# Time dilation due to space expansion

As we observe a remote galaxy, we see it with a redshift. The most distant galaxy discovered to date is GN-z11 visible with the redshift of $z=11.09$. For simplicity, let's assume no gravitational redshift.

In Special Relativity, the Doppler effect has two components, the Doppler component $1+\beta$ and the time dilation component, which is simply $\gamma$. The combined relativistic effect is $z+1=(1+\beta)\gamma$.

In case of the expanding universe, the Doppler effect would seem to have similar components, the Doppler component due to the galaxies recession speed and the time dilation component due to the space expansion. Some argue that there is no time dilation in this case, based on the grounds of comoving time. However, this argument holds neither logically, because the relative observed time is different from the cosmological time, nor practically, because without the time dilation component the maximum observed redshit would be $z=1$ for $\beta\approx 1$ near the particle horizon.

Could someone please clarify if there is a relative time dilation in the expanding universe? Do we observe time of remote galaxies moving slower? Otherwise, if there is no such a time dilation, then what additional factors make the Doppler effect redshift so significant for distant objects?

• Here is a derivation of the cosmological redshift (redshift due to cosmological expansion of space): en.wikipedia.org/wiki/Redshift#Expansion_of_space – Photon Jan 1 '18 at 9:09
• @Photon This is no doubt very helpful, thank you! However, the derivation there is not completely convincing. "The subsequent crest is again emitted from $r = R$", - not really, especially if the expansion speed is faster than the speed of light. Wouldn't it be more like $r=R+\beta\lambda\,$? where $\beta$ is the expansion speed in light speed units (with possibly $\beta\gt 1$)? – safesphere Jan 2 '18 at 1:33
• I don't understand, where should the additive term $\beta\lambda$ come from? – Photon Jan 2 '18 at 7:26
• @Photon Thanks for the reference. And I see now what you mean. The space increase if all attributed to the scale factor while the comoving distance is presumed constant. Got it, thanks! – safesphere Jan 2 '18 at 9:10
• "Could someone please clarify if there is a relative time dilation in the expanding universe?" For what it's worth: A cosmological time-dilation factor of (1 + z) is applied to the type Ia supernova data in The Supernova Cosmology Project (Perlmutter, et al.). The following paper from that project (arxiv.org/abs/astro-ph/9602124) claims to show "the first clear observation of cosmological time dilation for macroscopic objects". – D. Halsey Oct 4 '18 at 22:09

## 4 Answers

Here is a derivation of the cosmological redshift (redshift due to cosmological expansion of space): https://en.wikipedia.org/wiki/Redshift#Expansion_of_space

All the $\beta$ and $\gamma$ terms are coming from Lorentz transformations between inertial frames in the framework of Special Relativity. However, an expanding universe cannot be described by Special Relativity, you need the tools of General Relativity, more specifically, one of GR's solutions used in cosmology, the FLRW metric. If you have no knowledge of GR, I'd recommend you the book "Physical foundations of cosmology" by V. Mukhanov.

the Doppler component $$1+\beta$$ and the time dilation component [...] $$\gamma$$

That decomposition really makes no sense from a special or general relativistic perspective. The only reason you might write the Doppler shift that way is if you're trying to make a connection with nonrelativistic Doppler shift.

A manifestly covariant formula for relativistic redshift is $$1{+}z = \frac{p_\text{light} \cdot v_\text{detector}}{p_\text{light} \cdot v_\text{emitter}}$$

where the $$v$$s are four-velocities and the $$p$$ is the four-momentum of the light, or any nonzero scalar multiple of it (i.e., any four-vector pointing along the path of the light).

This formula works in general relativity too, in all cases, if you parallel transport the vectors to a common location along the path of the light before taking the dot products. All light frequency shifts in general relativity are aspects of the same underlying phenomenon. The decompositions into "cosmological redshift", "gravitational redshift" and so on are human inventions.

Some argue that there is no time dilation in this case

What John Rennie is saying there is just that the situation is symmetrical, like two people in Minkowski space who are moving apart. They see each other redshifted forever, but they're not mutually aging more slowly than each other. In some sense, they are aging "at the same rate" if you consider the symmetry. In another sense there's no way to compare the rates. The "universal time coordinate" (FLRW time) that he mentions is the proper time of the receding people, or the proper time of clocks moving with the Hubble flow.

Doppler component due to the galaxies recession speed and the time dilation component due to the space expansion

There is a redshift from the recession speed of the galaxies, and there's a redshift from the expansion of space, but they don't combine with each other, they're just equal to each other, because "recession speed" and "expansion of space" are different names for the same thing.

In the special case of linear expansion, $$a(τ) = τ/τ_0$$, spacetime is flat and you can actually put a global Minkowski coordinate system on it. The Minkowski coordinates $$(x,t)$$ are related to the FLRW coordinates $$(χ,τ)$$ by $$\begin{eqnarray} t &=& τ\,\cosh\,(χ/τ_0) \\ x &=& τ\,\sinh\,(χ/τ_0) \end{eqnarray}$$

and the cosmological recession is just SR relative motion and the cosmological redshift is SR redshift given by the SR formula. (As I said, you can always use the SR formula if you're willing to parallel transport the vectors, but in this case you don't have to transport them if you use Minkowski coordinates.)

You shouldn't attach too much physical significance to this coordinate system (if any at all), but it illustrates that there's no real difference between redshifts that are usually attributed to different physical mechanisms.

General relativity doesn't have a general definition of gravitational time dilation that applies to all spacetimes. This only works in a static spacetime. In a static spacetime, the metric is derivable from a potential $\Phi$, and a gravitational time dilation is of the form $e^{\Delta\Phi}$. Cosmological spacetimes aren't static.

A similar example is a Schwarzschild black hole, which is static outside the event horizon but not inside it. This is why we can't define a gravitational time dilation between a location inside the event horizon and a location outside it.

There is no reason to expect cosmological Doppler shifts to be analyzable into factors like the ones you used for your argument for the longitudinal Doppler shifts in SR. Actually GR doesn't have a way to define the relative velocities of distant objects, so there would be no way to define a $\beta$. When people talk about cosmological expansion in terms of the velocities of distant objects relative to us, that's just a popularized explanation.

Even in the case of SR, I don't think your derivation of the Doppler shift really works. The SR Doppler shift isn't a nonrelativistic Doppler shift multiplied by a correction factor. If it were, then we would have something like $[(1+\beta_o)/(1+\beta_s)]\gamma$, where $\beta_o$ is the velocity of the observer relative to the medium and $\beta_s$ is the source relative to the medium. But this is not in fact the form of the relativistic Doppler shift, in which there is only one velocity (of o relative to s).

• Thank you for the answer. The relativistic Doppler effect formula and derivation in my question are straight from Wiki: en.m.wikipedia.org/wiki/… - It was just an example anyway, not very important. My question is, what defines the actual observed Doppler effect of distant galaxies? In other words, what is the formula for the redshift in the expanding universe assuming the simplest case of a constant speed expansion with no acceleration and no local gravitational time dilation? Either flat or positively curved space. Thanks! +1 – safesphere Dec 26 '17 at 4:22
• what is the formula for the redshift in the expanding universe assuming the simplest case of a constant speed expansion with no acceleration and no local gravitational time dilation? You might want to ask this as a separate question. If you want the expansion to be at a constant rate, then you need a universe with no matter in it, and the spacetime is flat. This cosmology is just Minkowski space described in funny coordinates. – user4552 Dec 26 '17 at 14:57
• It's not a separate question, it's my original question that I am trying to phrase differently to make it as simple as possible to get a specific answer. The acceleration was discovered only recently, ignore it or not, I don't care. No local gravitational time dilation means that light does not come from a neutron star. All I want is the formula. Can you provide it? – safesphere Dec 26 '17 at 16:44
• @safesphere: The attitude shown in your comment doesn't motivate me to spend any more time trying to help you, but I'll correct your misconception for the benefit of other people who might come across this comment thread. The acceleration was discovered only recently, No, the acceleration was previously believed to be negative, but is now known to be positive. – user4552 Dec 26 '17 at 23:52
• Perhaps you could add some details to your answer now? Thanks! – safesphere Jan 2 '18 at 1:35

Great Question..

Because objects move slower through time as they move faster through space; Earth Time Dilation from its entropy (velocity) is approximately .9999979972 seconds per Second. That is if one stacks the orbital speeds of Hubble, Earth around sun, Sun through the Galaxy, and the Milky Way through the universe (approximately 600 Million mps).

That seems like a very small number but it's very large when one considers the affect over a distance from 30 to 300 megaparsecs. Put another way... the photons we collect with the Hubble Telescope live 100 Million Years (earth years that is) before being observed in the form of light from candles at approximately 30 parsecs away(Quadrillions of seconds). So Looking at .99999979972 seconds per second factors out to a huge amount of time (approximately 221 years or a 7 billion second differential).

While we observe candles moving forward through space very fast.. they are also moving forward through time (As seen from Earth); and the closer they are to earth.. The smaller the impact of Entropic Time Dilation.

Put another way; from a Cosmological Perspective, we are not looking at present time. We are always looking at the past; and when we see a candle move further away than we think it should, we are looking into that object's past and taking measurements of it movement forward through both time and space.. not merely space alone.

"If you eliminate the impossible, whatever remains, however improbable, must be the solution"

• Thanks for the answer. How do you account for the fact that distant galaxies move away from us three times faster than light? – safesphere Aug 16 '20 at 1:44
• By the time we observe light from an object at 30 megaparsecs the candle is already 220 years further into its future the moment we see it (Earthtime). An object 300 megaparsecs away takes 1000 million years to arrive. By the time we see photons from a candle 300 Mpcs away it moved 2200 years into its future. It did not move through space alone. It moves forward through time. Our time however remains 2,200 earth years behind (It's Time). Relatively speaking.. Nothing travels faster than the speed of light. – Richard Bradford Aug 16 '20 at 17:07
• When something is.. IS much more important than where something is. – Richard Bradford Aug 16 '20 at 17:11
• Hi Safesphere, Again thank you for your question. I am recalculating the data I shared with you yesterday in hopes of giving you more precise answer. The general answer remains however: - When we arrive at the conclusion that an object at 300+ megaparsecs moved further through space ; the object does so because it has larger amount of time to get there. It's not because it traveled a further distance. It is traveling for a longer amount of time at its known speed through space. If you have a specific "distant galaxy" you are referring to.. I will be more than happy clarify. – Richard Bradford Aug 17 '20 at 19:42