# Why don't galaxies move faster then c?

#### Fredrik

Staff Emeritus
Gold Member
We know that photons emitted from galaxies "receding" from us superluminally (at the time of emission) have reached us. Thus it must be possible for a slow spaceship to "catch up" with a galaxy with a high recessional velocity, or for a photon to "catch up" with a superluminally receding galaxy.
This is my understanding too, except I'm still not sure I understand it. I think I do now that I've had to think about the things I'm writing in this post.

However, our math shows that:

$D(t)=a(t)L-ut$, or
$\dot D(t)=\dot a(t)L-u$

Which for constant expansion or accelerating expansion implies we'll never reach the galaxy (even if we're a photon, i.e. u=c and recessional velocity > c).

Here's my guess at resolving the issue:
I think we're failing to take into account the fact that earlier in the universe,
$\ddot a(t)<0$. Take a look at any graph of a(t) versus time, (e.g. on http://www.astro.ucla.edu/~wright/cosmo_03.htm" [Broken], look at the graph with the magenta colored line) and you see that $\ddot a(t)<0$ for most of the universe's history. It's only recently that $\ddot a(t)>0$ (I believe cosmologists are a bit suspicious of the fact we are near the point of inflection of the a(t) vs. t graph). Thus, for photons being emitted from superluminal galaxies, D(t) would initially be increasing, that is, the recessional velocity of our galaxy, $\dot a(t)L$, would initially be greater than c. However, since, $\ddot a(t)<0$, at some later time t,
$\dot D(t)=\dot a(t)L-c<0$.

What do you think?
I think you're right again. For me, the most interesting piece of information on that page is that in the model I've been considering (flat space), we have $a(t)=(t/t_0)^{2/3}$ where $t_0$ is a constant, so

$$\dot D(t)=\dot a(t)L-u=\frac{2L}{3}\Big(\frac{t_0}{t}\Big)^\frac{1}{3}-u$$

The first term goes to zero as t goes to infinity, so even if the distance between the ship and the destination galaxy is increasing at first, it will at some point begin to decrease. This is pretty cool. It means that in the flat model, we can move at a constant velocity of say 1 m/s towards a distant galaxy that's moving away from us at a million times the speed of light, and get there in a finite time!

If we want to explain why we can see galaxies that are moving away from us at speeds >c, we'd have to do these calculations for a more realistic model, but the flat model is good enough to explain the principles.

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#### nwright

I think you are mistaken...

The first term goes to zero as t goes to infinity, so even if the distance between the ship and the destination galaxy is increasing at first, it will at some point begin to decrease. This is pretty cool. It means that in the flat model, we can move at a constant velocity of say 1 m/s towards a distant galaxy that's moving away from us at a million times the speed of light, and get there in a finite time!
I'm sorry, but I don't know where you've got this idea from so I don't know where your errors have arisen from. There are no galaxies moving faster than light, nothing can travel faster than light in any comparison of velocities.

If we want to explain why we can see galaxies that are moving away from us at speeds >c, we'd have to do these calculations for a more realistic model, but the flat model is good enough to explain the principles.
[/QUOTE]

Since no galaxy can be moving away from us at speeds >c we don't need to explain why we can see them. Could you please give a literature example of an observation of a galaxy moving faster than light? I don't think you'll find one!

#### Fredrik

Staff Emeritus
I think I have explained pretty well why superluminal "speeds" are possible, and why they don't contradict GR. It's actually pretty simple (compared to other problems in GR). If the metric is e.g. $ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2)$ (a flat FLRW spacetime), the distance L(t) to another galaxy at time t is just $L(t)=a(t)L$ where L is the distance at the time when a(t)=1. The galaxy is "moving away from us" faster than the speed of light if L'(t)>1. (I expressed the metric in units such that c=1). We can make $L'(t)>1$ simply by choosing to consider a galaxy with a large enough L. ($L>1/a'(t)$).