How long does something exist at relativistic speeds?

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So I want to include near-light-speed travel in my fiction, but I'm getting bogged down by the implications of it!

My understanding is:

If something travels very fast (close to the speed of light) time dilation will cause it to experience time slower, meaning that the journey, as observed by earth (the origin of the travel) would last for longer than it would to those on board. EG (number invented to numbers sake) earth sees it as taking 10 years, the people on board see it as taking 5 years.

What I'm curious about is what effect this has on how long the vessel has existed? so, for example, if your vessel contains radioactive elements with a half life of (in this example) 5 years, would they be 1/2 as active or 1/4 as active on arrival?

Now, if the vessel was set up to send a signal back to earth each time it decays (like some sort of excessively elaborate and very fast geiger counter) would earth, assuming they account for the ever increasing distance of the vessel from earth, register the decreased speed of time on the vessel? if the crew were to record a message and transmit it, would their voices be distorted, IE sped up or slowed down?

Appreciate any input here!
 

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Check out the Heinlein YA novel Time for the Stars, should give you some answers and ideas.
 
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So I want to include near-light-speed travel in my fiction, but I'm getting bogged down by the implications of it!

My understanding is:

If something travels very fast (close to the speed of light) time dilation will cause it to experience time slower, meaning that the journey, as observed by earth (the origin of the travel) would last for longer than it would to those on board. EG (number invented to numbers sake) earth sees it as taking 10 years, the people on board see it as taking 5 years.

What I'm curious about is what effect this has on how long the vessel has existed? so, for example, if your vessel contains radioactive elements with a half life of (in this example) 5 years, would they be 1/2 as active or 1/4 as active on arrival?

Now, if the vessel was set up to send a signal back to earth each time it decays (like some sort of excessively elaborate and very fast geiger counter) would earth, assuming they account for the ever increasing distance of the vessel from earth, register the decreased speed of time on the vessel? if the crew were to record a message and transmit it, would their voices be distorted, IE sped up or slowed down?

Appreciate any input here!

Using your 10 and 5 yr figures, this means that The ship would be traveling at ~0.866c relative to the Earth, and that, as measured from the Earth, the destination is ~8.66 ly away.

That 5 year value for the ship is valid for any means used to measure time on the ship, clocks, isotope decay, how much the metal of the ship fatigues, etc. In ever possible way, the trip took 5 years according to the ship and its occupants. A radioisotope with a half-life of 5 yrs, will be 1/2 as active upon arrival.

In your post, you say that the occupants "experiences" time dilation. This is not really an accurate statement. No one experiences time dilation. Time dilation is something you measure as happening to objects moving relative to you. As far as the ship and its occupants are concerned, there is a different reason that the trip only takes 5 yrs. From their perspective, it is the Earth and destination that is moving at 0.866 c, and as a result are undergoing length contraction. This length contraction also applies to the distance between them. Thus according to the ship, the distance between the Earth and destination is only 4.33 ly, and it take 5 yr to cross this distance at 0.866c.

As far as signalling between Earth and ship goes. There will be the time dilation effect on top of the standard Doppler effect caused by the increasing distance. Together they produce Relativistic Doppler shift. If you factor out the increasing distance effect, you would be left with the time dilation effect. This will be two way.

Remember above, how I said that according the ship it was the Earth and destination that was moving which resulted in a length contraction? Well it also results in the ship measuring time dilation for the Earth. Even after factoring out the propagation delay effect, the ship would measure signals coming from the Earth as being slow. ( This is something that Heinlein got wrong in the novel mentioned in the above post. He assumed that while in flight, the signals from Earth would seem sped up. While Heinlein was a good writer and for the most part strived to be scientifically accurate, he didn't quite get Relativity right.)

About this time, you are wondering how this can be. If the ship gets to its destination and stops, after 5 yrs its time, while measuring slowed down messages from the Earth the whole time, how can it conclude that 10 yrs have passed on Earth?

Let's work it out. We will use the full relativistic Doppler effect:
$$ f_o = f_s \sqrt{ \frac{1-B}{1+B}}$$

Where B = v/c
Fo is the observed frequency
Fs is the source frequency
So during the trip, the Earth will receive signals from the ship at a rate of ~0.268. It takes ten years for the ship to reach the destination and another 8.66 yrs for the light of the ship's arrival to reach Earth, which means that the Earth won't actually record the ship's arrival until 18.66 yrs after it left. If it was receiving signals the whole time it got 0.268* 18.66 = 5 yr of "ship time" transmissions. It will see the Five years passed on the ship during the trip.

Now consider the ship: It also receives signals from the Earth at a rate of 0.268. After fives years, its own time, it arrives and stops at the destination. It will have received ~1.34 worth of " Earth time" signals from the Earth. Since it has now come to a stop relative to the Earth, it knows the distance between it and the Earth is 8.66 ly and that the very last signal it got from the Earth took 8.66 y to reach him and therefore is is actually 8.66+1.34 = 10 y later on the Earth than when he left.

If our ship had not come to a stop at the destination, he would have concluded something else. At that moment the Earth would have been only 4.33 ly away (length contraction), but the light reaching him at that moment left when he was much closer to the Earth and took that much less time to reach him. Once he takes all that into account he would conclude that only 2.5 y have passed on the Earth since he left.

The key difference between these two assessments of how much time has passed on Earth lay in the fact that the ship came to a stop in one and didn't in the other. This means that in one scenario the ship has to undergo an acceleration when coming to a stop, and for that period of time is non-inertial. The basic time dilation/ length contraction equations are only valid when the measurements are made from inertial frames.
During the acceleration undergone when coming to a stop, the occupants are making their measurements from inside a non-inertial frame.
The rules are different here. For one, clocks separated along in the direction of the acceleration run at different rates even if they are at rest with respect to each other. The upshot is that during this period of acceleration the ship would conclude that "Earth time" ran much faster than "ship time", and even though Earth time was behind ship time at the start of the acceleration, it ends up being later by the time the ship comes to rest with respect to the Earth.

If you are just worried about communications between ship and Earth during the travel segments, use the Relativistic Doppler shift equation. (If the ship and Earth are approaching and not receding from each other, just reverse the sign for B.)
 
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Even after factoring out the propagation delay effect, the ship would measure signals coming from the Earth as being slow. ( This is something that Heinlein got wrong in the novel mentioned in the above post. He assumed that while in flight, the signals from Earth would seem sped up. While Heinlein was a good writer and for the most part strived to be scientifically accurate, he didn't quite get Relativity right.)
Huh. I never realized that.
 

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