Suppose we want to send a spaceship to Alpha Centauri and calculate the minimum possible time it will take for us here on Earth to have the spaceship back. We are not interested in how fast the time will pass for the people in the spaceship, but rather how long it will take for us here on Earth to have the spaceship back. Increasing the speed of spaceship will decrease the time of the roundtrip journey as measured by us waiting on Earth, but after the spaceship passing a certain fraction of the speed of light time dilation will start to be relevant. Is there a minimum time possible time for the spaceship to do the roundtrip journey (again, using Earth as reference)? As the spaceship accelarates, at which point the time dilation added due to the incrase of speed starts to increase the total time of the journey measured on Earth?
-
$\begingroup$ Note that the time dilation only affects time as measured by the rocket occupants. The Earth is considered stationary and hence not subject to time dilation. If you want to get picky the Earth will have some tiny amount of time dilation due to its motion in, say, a Sun centered frame, but this is not in any way affected by how fast the rocket travels. $\endgroup$– Eric SmithCommented May 19, 2022 at 20:22
-
$\begingroup$ Thanks for your response, Eric. I understand that time dilation would be measured by the rocket occupants (as the time will pass slower for them). However, I always read if we travelled $x\%$ close to the speed of light for $T$ amount of time (say, with $x\%$ and $T$ being enough to go for a round trip to a given point in space) and come back, on Earth it will have passed $T + \delta$ amount of time (example: quora.com/…). $\endgroup$– DouglasCommented May 19, 2022 at 21:02
-
$\begingroup$ It all depends on who is measuring the elapsed time. The linked quora question used time on the spaceship, and so of course something has to be added to convert to Earth time (which will be longer). If you measure time on Earth then the spaceship is irrelevant -- time on Earth is not affected by whether or not a spaceship is traveling somewhere! $\endgroup$– Eric SmithCommented May 19, 2022 at 21:08
-
$\begingroup$ In the example above of the quora question. Is it correct to say that on Earth it will have passed 138 years, but only 120 for the rocket occupant? If so, as we set the rocket velocity closer to the speed of light, more time would have passed on Earth, or not? If we could send a rocket to Alpha Centauri at 99.9999% of the speed of light, wouldn't the 8 year journey of the rocket occupants be observed on Earth as a 8 year + $delta$? Thanks again for your response. $\endgroup$– DouglasCommented May 19, 2022 at 21:10
-
$\begingroup$ Got it, Eric. Thanks for your reply again. $\endgroup$– DouglasCommented May 19, 2022 at 21:11
2 Answers
As the spaceship accelarates, at which point the time dilation added due to the incrase of speed starts to increase the total time of the journey measured on Earth?
Never. If the ship can accelerate, it will reduce the time measured on earth. There are diminishing returns, but it is never zero.
The limit of this process is exactly the time observed on earth by light to make the round trip. A massive object with no restrictions on its ability to accelerate can get arbitrarily close to that limit.
The minimum time, as measured on Earth, for anything to travel to Alpha Centauri and back is 8.74 years or thereabouts, if the something was massless. For something massive like a spaceship the time would be greater, and would depend on the mass of the object, upon how much energy you were prepared to expend in accelerating it, and how much acceleration it could stand without disintegrating.
