there is no experimental way of distinguishing the two "interpretations" you describe, so talking about either of them being "right" or "wrong" is meaningless. We don't use "interpretations" to make predictions in any case. We use the mathematical model.

It is possible, however, that you are thinking of some different mathematical model in which there is no expansion at all. If so, you need to provide a reference for such a model. Just waving your hands and saying "time ran slower in the past than now" instead of "change in the scale factor" is not sufficient.

Ok, in my world different interpretations of the observations are different mathematical models. (Discussions of the interpretation of a specific mathematical model is only allowed in the specific QM forum.)

What I intended to asked was could we experimentally differentiate between a mathematical model in which the cosmological red shift was taken to be due to a change in the length of the second ( i.e. the frequency of the relevant transition changed with time) to the one we currently use involving a changing scale factor.

I realize historically Hubble and co assumed it was due to change in space rather than time given the interpretation of red a blue shifts as due to "relative" velocities leading to the current model.

I am not clear why a change in the length of the second is a priori more problematic than one the spatial scale factor . That is what I am seeking to understand.

I am not trying to advocate such a cosmology (it would violate the PF rules).

Regards Andrew

I think the problem with "time ran slower in the past" is that one could trap a light beam in a small perfectly reflecting box (i.e. one whose walls are bound together somehow) for a couple of billion years and then release it. It would not be redshifted, so it isn't "the past" where time runs slowly.
How can you possibly know this without assuming the answer. If the length of the second was slower when trapped than when released it would appear to be red shifted. Regards Andrew

in my world different interpretations of the observations are different mathematical models.

Then you need to give a reference for the different mathematical model you are referring to with the phrase "time ran slower in the past". You can't just wave your hands and assume that such a model even exists. There might not be one. Can you give a reference to one?

What I intended to asked was could we experimentally differentiate between a mathematical model in which the cosmological red shift was taken to be due to a change in the length of the second ( i.e. the frequency of the relevant transition changed with time) to the one we currently use involving a changing scale factor.

This question is unanswerable unless and until we have a mathematical model of the first kind to do the comparison with. Can you give a reference to one?

I am not clear why a change in the length of the second is a priori more problematic than one the spatial scale factor .

I am not clear what "a change in the length of the second" even means. And unless and until you can show me a mathematical model that has such a property, there is no way to even begin to answer your question.

If you say "time ran slower yesterday". How are you defining "yesterday"? If you use comoving time that makes that a special reference frame in which the laws of physics hold, but not in other frames.

The expansion of the universe is a physical phenomenon: it says something about the spacetime of the universe. Relative time dilation is just a coordinate effect.

If you are saying that things like the Caesium clock slow down as they age, then two physical processes would only synchronise if their components had experienced the same proper time since the Big Bang. That would makes sense as an idea, but there's no evidence for it.

In any case, saying "time ran slower yesterday" or "time ran slower in the past" is meaningless.
And why can't you apply the same complaint about the change in scale factor? Rather than saying the expansion of the universe is a physical phenomenon you would say the change in the clock frequency was.

Regards Andrew

What I intended to asked was could we experimentally differentiate between a mathematical model in which the cosmological red shift was taken to be due to a change in the length of the second ( i.e. the frequency of the relevant transition changed with time) to the one we currently use involving a changing scale factor.
I think that if you intend your model to reflect reality then it isn't internally consistent. Light trapped locally in a box would not be redshifted while light traveling cosmological distances would be, so there's more going on than just "time used to run slowly".

So what you are describing seems to me to be a distinct spacetime (which may or may not be realisable) rather than just FLRW spacetime in funny coordinates.

How can you possibly know this without assuming the answer.

You're getting things backwards. It's not a matter of "assuming the answer". As you have already pointed out, any answer is model-dependent. That means that you need to show us the mathematical model you are using to make claims like the one I will quote below. It is not up to us to explain why we are using the model we are using; so far, it's the only model we have. You need to show us a different model before we can even begin to make any kind of comparison.

If the length of the second was slower when trapped than when released it would appear to be red shifted.

How can you possibly know this without having a mathematical model with which to make the prediction?

And why can't you apply the same complaint about the change in scale factor? Rather than saying the expansion of the universe is a physical phenomenon you would say the change in the clock frequency was.

Regards Andrew
Change in clock frequency with respect to what? That's the problem. The scale factor changes wrt comoving time. That's the simplest way to express the model, but the model describes a spacetime metric that can be transformed into any other coordinate system. And, it comes naturally out of an analysis using GR and the various densities of the universe.

What does a change in clock frequency even mean? The second is defined as so many transitions of the caesium atom. You're left speculating that either the universe has some sort of memory (that requires absolute time); or, that physical objects have some sort of memory (of which there is no evidence - e.g. the debunked "tired light" hypothesis.)

I realize I was asking about a non standard model. I had hoped you might have encountered such a model where changes in the time scale were discussed rather than in the standard scale factor.

Clearly not so thank you for your time. Regards Andrew

How can you possibly know this without assuming the answer. If the length of the second was slower when trapped than when released it would appear to be red shifted. Regards Andrew
In an FLRW universe, what I'm saying is clearly true. So if you have a model that says otherwise, there is a test you can do in principle.

It's worth noting that, since "light trapped in a box" is essentially a light clock, light redshifting inside it due to a change in "the rate of time" is a moderately complex concept. Does the light clock tick faster as the light redshifts? If so, why? Either the clock's length would have to change or the speed of light (or actually the fine structure constant, since dimensional constants changing is physically meaningless) must vary too.

People do research on time dependence of the fine structure constant. You could look into that. It's distinct from cosmological redshift, though, as far as I'm aware.

In an FLRW universe, what I'm saying is clearly true. So if you have a model that says otherwise, there is a test you can do in principle.

I agree.

It's worth noting that, since "light trapped in a box" is essentially a light clock, light redshifting inside it due to a change in "the rate of time" is a moderately complex concept. Does the light clock tick faster as the light redshifts? If so, why? Either the clock's length would have to change or the speed of light (or actually the fine structure constant, since dimensional constants changing is physically meaningless) must vary too.

You misunderstood me but I don't think it is worth pursuing further. Thanks for your comments. Regards Andrew