# Does an object travelling at speeds approaching $c$ observe an increase in the expansion rate of the universe?

According to the theory of special relativity as an object approaches light speed it experiences length contraction. This means that the distance it observes between it and it's destination reduces as it's speed increases.

We also know that the universe is expanding. And that the expansion rate depends on the distance between two objects. The further the distance the faster the space between them is expanding. This means that if an object is far enough away the expansion rate of the universe can exceed it's speed meaning that it can never arrive. Beyond the cosmic event horizon space between us and it is expanding at a rate greater then speed of light so even light can never reach us.

However if an object was to travel towards earth from just beyond the cosmic even horizon at speeds approaching the speed of light the length contraction that it observes would bring earth within the cosmic even horizon. In the objects frame of reference it will arrive on earth. But in earths frame of reference it will never arrive.

How does one resolve this paradox?

Does the observed expansion of the universe speed up as an object approaches the speed of light? Do we need to add expansion acceleration to time dilation and length contraction in special relativity?

Is there some other way to resolve this paradox?

• It's not quite right to say that photons experience 'no time' - since time intervals are undefined at $c$ (not defined to be zero). And so it seems to me to be a meaningless question since SR is not equipped to speak of things from the point-of-view of a photon. – lemon Feb 13 '17 at 9:15
• If it would help then think of an object travelling at a speed very close to C. Then change arrives instantly to arrives very quickly (or slowly, actually arriving at all is enough to cause the paradox). It just needs to be going fast enough to cause length contraction to bring the earth within the cosmic event horizon, seeing as in earth's reference frame it doesn't arrive ever, the paradox still exists, so the question still stands. – trampster Feb 13 '17 at 9:23
• Length contraction has a limit. It can make the horizon appear closer but you still cannot reach the horizon because it's continuously moving away from you. You seem to be imagining the horizon as static... – lemon Feb 13 '17 at 9:45
• But for this to happen then you would observe a higher rate of expansion of the universe then when you where stationary. This is required to make up for the fact that you observe less distance between you and earth. – trampster Feb 13 '17 at 9:51
• I don't see a paradox here - photons don't have "perception", there is no reference frame moving at the speed of light, so the question of what they (or anything else moving at the speed of light) "perceive" is meaningless. – ACuriousMind Feb 13 '17 at 12:55