# 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

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.

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