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I have read that the object, the light of which is reached us in 13.8 billion years is actually about 46 billion light years away from us now, due to the expanding Universe.

Lets assume that we started to travel to the object which is at current time 13 billion light years away.

What happens after 13 billion years from the view of an observer on the Earth?

  1. Will the traveller reach the destination?

  2. If Yes, will he also (feel that he) spend 13 billion years?

  3. If no, what will be distance from the Earth to the traveller and from the traveller to the destination?

  4. As the Universe is expading in all directions, does it mean that Earth is moving away (from the view of the traveller) faster than the speed of light?

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Everything within the so-called cosmic event horizon is, in principle, possible to reach in a finite time. Given the various observable parameters that enter the Friedmann equation, it is possible to calculate its distance to roughly 16.5 billion lightyears (Glyr).

That is, everything that today is less than 16.5 Glyr away is possible to reach in a finite time (at least for a light ray). Everything that is farther away will be carried away by that expansion of the Universe at a faster rate than even light may catch up with.

So, the answer to your question 1 is "yes". The answer to Q2 is, as descheleschilder has already answered, no — the faster you go, the less time will pass in your reference frame.

The answer to your Q4 is also yes; the farther the traveler is from Earth, the faster the expansion will make her recede. Since this motion is not a motion through space, the recession velocity is not limited by special relativity, and thus may increase without bounds.

For instance, a photon that leaves Earth now, recedes at $v=c$, but as soon as it's a few million lightyears away, out of the Local Group of galaxies that are gravitationally bound to the Milky Way, it picks up speed and starts having $v>c$. This velocity is with respect to Earth — locally, it's speed is always $c$.

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  • $\begingroup$ If I wanted a return trip would the "today" maximum be less than 16.5 Glyr? (Not asking you to calculate the number.) $\endgroup$ – Keith McClary Apr 14 at 22:35
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    $\begingroup$ @KeithMcClary Yes, definitely. The 16.5 Glyr is the maximum distance right now for an object you want to reach; that is, you will only reach it in "almost infinite time", meaning that if you want to be able to go back, you must return before that. The point of return is half that distance, i.e. ~8.3 Glyr today, but by the time you get there, that point will be roughly twice this distance from the Milky Way, due to expansion. Still, you will just have time enough to get back to the MW before infinite time (so to speak). $\endgroup$ – pela Apr 16 at 19:27
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Welcome to the "family", Komil!

Let me try to give an answer. By analogy with a car traveling on a road in a world where there is an maximum speed for light much less than in the real world.

1) Yes, the traveler will reach the destination (maybe a city he's driving to). Say the traveler rides with the maximum possible speed (say $30(\frac {km} {h})$). When the road beneath him is stretching the distance traveled is also stretched.

2) Because he's traveling at the maximum possible speed no time will have passed for him or her (or both), just like photons don't experience the passage of time (all the photon's velocity is through space, none through time).

3) The distance traveled by the car is $\frac{46}{13,8}$ times the distance if the road wouldn't be stretching (or less if the road were contracting) to the family.

4) For the traveler the place where he started his journey will indeed be moving away with a speed greater than the speed of light (in this case $30(\frac{km}{h})$. But this does not mean the car is traveling faster than light. The expansion of the road makes it look like that though.

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First, you are right, that as the universe expands, space itself is expanding, and light that travels for 13 billion years will not be able to reach objects that were originally 13billion lightyears away. This is because the space between the object and the original emission place of the light is expanding in the 13 billion years as the light travels. In the last 13 billion years to date, this came out to be 46 billion lightyears distance.

Now today, if you stared this, it would be even more then 46 billion lightyears, because the expansion is accelerating.

  1. No, the traveler will not reach the destination

  2. if he looks at his own clock, it will tick normally to him. If he compares his clock with a clock on earth, then it will depend on his traveling speed. If he travels near the speed of light, then because of SR, he will see his own clock tick slower then the clock on earth. (unfortunately, because of simultaneity of relativity, the person on the earth would see the same thing on his own clock)

  3. it depends on the speed of the traveler, let's assume almost light speed. The distance of the traveler from earth will be more then 13 billion light years, because the space between the traveler and the earth was expanding for 13 billion years. The distance between the traveler and the destination is more then 13 billion light year too. This is because even with the past example, 46billion light years takeaway 13billion (this is a very simplified math) is still 33billion. And the expansion of space is accelerating.

  4. The expansion is accelerating, and at a point in time, yes everybody will see the rest of the universe to recede at a speed faster then light.

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