Length of ruler in expanding Universe

[johne1618]

Imagine a standard ruler (made of atoms) at the present epoch.

Assume its comoving length is $dx=x_1 - x_2$ where $x_1$ and $x_2$ are the comoving coordinates of its ends at the present time.

As the scale factor $a=1$ then its proper length $ds=a \ dx$ is equal to its comoving length $dx$.

Now imagine that ruler persists to a later epoch with $a=2$.

As the ruler doesn't expand with the Universe is it correct to say that its proper length $ds$ is still equal to its comoving length $dx$ even though the space around it has expanded by a factor of two?

 

[Naty1]

An expanding universe depends on homogeneous and isotropic characteristics assumed in our cosmological model. The model doesn't apply at even at galactic size distances.

So yes to your question.

Some research papers discussed in these forums argue that the change in distance is infinitesimally small, much too small to be detected; but those are simplified models with different assumption and don't seem to match those of the cosmological model.

A practicing cosmology Wallace had this to say in a post:
thread # 162727

The FRW metric is the inevitable result of the cosmological principle, CP. which is that the universe is homogeneous and isotropic. The metric is only valid if these principles hold. Consider now a galaxy, solar system or planet. Does the CP hold? No. Is it a remotely useful approximation? Not at all! Unsurprisingly then the dynamics of bodies in these systems and on these scales bears no resemblance to the dynamics of galaxies. So for instance, there is no redshift of light due to a(t) when we observe light from the other side of our galaxy, or from say Andromeda. The FRW metric simply is not valid on these scales.

….. The better way to look at it is that the presence of the mass in the galaxy gives the metric of space-time around this mass a form that would look much more like a Schwarzschild metric than FRW (though we cannot fully solve GR for a galaxy.). The point is though that there is not expansion to ‘overcome’ since the ‘expansion’ is merely the result of the metric [variable over time] formed by a homogeneous and isotropic mass distribution. If the mass doesn’t obey these principles we shouldn’t be surprised that we don’t see any ‘expansion’.

Some may dissent from this view...we'll have to wait and see.

 

[Mordred]

In regards to Naty's reply. The cosmological constant has an energy density of 6.0^-10 joules per m³.
The strong nuclear force of the ruler easily has a higher energy density so can easily overpower expansion energy, in the same manner that gravity does with gravitationally bound objects.

This is often a confusing point, DE exists everywhere in a homogeneous and isotropic distribution, however the amount of expansion in a given region is a sum of energy density or pressure.

 

[johne1618]

An expanding universe depends on homogeneous and isotropic characteristics assumed in our cosmological model. The model doesn't apply at even at galactic size distances.

Does the FRW model apply for an astronaut floating in space far away from any planets, stars or galaxies?

If it does is the proper length of his ruler always equal to its comoving length?

 

[Naty1]

Does the FRW model apply for an astronaut floating in space far away from any planets, stars or galaxies?

That's the only place it really does apply...read Wallace's post which I quoted above.

 

[johne1618]

That's the only place it really does apply...read Wallace's post which I quoted above.

Then is it correct to say that the proper length of the astronaut's ruler is always equal to its comoving length?

 

[Chalnoth]

Then is it correct to say that the proper length of the astronaut's ruler is always equal to its comoving length?

It's more correct to say that applying a comoving length to something that isn't expanding doesn't make any sense. It's just plain "length". Comoving length is for things that expand with the universe, such as the typical separation between galaxies that is used for the Baryon Acoustic Oscillation measurement (also a standard ruler measurement, though this one is a comoving ruler).

 

[johne1618]

It's more correct to say that applying a comoving length to something that isn't expanding doesn't make any sense. It's just plain "length". Comoving length is for things that expand with the universe, such as the typical separation between galaxies that is used for the Baryon Acoustic Oscillation measurement (also a standard ruler measurement, though this one is a comoving ruler).

The reason for my question is that I want to investigate what happens if one uses a rigid ruler to define an interval of time (one's "second").

I want to define a unit of time $d\tau$ as the time it takes light to travel along a rigid ruler of proper length $ds = dx$. If we assume $c=1$ we have
$$d\tau = ds = dx$$
In an expanding metric light travels on a null geodesic given by
$$\frac{a dx}{dt} = 1$$
If I substitute $d\tau$ for $dx$ in the above equation I get
$$d\tau = \frac{dt}{a}$$
Thus if the astronaut uses a rigid ruler to define his time scale, which seems a reasonable thing to do, then his time will speed up relative to cosmological time as the Universe expands. His "second" will get shorter and shorter compared to ours now. This seems an interesting and surprising result to me.

 

[timmdeeg]

Thus if the astronaut uses a rigid ruler to define his time scale, which seems a reasonable thing to do, then his time will speed up relative to cosmological time as the Universe expands. His "second" will get shorter and shorter compared to ours now. This seems an interesting and surprising result to me.

I don't think that the astronaut defines his time, he rather collects information regarding the expansion of the universe.

 

[johne1618]

I don't think that the astronaut defines his time, he rather collects information regarding the expansion of the universe.

Well I think measuring time by timing light along a fixed length is a reasonable model of a clock.

The concept of a light clock has been extensively used in explanations of time dilation due to special relativity.

This is just another use of the concept but this time in a cosmological context.

 

[Naty1]

John, you have progressed 'above my paygrade'...!

There are too many subtleties for me to be reasonably certain about a reply...

[My novice reaction is that ds =dx likely isn't the FLRW metric required in cosmology for
OUR universe...]

This discussion might enable you to draw some conclusions about your approach: https://www.physicsforums.com/showthread.php?t=606843

 

[Naty1]

By chance I stumbled across some prior posts which may apply...they reflect my concern about 'ds'...

The usual notion of distance in cosmology (“proper distance” measured at an instantaneous fixed and uniform cosmological time) defined in this manner (measuring the distance along a curve of constant cosmological time) does not actually measure the distance along a straight line (or the equivalent of a straight line in a curved space-time, a space-like geodesic). This is the convention used in Hubble Distance where v =HD.

The so called ‘physical’ distance in cosmology doesn’t have the status of invariance (independence of coordinate systems) like the line element ds^2 because the ‘physical’ distance is a coordinate quantity.

Hope this helps..it's funny that some questions elicit pages of expert discussion, others so little.

 

[Chalnoth]

The reason for my question is that I want to investigate what happens if one uses a rigid ruler to define an interval of time (one's "second").

I want to define a unit of time $d\tau$ as the time it takes light to travel along a rigid ruler of proper length $ds = dx$. If we assume $c=1$ we have
$$d\tau = ds = dx$$
In an expanding metric light travels on a null geodesic given by
$$\frac{a dx}{dt} = 1$$
If I substitute $d\tau$ for $dx$ in the above equation I get
$$d\tau = \frac{dt}{a}$$
Thus if the astronaut uses a rigid ruler to define his time scale, which seems a reasonable thing to do, then his time will speed up relative to cosmological time as the Universe expands. His "second" will get shorter and shorter compared to ours now. This seems an interesting and surprising result to me.

Sorry for the lack of response, been busy.

The first thing I'd point out is that this can't be correct, because it would imply that the speed of light changes over time in proportion to the scale factor. And we know that isn't the case.

 

[johne1618]

Sorry for the lack of response, been busy.

The first thing I'd point out is that this can't be correct, because it would imply that the speed of light changes over time in proportion to the scale factor. And we know that isn't the case.

But I would say that I am assuming that the speed of light is constant.

It is our definition of time that must change inversely with the scale factor in order to maintain the constancy of the speed of light.

 

[johne1618]

By chance I stumbled across some prior posts which may apply...they reflect my concern about 'ds'...
Hope this helps..it's funny that some questions elicit pages of expert discussion, others so little.

I guess I'm saying that if we take a co-ordinate system such that the spatial interval is $dx$ then we should take the time interval as $d\tau=dt/a(t)$ to ensure that all future observers, using that co-ordinate system, measure the speed of light to be constant.
$$\frac{dx}{d\tau}=\frac{dx}{dt/a(t)}=\frac{a(t)\ dx}{dt}=c$$

Actually, contrary to what I have stated in earlier posts, I now recognize that what I'm describing is a cosmological time dilation effect analogous to the time dilation effects in special and general relativity. From our perspective, at the present time, clocks in the future will tick slower and slower as the Universe expands.

 

[timmdeeg]

Actually, contrary to what I have stated in earlier posts, I now recognize that what I'm describing is a cosmological time dilation effect analogous to the time dilation effects in special and general relativity.

Well, and the expansion of the universe is the cause of cosmological time dilation. This simply what the astronaut measures, nothing else. And this is the difference to flat spacetime, there you are right.

 

[Naty1]

John, your a[dx]/t = 1 seems to set distance equal to time...the units seem wrong...


Check out how Leonard Susskind uses a[t] in this youtube lecture...

He says the 'x' designations don't change, a[t] changes...and is a function of time...

This is because the 'x's designate comoving points...points that expand according to the scale factor. They are not the usual axis coordinates.



Try the first 15 or 20 minutes...

Post any conclusions you reach...

 

[Naty1]

Well, and the expansion of the universe is the cause of cosmological time dilation.

What do you mean by this?

The purpose of co moving observers is to set some standard of time that does NOT change. these are the preferred observers with dipole symmetry in the CMB….at rest with respect to the CMBR. Such observers agree with one another on the amount of clock time since the Big Bang.

 

[Chalnoth]

But I would say that I am assuming that the speed of light is constant.

It is our definition of time that must change inversely with the scale factor in order to maintain the constancy of the speed of light.

I figured it out. The issue is a misunderstanding of the coordinates. Take the FRW metric, ignoring two spatial dimensions and setting $c=1$:

$$ds^2 = dt^2 - a^2 dx^2$$

Consider, for a moment, what this means: if we have two objects that are at constant $x$ coordinate over time, when the universe expands from $a=1/2$ to $a=1$, the proper distance will have doubled.

So the distance across a rigid rod is not a constant distance in coordinate $x$, but a constant distance in $ax$: for a rigid rod as the universe expands, $a$ increases while the coordinate distance $\Delta x$ decreases, leaving the product $a\Delta x$ unchanged. Use this as your distance measure for a rigid rod, and you'll find that the light time travel distance across the rod does not change with expansion.

 

[timmdeeg]

What do you mean by this?

The purpose of co moving observers is to set some standard of time that does NOT change. these are the preferred observers with dipole symmetry in the CMB….at rest with respect to the CMBR. Such observers agree with one another on the amount of clock time since the Big Bang.

Perhaps I miss, what you want to say.

The cosmological time dilation, johne talked about, depends on a(t). The time the light travels along the rigid ruler yields the relative increase of a during this time.

There is no disagreement with the proper time of comoving observers.

 

[johne1618]

I figured it out. The issue is a misunderstanding of the coordinates. Take the FRW metric, ignoring two spatial dimensions and setting $c=1$:

$$ds^2 = dt^2 - a^2 dx^2$$

Consider, for a moment, what this means: if we have two objects that are at constant $x$ coordinate over time, when the universe expands from $a=1/2$ to $a=1$, the proper distance will have doubled.

So the distance across a rigid rod is not a constant distance in coordinate $x$, but a constant distance in $ax$: for a rigid rod as the universe expands, $a$ increases while the coordinate distance $\Delta x$ decreases, leaving the product $a\Delta x$ unchanged. Use this as your distance measure for a rigid rod, and you'll find that the light time travel distance across the rod does not change with expansion.

Hmm - sounds plausible.

That might well be the answer.

Thanks.

 

[Naty1]

John..if you watch the Susskind video you'll see he explains the complementary expansion distance D around minute 8 of the video: D = a[t] (delta x)

I was not so clever as Chronos as to be able to make the simple translation from Susskind's description to yours...

a fun exercise...good luck...

 

[tiny-tim]

hi johne1618!

As the ruler doesn't expand with the Universe …

i think it does

using the familiar balloon analogy:

the individual galaxies stars or atoms are not fixed to a particular point on the fabric of the expanding balloon … although they are forced to move with the surface, they are free to move along the surface, and their relative positions will depend on their mutual interactions, and the way those interactions decrease with distance

imagine a circle ABCDEF with six springs on it, three very weak ones initially of length 119° (AB CD and EF), and three very strong ones (in between) of length 1° (BC DE and FA) (total 360°)

expand the circle ten times: obviously, the lengths AC CE and EA will always be 120°

but the very strong springs will now be a lot less than 1° …

if we use them as our "rulers", then we measure the long springs as being (almost) ten times as long as before!

so yes, our one-metre rulers are expanding as the universe expands, but much more slowly (because gravity is a "spring" whose strength decreases with distance), and so we do measure an expansion of distances between galaxies

 

[Naty1]

expand the circle ten times:

But that does not happen even at scales as large as galactic scales...

There are several extensive discussions on this issue in the forums.

 

[tiny-tim]

expand the circle ten times

But that does not happen even at scales as large as galactic scales...

ok, expand the circle 1.000000001 times …

the weak springs will still expand more, proportionately, than the strong springs

 

[Naty1]

ok, expand the circle 1.000000001 times …

What I was saying is that there IS no such expansion as far as I have been able to tell.
Wallace's quote in post #2 describes what I believe is the best knowledge so far.

In previous discussions in these forums there were research papers discussed with assumptions different from those in the cosmological principle...and different from realistic local conditions. Under different assumptions such expansion does appear possible. But in fact as far as those forum discussions concluded, nobody has been able to solve local [say even as big as galactic size] realistic equations in GR reflecting a lumpy situation.

But since nobody knows what causes expansion, I remain open about possibilities. First it was the cosmological constant responsible, then dark energy, now it seems spacetime curvature without dark energy seems to becoming more popular based on posts and papers discussed here.

The thread referenced by # in post #2 is one of my favorite discussions...check it out if interested. Contrary views are posted.

 

[johne1618]

I figured it out. The issue is a misunderstanding of the coordinates. Take the FRW metric, ignoring two spatial dimensions and setting $c=1$:

$$ds^2 = dt^2 - a^2 dx^2$$

Consider, for a moment, what this means: if we have two objects that are at constant $x$ coordinate over time, when the universe expands from $a=1/2$ to $a=1$, the proper distance will have doubled.

So the distance across a rigid rod is not a constant distance in coordinate $x$, but a constant distance in $ax$: for a rigid rod as the universe expands, $a$ increases while the coordinate distance $\Delta x$ decreases, leaving the product $a\Delta x$ unchanged. Use this as your distance measure for a rigid rod, and you'll find that the light time travel distance across the rod does not change with expansion.

Actually I still have a query about all this. Here is another way of putting my argument.

Light travels on a null geodesic, ds=0, so that its path obeys the relationship
$$a \ dx = dt$$
So at the present time with $a=1$ light travels 1 light-second in 1 second of cosmological time.

But at a future time with $a=2$ light travels 2 light-seconds in 1 second of cosmological time.

If a future observer is going to measure a constant speed of light then his time scale must change so that light travels the 2 light-seconds in 2 of his seconds.

Therefore the future observer's clock must run twice as fast as our present clock (which by definition runs on cosmological time).

 

[timmdeeg]

Johne, you can't take a as static, not even for a second. The scalefactor a increases as the light travels along the rigid (not expanding) ruler. So, it takes more time to arrive at the ruler's other end compared to the static case. The light "works" against expansion, whereby it nevertheless travels locally with c. Regarding the extreme case of superluminal expansion, the light would recede from the other end, though traveling locally in it's direction.
Whether you do the measurement now or in future doesn't matter. What matters is the change of a(t) during the time the light travels along the ruler.

 

[Naty1]

the units are wrong so you know you have made an incorrect assumption.

 

[johne1618]

the units are wrong so you know you have made an incorrect assumption.

But I'm assuming $c=1$ so space and time have the same dimensions.

I take $a$ to be dimensionless.

 

[johne1618]

Actually I still have a query about all this. Here is another way of putting my argument.

Light travels on a null geodesic, ds=0, so that its path obeys the relationship
$$a \ dx = dt$$
So at the present time with $a=1$ light travels 1 light-second in 1 second of cosmological time.

But at a future time with $a=2$ light travels 2 light-seconds in 1 second of cosmological time.

If a future observer is going to measure a constant speed of light then his time scale must change so that light travels the 2 light-seconds in 2 of his seconds.

Therefore the future observer's clock must run twice as fast as our present clock (which by definition runs on cosmological time).

Actually I now accept my argument is wrong after all for the reason that Chalnoth pointed out!

The null geodesic for light gives
$$a(t) \ dx = c \ dt$$
where I am retaining $c$ for clarity.

$dx$ is an interval of comoving distance and $dt$ is an interval of cosmological time.

Now an interval of proper distance $ds$ at any time $t$ is given by
$$ds=a(t) \ dx.$$
Therefore the above geodesic can be written simply as
$$ds = c \ dt$$
which is true for any cosmological time $t$.

Thus light travels at a constant velocity $c$ for all observers who simply use cosmological time.

To illustrate the situation as I now see it:

For $a=1$
$$dx = c \ dt \\ ds = dx = c \ dt$$
For $a=2$
$$2 \ dx = c \ dt \\ dx = c \ dt / 2 \\ ds = 2 \ dx = 2 \cdot (c \ dt / 2) = c \ dt$$