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  1. According to theory of time dilation, flow of time slows down significantly at the speed of light.Is there any conditions practically or theoretically when flow of time is reduced to zero means it comes to still?

  2. Whether time & space are infinite?

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2 Answers 2

The answer to (2) is simply that no-one knows, and further that it's unlikely we will ever know. It's impossible to prove that the universe is infinite, but it's just possible we might prove it closed and therefore finite if the length scale is around the size of the currently observable universe. The paper Topology of the Universe: Theory and Observations describes some ways we might observe a closed topology. However I must emphasise that this seems basically unlikely and we will never know the answer to your question.

Your question (1) is more subtle because time dilation is a widely misunderstood phenomenon. It's certainly true that an observer on Earth would measure the clock on a fast moving spaceship to be ticking slowly. However if you were on the spaceship you'd see the clock ticking normally. There is no absolute sense in which time runs more slowly on the spaceship - it is just running slowly relative to a clock on Earth as observed by the observer on Earth.

The confusion arises because in relativity there is no absolute time. There is no special reference clock that you can compare all other clocks to. Every observer has there own time scale and all time scales are equally valid.

Having said this, the answer to your question is that no observer will ever see another clock slow completely to a halt i.e. see time stop completely. The best we can do is see time slow asymptotically towards a stop, that is the clock we are observing will get slower and slower but would take an infinite time to come to a complete stop. An example of this is if we drop a clock into a black hole. We'd see the clock slow down as it approached the event horizon, but we would have to wait an infinite time before we could see it stop completely.

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…but note that a photon's Proper time (Eigenzeit) is indeed $\tau=0$. –  Ansgar Esztermann Feb 20 at 9:42
    
Do I have to observe clock travelling at infinite speed, to see it's time to stop (approach zero)? Or to rephrase it, is time dilation inverse to speed? –  jnovacho Feb 20 at 10:08
    
@jnovacho: in special relativity you need to observe the clock travelling at the speed of light. Since nothing (except photons as Ansgar says) can reach the speed of light you'll never see the clock stop. In GR the situation is more complicated. In the example I gave of a black hole the clock actually slows to a halt at the event horizon, but it still takes an infinite time to slow to a stop. –  John Rennie Feb 20 at 10:22
    
"...it is just running slowly relative to a clock on Earth as observed by the observer on Earth." Does that mean observers from other reference frames might disagree about whether the Earth clock is running slower than the spaceship clock or vice versa? –  mbeckish Feb 20 at 14:18
    
@mbeckish: yes indeed! For example if you're on the spaceship then in your frame you are stationary and it's the Earth that's moving. In that case you would see clocks on Earth running slowly compared to your clock on the rocket. This is the origin of the infamous twin paradox. –  John Rennie Feb 20 at 15:02

Time is relative. When it comes to Time Dilation, you actually see dilated time of another observer. So, your own time flow won't get frozen in any case.

Hypothetically, you can see another one's time frozen if she is traveling at speed of light (time dilation by speed) or she is at event horizon of Black Holes (gravitational time dilation). Unfortunately, both are not possible in the framework of Relativistic Physics.

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