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I’m planning a hard science fiction novel, thus there is no Faster-Than-Light travel, but I do present the existence of a means of transportation that is close to the speed of light. For example, it can travel half the speed of light, my two questions are:

a) Am I correct in thinking that if a ship travels half the speed of light would reach the closer stars at the double of time the light travels from them to us? For example, it will reach Proxima Centauri (4.24 light years away) in 8-9 years, Barnard Star (5.96) in 10-11 years, Wolf 375 (7.78) in 14, etc.

b) What would be the relativistic effects, if any, for the crew in respect to the rest of the Universe and the people on Earth? Should be notice that in this setting humans live for hundreds of years due to the advancement in medical tech, thus decades-long space travel is not such an issue, however the difference of aging between people left on Earth and space travelers is a plot point.

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The important thing to note here is that to say "it takes 8-9 years" doesn't make sense without specifying who it applies to. When relativistic effects start to apply, it's not the same in all reference frames.

Let's take your Proxima Centauri example.

At half the speed of light, the spacecraft reaches Proxima Centauri in 8.5 years, and if it immediately turns and comes back, it will come home 17 years after the mission started as observed from Earth. But the light from Proxima is delayed, so we will only be seeing the turnaround at year 12.75

The crew onboard on the other hand, will have experienced time dilation. Time goes slower for them, so we have to divide the 17 year roundtrip time by the Lorentz factor:

$$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$

For $v = \frac{1}{2} c$, the Lorentz factor $\gamma = 1.15$, cutting 17 years down to 14.7 for the crew.

For higher velocities, the Lorentz factor grows accordingly. For instance, at around 99% of $c$, the time ratio is about 7.

So in essence:

a) Yes, you can assume that the travel time, as seen from the target or destination, is the distance in light years divided by your speed measured in fractions of $c$. But you still have to keep in mind that the "news" (light/radio waves ) of arrival will still need some additional time to get back.

b) The crew will stay younger/experience less time than the rest of the universe. This effects gets stronger the faster you go, but isn't really noticeable before you reach at least a quarter of the speed of light.

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  • $\begingroup$ To clarify a bit, because this rapidly expands into paradox: Funny things happen during acceleration. $\endgroup$
    – ikrase
    Sep 25, 2019 at 7:07
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Answer from Hohmannfan is accurate, but also please note:

a) That calculation require instant acceleration to 0.5c to apply. And since we talk for 'hard science fiction', such thing is quite... embarrassing at least. You have to use a different formula/calculator to calculate time needed based at some acceleration - unless you decide that your starship accelerates (and decelerates!) to 0.5c at no considerable time. 0.5c is about 150 million meters per second(!) so hypothetical thrusters that are capable of get that velocity starting from 0 m/sec at, lets say, 1 day or even 1 month only, are highly likely to also get you to even higher velocities. If, on the other hand, you assume e.g. 1 year accelerating and 1 decelerating, something also 'too much' but definitely more 'realistic', you need 2 years + the above formula for all distance minus the distance traveled while accelerating and decelerating.

b) When it comes to relativistic time, calculations are easy but what that exactly means for 2 different frames of reference is extremely hard for the average mind to comprehend, if both frames have humans. See the larger scale: A proton takes 100.000 years to cross the galaxy for us to observe(from earth frame of reference), but only takes 296 seconds for the proton itself(own frame of reference). https://users.physics.ox.ac.uk/~rtaylor/teaching/lectur345%20text.pdf So for a go and return, 17*2=34 years have passed for earth observant and 14.7*2=29.4 for the traveler. 4.6 years difference or 'difference feeling'. This does not necessarily means that travelers are 4.6 years younger biologically.

Extra info: Lots of comments so i clarify some things to help the writer: First, to clear again my position, i add 2 things: Acceleration and deceleration time should be considered, it makes it closer to hard fiction Time dilation is not experimentally proven that affects aging of humans, and that means that it is a small window for the writer that actually helps.

Now about time dilation and how much time has been elapsed for different frames of reference: The static earth based frame of reference needs more time to get information that has happened. Imagine it as a network lag or latency. The starship will get time X to get to destination no matter what. This time X is X for the staship, for everyone in the starship, and for an observer travelling next to the starship and has a nice external view all the way. For an observer on earth, since the starship is constantly moving AWAY, it will take more and more time for the INFORMATION, the light the ship sends back, to reach earth. This means what you see from earth at any given time is the past position of the ship. The ship is really much far away than you see, again at any time. That is why it takes 'more time' for earth, the ship eventually reach destination at time X, but earth observer at time X will see the ship still going, and reach destination at X+Y, where Y is the time the light showing the ship in orbit takes to go back in earth. To change the scale may help: Changing speed: If the ship exceeds the speed of light it will simply disappear from view from earth, because at any given time, the light the ship emits to earth travels AWAY from earth, as simple as that. Changing journey distance: The ship goes to another galaxy. The distances are impossible to comprehend but if the ship goes with 05.c all the way the time difference between the arrival of the ship and when earth sees that arrival will be the actual travel time plus the time light needs to reach us from that galaxy. That is, the ship will reach at some time X and we know it X+some millions of years later, e;g; 12 for those we have observed at galaxies within 3.8 megaparsecs. And a final example: When ship reach destination, enter a fantastic wormhole and INSTANTLY is placed at earth orbit, happily observe itself travelling to destination and enter the wormhole!

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    $\begingroup$ I'm not quite sure what your last point is. Are you saying that our biological clock is somehow independent of time itself? Or that there are biological factors that somehow only apply at high velocities? $\endgroup$
    – SE - stop firing the good guys
    Sep 24, 2019 at 14:31
  • $\begingroup$ Assuming reasonable accelerations (on the order of 1g) and a desired travel speed (ie. an accelerate-coast-decelerate profile rather than an accelerate to midpoint, decelerate to target profile), acceleration times can be ignored for all but the shortest trips. For example, travel to Proxima Centauri at 0.5c will involve a six-month acceleration phase as viewed from Earth, a 7.5-year coast phase, and a six-month deceleration phase. $\endgroup$
    – Mark
    Sep 24, 2019 at 21:00
  • $\begingroup$ Yes and no, accelerating is not the same as velocity increases. Going from 0 to 0.1c requires tremendous less thrust and fuel than going from 0.4 to 0.5 . 5 months accelerating means 0.1c per month, and thrusters capable of accelerating from 0.4 to 0.5 within a month will definitely be capable of accelerating even further - so why max velocity = 0.5c? With gradual acceleration. more accurate and more logical, means that the ship will reach 0.4c within 1 month to be able to reach the next 0.1 within the rest 4 months. $\endgroup$
    – Stefanos Zilellis
    Sep 25, 2019 at 6:39
  • $\begingroup$ Too much calculations here - to keep it simple i believe acceleration and deceleration times should be considered. even roughly as some % over the plain x distance * y velocity formula. $\endgroup$
    – Stefanos Zilellis
    Sep 25, 2019 at 6:39
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    $\begingroup$ "it is not proven that those experience less time will age less" -1 from me too then. This is misinformation. $\endgroup$
    – SE - stop firing the good guys
    Sep 25, 2019 at 11:18

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