hubble law&galaxies w speeds faster than light

Alain De Vos

As speed of galaxies is proportional to distance.
Can we assume some galaxies have speeds grater than c?
And do they have a negative time with a reference frame bound to earth?
And the light they emit does it have red shift below cosmic background radiation?
...

zhermes

There is a distance at which the relative velocity becomes greater than the speed of light, yes. This is what defines the cosmological event horizon---we cannot see beyond that point (where the relative velocity reaches c).

They don't have a time with respect to the earth, because they are outside of our event horizon.
As objects approach the cosmological event horizon, their redshift approaches infinity. The redshift beyond the horizon is undefined.

Chalnoth

Sort of. Basically, in General Relativity, relative speed only has definitive meaning at a single point. Depending upon the coordinates I use, I can define the relative speed between us and faraway galaxies as being zero or greater than the speed of light.

But, by what is perhaps the most intuitive coordinate system, yes, faraway galaxies are definitely moving faster than light compared to us.

Nope. The furthest galaxies that we can observe are at around a redshift of 10 or so. At a redshift of 10, we observe time moving slower at that galaxy by a factor of 11 relative to our own clocks. And yet, by inferring the recession velocity by comparing the Hubble expansion rate and the distance to this object, we get a velocity that always has been and always will be receding faster than light.

When the CMB was emitted, no galaxies had yet formed. So nothing can have a redshift below the CMB.

Chalnoth

This isn't actually true. The majority of observable galaxies always have had and always will have recession velocities greater than the speed of light (I forget the exact crossover point, but it's somewhere between a redshift of 1 and 2). The reason why they aren't outside of our cosmological horizon is that for much of the history of our universe, the expansion was slowing down. With the larger expansion that existed early-on, a light wave far enough away would have started in our direction, but would have actually lost ground due to the intervening expansion between us and that light wave.

However, as the expansion slowed, this slowing allowed the light wave to start gaining ground again, eventually reaching us. But by the time this happened, the light wave may have departed its original source long before, so the fact that the original galaxy is still receding at faster than light has no bearing: the space where the light wave is now is no longer receding at faster than light, and so it manages to get to us.

It is only if you have a constant Hubble expansion rate that the cosmological horizon equals the distance at which objects recede at the speed of light.

Lost in Space

So similarly we're moving faster than the speed of light relative to them as well?

Chalnoth

With this sort of coordinate system, yes, absolutely.

That should drive home just how unphysical that sort of recession velocity is. In General Relativity, you can define the velocities of faraway objects however you want.

Lost in Space

Interesting. What would that say about the relative time between those objects? If objects travel faster than the speed of light don't they go backwards in time? Are those objects travelling backwards in time relative to us?

Chalnoth

Well, again, that depends upon how you measure things. If you simply consider the FRW coordinates from which this idea of recession velocity is defined, there is no difference in relative time whatsoever.

Alternatively, you could instead ask how much slower time appears to pass in the image of the object we observe. In that instance, your answer is simply the cosmological redshift (plus an additional correction from the object's motion relative to the average). So if the object is at a redshift of z=10, then it will have a time dilation factor of 11: its image will appear to evolve at 1/11th the speed as an identical object nearby. With this latter definition, asking about the relative time for objects that are beyond our horizon (for which there is no redshift), the relative time is simply undefined.