B Hubble flow and velocity addition

[Stephanus]

Dear PF Forum,
I've been wondering about this thing.
The universe's radius is 46 gly. But a galaxy in 13.8 gly from us cannot be seen because it travels faster than the speed of light.
So is there a galaxy in, say, 30 billions light years away from us?

But before that, I'd like to know about this question below.
According to
https://en.wikipedia.org/wiki/Hubble's_law#Observed_values
Hubble constant is 67.6 km/s per mega parsec as of 13th July 2016 observed by https://en.wikipedia.org/wiki/Sloan_Digital_Sky_Survey
Wow, is it last week?
Now, what I want to know is this.
A. Is it true that a galaxy at $300000/67.6 = 4437.87 mega parsec$ or $4437.87 * 3.26 = 14467$ or 14.467 giga light years away from us travels almost at the speed of light wrt us? Let's call it galaxy A
B. Is it true that a galaxy at ##$150000/67.6 = 2218.94 mega parsec$ or $2218.94 * 3.26 = 7233$ or 7.233 giga light years away from us travels almost at 0.5c wrt us? Let's call it galaxy B
3. Now relativity velocity addition would tell us that.
$v1=\frac{v3+v2}{1+v3.v2}$ or
$v3=\frac{v2-v1}{v1.v2-1}$

Let
v1 = speed of galaxy A = 1 wrt us
v2 = speed of galaxy B = 0.5 wrt us
v3 = speed of galaxy A wrt galaxy B
Now if we put the number into equation.
$v3 = \frac{v2-v1}{v1.v2-1}$
$v3 = \frac{0.5-1}{0.5-1} = 1$
So? The speed of galaxy A wrt B is 1?
So, actually, because length is so contracted, then there's no galaxy at say 20 gly away from us? They are all compacted at around 14.4 giga light years away from us?

Thank you very much.

 

[PeterDonis]

Is it true that a galaxy at $300000/67.6 = 4437.87$ mega parsec or $4437.87 * 3.26 = 14467$ or 14.467 giga light years away from us travels almost at the speed of light wrt us?

Only if we define "speed" in terms of cosmological coordinates--the rate of change of "distance" in cosmological coordinates with respect to "time" in those coordinates. This "speed" is not the same as a relative speed in SR and does not obey the same rules; for example, it is not limited to the speed of light.

relativity velocity addition

Does not apply in this context. The coordinates in which these "velocities" are defined are not SR inertial coordinates, so you can't use the SR velocity addition formula.

because length is so contracted,

This doesn't apply either, for the same reason as above. You are trying to apply SR in a situation where SR is not applicable.

 

[GeorgeDishman]

A. Is it true that a galaxy at 300000/67.6=4437.87megaparsec300000/67.6 = 4437.87 mega parsec or 4437.87∗3.26=144674437.87 * 3.26 = 14467 or 14.467 giga light years away from us travels almost at the speed of light wrt us? Let's call it galaxy A
B. Is it true that a galaxy at 150000/67.6 = 2218.94 mega parsecor or 2218.94 * 3.26 = 7233## or 7.233 giga light years away from us travels almost at 0.5c wrt us? Let's call it galaxy B

You have to be vary careful because the light we see from such distant sources left them a long time ago. The universe has expanded and the Hubble Constant has changed its value since then.

A galaxy we see with a redshift of z=1.597 would have been receding at the speed of light at the time the light was emitted, it is now receding at 1.054c. It would have been 5.85 billion light years from us at that time and is now 15.18 billion light years away (the distance ratio is 1+z ~ 2.6). The light was emitted 9.739 billion years ago.

For z=0.685, the speeds are 0.5c then and 0.573c now, distances are 4.898 billion light years then and 8.253 billion light years now. The light was emitted 6.392 billion years ago.

So? The speed of galaxy A wrt B is 1

You are trying to compare galaxies seen at completely different times so the question is at what time are you trying to calculate their relative separation rate?

So, actually, because length is so contracted, then there's no galaxy at say 20 gly away from us? They are all compacted at around 14.4 giga light years away from us?

The most distant known galaxy, GN-z11, has a redshift of z=11.09, we see it in light that was emitted 13.38 billion years ago, it was receding at 4.32c at that time and was 2.663 billion light years away then. It is now 32.2 billion light years away and is receding at 2.235c.