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If one travels in a spaceship at speed V, the time elapsed for the traveller relative to an observer on earth is dilated by,

$$ t' = t(1 - v^2/c^2)^{1/2} $$

Does this mean that a space traveller, who gradually accelerated to a speed very close to the speed of light can then travel 10 billion light years in a few minutes?

Relative to the travelling observer, does this mean that effectively there is no speed limit? Even though this can be translated as a space contraction, what difference does that make - the traveller effectively travelled 10 billion light years in say 6 minutes - meaning an effective speed of 100 billion light years per hour (876 trillion times the speed of light).

Does this mean that time travel into the future is possible? Would it be possible for a traveller making a round trip away and back to earth to travel to the year 3000 in 5 years?

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  • $\begingroup$ PLease state added sub-questions in the comments below the answer, to make it clear which answer you are commenting on. Also, it is a bit difficult to answer a question, which later gets substantially changed, in a meaningful way :) $\endgroup$ – Thriveth Feb 1 '14 at 0:18
  • $\begingroup$ But, as @dmckee states in the comment below my answer, there are practical reasons why this is only in principle. But yes, in an empty Universe with an infinite energy supply, any acceleration of the ship becomes enhanced time dilation for the traveller. $\endgroup$ – Thriveth Feb 1 '14 at 0:21
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The traveller can travel 10 billion light years arbitrarily fast in his/her own experienced time. However, to the observer who stays at home on Earth, the traveller's speed will simply get asymptotically closer to the speed of light.

This does, if you think about it, line up very fine with the Twin "Paradox" (which is not really a paradox at all). As the traveller keeps accelerating the engine, when $v \rightarrow c$, the time his travel takes will not be shortened to the twin who stays at home. A journey of, say, 40 light years will mean that the twin who stays home will always have aged at least 80 years when the traveller returns home.

The traveller, however, can experience an arbitrarily short travel time, tending asymptotically to zero as $v$ tends to $c$.

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    $\begingroup$ "The traveller can travel 10 billion light years arbitrarily fast in his/her own experienced time." This is very much a "in principle" answer. In fact, a $\gamma \approx 10^{14}$ would pose some serious problems; for instance, the blue-shifted CMB would vaporize you and your ship. But, what the heck you have already assumed access to unreasonable amounts of energy... $\endgroup$ – dmckee Jan 31 '14 at 23:55
  • $\begingroup$ We are obviously leaving out any petty practicalities like that here :) Space is empty, and energy supply is infinite. Obviously ;) Special relativity thought experiments are not exactly known to be hyper realistic. $\endgroup$ – Thriveth Feb 1 '14 at 0:16
  • $\begingroup$ @dmckee you wouldn't be able to experience that anyways. "10 billion light years in a few minutes" translates to some pretty impressive acceleration. I don't think it's possible to accelerate the spaceship this fast without crushing the passengers in the process. Maybe using a warp bubble, but my predictions are negative as for the existence of negative-energy matter. $\endgroup$ – John Dvorak Feb 1 '14 at 5:57
  • $\begingroup$ @Jan I simply assumed that was the ultimate speed at the end of a looooooooong run-up. $\endgroup$ – dmckee Feb 1 '14 at 5:58
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    $\begingroup$ @JanDvorak Related: physics.stackexchange.com/questions/6435/cosmic-ray-hazards $\endgroup$ – dmckee Feb 1 '14 at 17:50
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Time travel to the future is not only possible but is occurring as we sit as part of our normal thermodynamic evolution. When it comes to relativistic speeds, two observers who are traveling away from each other both see the other observer's thermodynamic evolution as slowing down as part of time dilation.

At the instant one or the other observer decides to turnaround and meet, the symmetry between observers is broken and one or the other observer will appear to have aged very rapidly. What breaks the symmetry is the acceleration associated with maneuver that brings the two observers back into contact after having been separated. The observer that has undergone the most "accelerations" or "boosts" will appear to have aged less. If both observers perform exactly symmetric maneuvers both will appear to have aged the same.

It is important to emphasize the aging of the observers is the result of thermodynamic processes and in general those will be irreversible unless additional energy is expended as governed by the second law of thermodynamics.

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