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I've hear several theories stating that the universe is expanding faster than the speed of light, others claim that the universe expands faster the further away you measure it. Which of this is correct and how do you prove it (mathematically)? Furthermore, does this correctly imply, then, that eventually galaxies will be so far away, and moving so fast, that we will never see them again?

Thank you in advance!

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This was supposed to be more a comment than an answer, but since I can not comment due to reputation lack I will spend few words here. First of all, the "theories" you mentioned are not inconsistent each other.

We know the simple Hubble law:

$v = H D$

where $v$ is the receding velocity of a galaxy, $H$ is the Hubble constant, $D$ is the distance of the considered galaxy. This means that the further is the galaxy you observe, the faster this galaxy is receding. At some point it will become faster than light (or superluminal). At some point, the space between us and the light emitter will grow so fast that the light can never reach us, and this will make those objects invisible. Indeed, all we can observe is by definition our observable universe. This is growing with time, but still some objects will stay invisible forever. The very first thing you mention, I suppose you should put it more correctly, since it should be better to talk about expansion rate of the universe (instead of velocity), and this is given itself by the Hubble constant, around $70 km/s/Mpc$. Take care of the units of this "constant", and you will grasp why this argument is not so intuitive. Please, wait for more experienced people, since this was just a very rough summary of cosmology concepts.

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And relativity does not apply here? I suppose that at some moment, the objects should appear to have the same speed regardless the distance thanks to the relativity concept. – Tomáš Zato Mar 1 '14 at 13:59

Both are correct, although the first can be further explained a bit. I won't give you a mathematical proof, though; Instead I'll play with characters.

Let's assume, for the sake of gedankenexperiment, that:

  • The speed of light is 5 characters/sec;
  • our universe is expanding at 1 character per 5 characters per sec.

This is our current universe, and we launch a photon from body A aiming at body E. (Space generated each second is marked with a # symbol.)

T=0s A----B----C----D----E           Bodies
     *                               Photon     - 19 chars to E
T=1s A--#--B--#--C--#--D--#--E 
      --#--*                                      17 chars to E
T=2s A--#---B-#----C#-----#D----#-E 
      --#-----#-*                                 17 chars to E
T=3s A--#----B#-----#-C---#----D#-----#-E 
      --#-----#-----#*                            18 chars to E
T=4s A--#-----#B----#----C#-----#---D-#-----#--E 
      --#-----#-----#-----#*                      19 chars to E

I know, the graph isn't too granular, and the space generation isn't evenly distribute. I apologize for that, but it's for the sake of demonstration.

Notice that at T=2 some space is already generated between A and the photon. But that's irrelevant: E is sitting at the event horizon, and will never be reached by the photon, because the amount of space being generated between photon * and body E is equal, or superior, to the speed of light.

Given any positive expansion rate, there will be an event horizon - a point where the accumulated dilation of space is more than the amount of space a particle moving at the speed of light can travel.

A galaxy sitting initially at say, 1000 characters from A at T=0, will be at staggering 1200C at T=1 - that's 40 times our speed of light.

At T=16s, B (that was passed by the original photon at T=1) will be sitting exactly where E was relative to A, and at T=17 will fall out of our event horizon. A new photon emitted from A will never reach it.

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I'm being very suspitions about this. It reminds me the most famous Zeno's paradox about Achilles and the tortoise. I'm quite curious what function of distance in time would look like. Sounds like some weird periodical function. – Tomáš Zato Mar 1 '14 at 14:01

The universe expands with about 70 km/s per Mega parsec, due to Hubble's law. This means, that the velocity two objects move away from each other is proportional to their distance. At one Mega parsec it's 70 km/s. That's an average value, which doesn't need to hold for each single object.

By dividing the speed of light of about 300,000 km/s by the Hubble constant, you get, that objects further away than about 4300 Mega parsecs move faster away from each other than the speed of light.

The expansion is measured by the redshift of spectra, meaning absorption and emission lines are shifted. Together with distance estimates, based on several methods, the expansion per distance i.e. the Hubble constant, can be estimated.

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