Is Cosmological Time Dilation real?

[johne1618]

The cosmological redshift can be understood in terms of time dilation.

In an expanding Universe light travels on a null-geodesic (ds=0) so that:
$$dr = \frac{c\ dt}{a(t)},$$
where $dr$ is an element of co-moving distance along its path, $dt$ is an element of time and $a(t)$ is the Universal scaling factor.

Thus if a photon is emitted from a co-moving galaxy it starts out with an element of co-moving distance $dr$ given by
$$dr = \frac{c \ \delta t_{em}}{a(t_{em})}$$

By the time we observe the photon an element of co-moving distance $dr$ is given by
$$dr = \frac{c \ \delta t_{ob}}{a(t_{ob})}$$
If we equate the two expressions for $dr$ we find an expression for an element of time now when we observe the photon, $\delta t_{ob}$, in terms of an element of time when the photon was emitted, $\delta t_{em}$ :
$$\delta t_{ob} = \frac{\delta t_{em}}{a(t_{em})}$$
where I take the current scale factor $a(t_{ob})=1$.

Thus if $a(t_{em})=1/2$ when the photon was emitted in the past then one second at time $t_{em}$ is equivalent to 2 seconds now at time $t_{ob}$.

Therefore my clock now is running twice as fast as the same clock at time $t_{em}$.

This interpretation seems at least as valid as the redshift interpretation. Instead of photons being somehow stretched by expanding space as they travel it seems that the passage of time itself is speeding up. I personally could only imagine photon wavelengths being stretched if one had standing waves in an expanding box.

Furthermore if all atomic frequencies are twice as high now as they were at time $t_{em}$ then surely all energies are twice as high now as at time $t_{em}$?

Thus the redshift that we observe when we observe photons emitted at time $t_{em}$ is due to our energy scale at time $t_{ob}$ being higher than the energy scale at time $t_{em}$.

 

[PAllen]

Generally speaking, there is nothing wrong with your line of reasoning. Redshift (which is invariant) can always be related to time dilation and energy (which are frame variant). Any time red shift is observed, a corresponding clock will be observed to be slow.

Where I would challenge you is that the direct observable here is the red shift. That is as real as anything gets in physics. Frame variant interpretations (time dilation, energy scale, stretching waves) have, at best, pedagogical value for certain purposes.

 

[PeterDonis]

[Edit: corrected some things below, the original statements I made weren't strong enough.]

The cosmological redshift can be understood in terms of time dilation.

I'm not sure I agree.

If we equate the two expressions for $dr$

This seems to me to be a key [STRIKE]missing piece[/STRIKE] error: [STRIKE]what justifies[/STRIKE] equating the two expressions for $dr$ is not justified. Since you are using the standard FRW chart, in which the time coordinate $t$ is the same as proper time for all "comoving" observers, and you are modeling both observers as "comoving" (since they are both at rest in the chart), [STRIKE]surely the obvious assumption[/STRIKE] the correct statement is that $\delta t_{ob} = \delta t_{em}$, and [STRIKE]that[/STRIKE] any change in the photon's observed wavelength is due to a change in $a(t)$ as it multiplies $dr$ -- in other words, it's a change in how much actual proper length corresponds to a given increment of the $r$ coordinate. The FRW metric also bears this out, since the scale factor $a(t)$ multiplies the spatial part of the metric; the time part of the metric is unchanged.

In other words, your statement that $dr$ is the "comoving distance" is not correct; the comoving distance is $a(t) dr$.

Thus the redshift that we observe when we observe photons emitted at time $t_{em}$ is due to our energy scale at time $t_{ob}$ being higher than the energy scale at time $t_{em}$.

I also don't see any justification for this interpretation. First, there is the issue I stated above. Second, there is no global notion of "energy scale" in a non-stationary spacetime, which this is.

 

[PeterDonis]

Where I would challenge you is that the direct observable here is the red shift. That is as real as anything gets in physics. Frame variant interpretations (time dilation, energy scale, stretching waves) have, at best, pedagogical value for certain purposes.

As you can see from what I just posted, I would challenge even more than that; but I agree with what you say here about the redshift being a direct observable vs. frame-variant quantities.

 

[johne1618]

[Edit: corrected some things below, the original statements I made weren't strong enough.]
This seems to me to be a key [STRIKE]missing piece[/STRIKE] error: [STRIKE]what justifies[/STRIKE] equating the two expressions for $dr$ is not justified. Since you are using the standard FRW chart, in which the time coordinate $t$ is the same as proper time for all "comoving" observers, and you are modeling both observers as "comoving" (since they are both at rest in the chart), [STRIKE]surely the obvious assumption[/STRIKE] the correct statement is that $\delta t_{ob} = \delta t_{em}$, and [STRIKE]that[/STRIKE] any change in the photon's observed wavelength is due to a change in $a(t)$ as it multiplies $dr$ -- in other words, it's a change in how much actual proper length corresponds to a given increment of the $r$ coordinate. The FRW metric also bears this out, since the scale factor $a(t)$ multiplies the spatial part of the metric; the time part of the metric is unchanged.

The calculation I use is a short-hand for the standard derivation of cosmological redshift using two integrals to express the same co-moving distance between an emitting atom and an observer at times that are one period apart. i.e.
$$\large r = \int dr = \int^{t_{ob}}_{t_{em}} \frac{cdt}{a(t)} = \int^{t_{ob}+\delta t_{ob}}_{t_{em} + \delta t_{em}} \frac{cdt}{a(t)}$$
This leads to the same expression relating time intervals:
$$\frac{c \delta t_{em}}{a(t_{em})} = \frac{c \delta t_{ob}}{a(t_{ob})}$$
There is no stretching of wavelength due to space expansion in this derivation. The observed redshifted wavelength is determined by the dilated oscillation period.

 

[johne1618]

Generally speaking, there is nothing wrong with your line of reasoning. Redshift (which is invariant) can always be related to time dilation and energy (which are frame variant). Any time red shift is observed, a corresponding clock will be observed to be slow.

Where I would challenge you is that the direct observable here is the red shift. That is as real as anything gets in physics. Frame variant interpretations (time dilation, energy scale, stretching waves) have, at best, pedagogical value for certain purposes.

My interest in the idea that cosmological time dilation implies an increase in energy scale stems from the fact that this might affect the mass/energy density of matter.

Normally one argues that the mass/energy density of matter goes like $1/a^3$ but if energy is increasing with the scale factor then the mass/energy of matter should go like $a/a^3=1/a^2$ instead. A matter dominated Universe would then scale linearly with time rather than the standard $t^{2/3}$ dependence.

 

[PeterDonis]

There is no stretching of wavelength due to space expansion in this derivation.

Yes, there is, because $r$ is not a proper distance; it's a coordinate distance. Moving the $a(t)$ factor to the RHS of the integral doesn't make $r$ a proper distance. And the fact that $a(t)$ changes during the light's travel means that space does expand; the proper distance between the emission and observation points is larger when the light is observed than when it is emitted.

 

[PeterDonis]

Normally one argues that the mass/energy density of matter goes like $1/a^3$

You appear to think that there is some wiggle room in this, but there isn't. The fact that the mass/energy density of matter goes like $1/a^3$ is not "argued" based on an interpretation that could be accepted or rejected; it's read directly off the stress-energy tensor. The SET component $T_{00}$, which is the mass/energy density of matter, goes like $1/a^3$, not $1/a^2$. There's no room for interpretation there.

 

[PeterDonis]

The SET component $T_{00}$, which is the mass/energy density of matter, goes like $1/a^3$, not $1/a^2$.

Strictly speaking, it goes like $1/a^3$ for non-relativistic matter; it goes like $1/a^4$ for highly relativistic matter and radiation; and it is constant for dark energy.

 

[PeterDonis]

the correct statement is that $\delta t_{ob} = \delta t_{em}$

On re-reading, I realized that I was wrong here; if this condition were true there would be no redshift!

There is a good discussion of cosmological redshift in this thread; in particular, this post by George Jones goes through the same derivation, but using hyperspherical coordinates ($\chi$ instead of $r$), which makes it clear that the coordinate interval $\chi$ is not a proper distance.

 

[johne1618]

Apparently cosmological time dilation has been observed.

A supernova that takes 20 days to decay will appear to take 40 days to decay when observed at redshift z=1.

See http://www.astro.ucla.edu/~wright/cosmology_faq.html#TD

This is more than expansion of photon wavelength.

All my physical processes are running twice as fast as those same processes back at the time when the Universe was 1/2 the present size.

 

[jartsa]

Apparently cosmological time dilation has been observed.

A supernova that takes 20 days to decay will appear to take 40 days to decay when observed at redshift z=1.

See http://www.astro.ucla.edu/~wright/cosmology_faq.html#TD

This is more than expansion of photon wavelength.

A supernova that takes 20 days to decay will appear to take 40 days to decay when observed at redshift z=1.

A supernova that takes 20 days to decay will appear to take 40 days to decay when observed at relativistic Doppler redshift z=1, from a window of a spaceship that is escaping at approximate speed 0.5 c, relative to the exploding star.

 

[PeterDonis]

All my physical processes are running twice as fast as those same processes back at the time when the Universe was 1/2 the present size.

By this same logic, your physical processes are running twice as fast as those of an person that is receding from you at 0.866c (the relative velocity at which the time dilation factor is 2).

 

[PeterDonis]

at approximate speed 0.5 c

It's not 0.5c, it's 0.866c; that's the relative speed for which $\gamma = 1 / \sqrt{1 - v^2 / c^2}$ is 2.

 

[PeterDonis]

All my physical processes are running twice as fast as those same processes back at the time when the Universe was 1/2 the present size.

Also, if this were true, there should be evidence that physical processes on Earth now are running twice as fast as physical processes on Earth a billion or so years ago (however long ago light from supernovas at z = 1 was emitted). I'm not aware of any evidence supporting that, and there's quite a bit of evidence against it.

 

[johne1618]

By this same logic, your physical processes are running twice as fast as those of an person that is receding from you at 0.866c (the relative velocity at which the time dilation factor is 2).

In your example the two observers are indeed equivalent.

But in the expanding Universe example the emitter and the observer are not equivalent. The observer is in the emitter's causal future and not the other way round.

 

[PeterDonis]

But in the expanding Universe example the emitter and the observer are not equivalent. The observer is in the emitter's causal future and not the other way round.

That is also true for you when you receive Doppler shifted light signals from someone moving away from you; the event of you receiving the signal is in the causal future of the event of the other person sending it. But you are not in the causal future of the other person "now".

Similarly, we on Earth, receiving light from a supernova with z = 1, are in the causal future of the event when the supernova emitted that light. But we are not in the causal future of the supernova "now".

(The point being that the relationship of being in the causal past or causal future is a relationship between events, not objects or observers.)

Also, we here on Earth now are in the causal future of the Earth a billion years ago (or whenever the light from supernovas at z = 1 was emitted), so my other point, that there should be evidence of physical processes on Earth now running twice as fast as a billion years ago, still stands even if we agree that the causal past/causal future relationship is relevant.

 

[johne1618]

Also, if this were true, there should be evidence that physical processes on Earth now are running twice as fast as physical processes on Earth a billion or so years ago (however long ago light from supernovas at z = 1 was emitted). I'm not aware of any evidence supporting that, and there's quite a bit of evidence against it.

I think the only physical evidence would be the total number of oscillations of a physical system.

 

[jartsa]

It's not 0.5c, it's 0.866c; that's the relative speed for which $\gamma = 1 / \sqrt{1 - v^2 / c^2}$ is 2.

Yes, but the z is 1 when the v is about 0.5 c. There is a 100 % frequency change, although there are almost no relativistic effects.


http://arxiv.org/abs/0808.1081v2
http://arxiv.org/abs/0707.0380v1

Now if we interpret cosmological redshift to be just kinematical redshifts, which is possible according to the two papers above, then no relativistic effects are involved.

 

[PeterDonis]

Yes, but the z is 1 when the v is about 0.5 c.

The redshift formula is

$$
1 + z = \sqrt{\frac{c + v}{c - v}}
$$

where $v$ is the recession velocity. So $1 + z = 2$ when $c + v = 4 ( c - v )$ or $v = 3/5 c$. So it's somewhat larger than 0.5c. But you're correct that the key parameter is the observed redshift, not the time dilation factor; in my previous post I incorrectly used a time dilation factor of 2 to derive $v$, I should have used a redshift $1 + z$ of 2, as I did here.

 

[PeterDonis]

I think the only physical evidence would be the total number of oscillations of a physical system.

Total number of oscillations relative to what? How would you count them?

 

[PeterDonis]

Now if we interpret cosmological redshift to be just kinematical redshifts, which is possible according to the two papers above, then no relativistic effects are involved.

I don't agree with that: the kinematical redshifts have to use the relativistic Doppler formula (as I did in my previous post).

 

[harrylin]

Also, if this were true, there should be evidence that physical processes on Earth now are running twice as fast as physical processes on Earth a billion or so years ago (however long ago light from supernovas at z = 1 was emitted). I'm not aware of any evidence supporting that, and there's quite a bit of evidence against it.

What is that evidence? Indeed, how can one measure that?

 

[johne1618]

Total number of oscillations relative to what? How would you count them?

Maybe one could perform an experiment with an atomic clock.

My claim is that the frequency of atomic systems increase with the Universal scale factor $a$.

Let us suppose we use an atomic clock to count the number of atomic oscillations, $N$, during the time that light takes to travel along a length $L$ of optic fiber.
$$N(a) = \frac{L}{c}a f_0$$
where $f_0$ is the frequency of the atomic clock now.

Let us assume that:
$$f_0 = 10^{16}\ Hz\\ L = 100\ km$$

Thus today with $a=1$ the number of oscillations is:
$$N_0 = \frac{10^5\ m}{10^8\ m/s} \times 10^{16}\ Hz = 10^{13}$$
We repeat the same experiment in a year's time.

The time light takes to travel the fixed proper distance $L$ should be the same.

The change in the scale factor $a$ can be derived from the definition of the Hubble parameter:
$$\frac{da/dt}{a} = H_0 \\ da = H_0\ a\ dt$$
If $H_0=10^{-18}\ sec^{-1}$, $a=1$ and $dt=10^7\ secs$ (1 year) then $da=10^{-11}$.

The change in the number of oscillations $dN$ is given by:
$$dN = \frac{L}{c}da f_0 \\ dN = \frac{10^5\ m}{10^8\ m/s} \times 10^{-11} \times 10^{16}\ Hz = 100$$

This change in the number of atomic oscillations should be detectable.

 

[PeterDonis]

What is that evidence?

All the evidence that, for example, the fine structure constant has not changed significantly in the past few billion years, since the fine structure constant affects the frequency of atomic oscillations.

 

[PeterDonis]

My claim is that the frequency of atomic systems increase with the Universal scale factor a.

This amounts to saying that the fine structure constant must depend linearly on the scale factor, so that it was [STRIKE]twice[/STRIKE] half as large a billion years or so ago (whenever light at redshift z = 1 was emitted). Various experiments have tested this and found, at most, very small variations (one part in 10^5 or less); a recent one is here:

http://arxiv.org/abs/1202.6365

Let us suppose we use an atomic clock to count the number of atomic oscillations, $N$, during the time that light takes to travel along a length $L$ of optic fiber.

The fine structure constant governs both atomic clock oscillations and the speed of light, so if one changes, they should both change. So I don't think this method of measuring what you want to measure will work anyway. But there are other methods of estimating what the fine structure constant was in the past, and they show at most a very small change (see above).

 

[johne1618]

This amounts to saying that the fine structure constant must depend linearly on the scale factor, so that it was [STRIKE]twice[/STRIKE] half as large a billion years or so ago (whenever light at redshift z = 1 was emitted). Various experiments have tested this and found, at most, very small variations (one part in 10^5 or less); a recent one is here:

http://arxiv.org/abs/1202.6365



The fine structure constant governs both atomic clock oscillations and the speed of light, so if one changes, they should both change. So I don't think this method of measuring what you want to measure will work anyway. But there are other methods of estimating what the fine structure constant was in the past, and they show at most a very small change (see above).

But gravitational time dilation doesn't change the fine structure constant.

Imagine that I start off with two identical clocks A and B. I lower clock B into a gravitational potential well, keep it there for a while and then pull it back up. When I compare the clocks I will find that clock A has advanced compared with clock B. The effect is due to gravitational time dilation rather than a change in fundamental constants.

 

[Dale]

But gravitational time dilation doesn't change the fine structure constant.

I agree, but I think that Peter Donis is right that your experiment seems to test changes in the fine structure constant. I.e. I don't think your experiment tests what you really want to test (I don't think there is an experiment to test what you want).

 

[PeterDonis]

Imagine that I start off with two identical clocks A and B. I lower clock B into a gravitational potential well, keep it there for a while and then pull it back up. When I compare the clocks I will find that clock A has advanced compared with clock B. The effect is due to gravitational time dilation rather than a change in fundamental constants.

Yes, but there are at least three key differences between this scenario and the cosmological redshift scenario:

(1) The two clocks A and B are at rest relative to each other (except for the periods when B is being lowered and raised, but the effect of that can be made negligible by lowering and raising B slowly enough). Comoving observers in our expanding universe are not at rest relative to each other.

(2) The two clocks are brought back together and compared, so the difference in their elapsed times is a direct observable. There is no corresponding direct observable in the case of us here on Earth and the faraway object whose light we receive and measure to be redshifted.

(3) The frequency shift in light going from B to A, vs. light going from A to B, is not symmetric; A observes B's light to be redshifted, but B observes A's light to be blueshifted. An observer in a distant galaxy, whose light we observe to be redshifted, would also observe light coming from us to be redshifted.

 

[nitsuj]

The fine structure constant governs both atomic clock oscillations and the speed of light, so if one changes, they should both change. So I don't think this method of measuring what you want to measure will work anyway. But there are other methods of estimating what the fine structure constant was in the past, and they show at most a very small change (see above).

This fine structure constant sounds really neat. Just double checking what wiki says of it, that it is about "namely the coupling constant characterizing the strength of the electromagnetic interaction." and goes on to call it a dimensionless quantity (which is literal).

How is c governed by the fine structure constant in the sense of; If it's "stronger" then is c faster?

Was John asking if we can test to see if time is speeding up?

 

[PeterDonis]

How is c governed by the fine structure constant in the sense of; If it's "stronger" then is c faster?

On our current understanding, dimensionless constants like the fine structure constant are actually the more "fundamental" quantities; dimensionful quantities like $c$ are actually artifacts of our system of units (after all, we can always choose units to make $c = 1$). So it's not so much that increasing the fine structure constant makes the speed of light "faster", as that increasing the fine structure constant, which increases the strength of the electromagnetic interaction, changes the behavior of the things we use to measure how "fast" light travels.

In the case of johne618's proposed experiment (which I'll modify slightly to eliminate any issues with measuring time intervals at spatially separated locations), if we emit a beam of laser light that reflects off a mirror a distance $L$ away and comes back and is detected, and we count the number of atomic clock oscillations between the emission and detection of the beam, if $\alpha$ changes, that changes both the interaction strength that governs the atomic clock oscillations, *and* the interaction strength that governs the measuring tools we used to measure the distance $L$.

Heuristically, increasing $\alpha$ makes the atomic clock oscillations "faster" (electrons are pulled towards the nucleus more strongly, so they have to orbit faster to maintain a stable orbit), and it "shortens" the measuring tools (by strengthening the interactions that hold them together) that are used to determine $L$, and the two effects should cancel each other out, at least to a first approximation, which is why I said I didn't think this experiment would work anyway, because I don't think it would give different results even if the fine structure constant *did* change. (But, as I said, there are other ways to estimate what the fine structure constant was in the past.)

 

[PeterDonis]

I think that Peter Donis is right that your experiment seems to test changes in the fine structure constant.

Actually, I was saying that I don't think his experiment even tests that; I don't think its results would change even if the fine structure constant changed.

 

[nitsuj]

On our current understanding, dimensionless constants like the fine structure constant are actually the more "fundamental" quantities; dimensionful quantities like $c$ are actually artifacts of our system of units (after all, we can always choose units to make $c = 1$). So it's not so much that increasing the fine structure constant makes the speed of light "faster", as that increasing the fine structure constant, which increases the strength of the electromagnetic interaction, changes the behavior of the things we use to measure how "fast" light travels.

In the case of johne618's proposed experiment (which I'll modify slightly to eliminate any issues with measuring time intervals at spatially separated locations), if we emit a beam of laser light that reflects off a mirror a distance $L$ away and comes back and is detected, and we count the number of atomic clock oscillations between the emission and detection of the beam, if $\alpha$ changes, that changes both the interaction strength that governs the atomic clock oscillations, *and* the interaction strength that governs the measuring tools we used to measure the distance $L$.

Heuristically, increasing $\alpha$ makes the atomic clock oscillations "faster" (electrons are pulled towards the nucleus more strongly, so they have to orbit faster to maintain a stable orbit), and it "shortens" the measuring tools (by strengthening the interactions that hold them together) that are used to determine $L$, and the two effects should cancel each other out, at least to a first approximation, which is why I said I didn't think this experiment would work anyway, because I don't think it would give different results even if the fine structure constant *did* change. (But, as I said, there are other ways to estimate what the fine structure constant was in the past.)

Wow flippin' remarkable! Thanks for the clear explanation too.

 

[harrylin]

All the evidence that, for example, the fine structure constant has not changed significantly in the past few billion years, since the fine structure constant affects the frequency of atomic oscillations.

I still don't see how that that makes it detectable; and next you seem to also not see how:

[..]
Heuristically, increasing $\alpha$ makes the atomic clock oscillations "faster" (electrons are pulled towards the nucleus more strongly, so they have to orbit faster to maintain a stable orbit), and it "shortens" the measuring tools (by strengthening the interactions that hold them together) that are used to determine $L$, and the two effects should cancel each other out, at least to a first approximation, [..]
I don't think it would give different results even if the fine structure constant *did* change. (But, as I said, there are other ways to estimate what the fine structure constant was in the past.)

So I ask again: what other ways?

 

[PeterDonis]

So I ask again: what other ways?

Did you read the paper I linked to? It uses details about spectral lines to estimate what the fine structure constant was when the spectra were emitted.

The Usenet Physics FAQ also gives a brief overview with references:

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/constants.html