Time Dilation in Hypothetical Faster-than-Light Travel

[JEJoll]

This sounds like a homework question, but I promise you it's not (I'm not even a student). To be honest, the question is simply for my own personal curiosity, to test a theory about a video game.

My question is this:

If an astronaut traveled for 15 years (their time) at twice the speed of light (supposing that were possible), how much time would have passed on earth? Based on my basic understanding of relativity, I am guessing I might get some responses that it can't be calculated (due to infinity possibly showing up somewhere).

I'm not great at advanced math, but if no one wants to solve this, I'd settle for an equation or a rough idea.

 

[phinds]

The question you have asked is exactly equivalent to asking "if the laws of physics do not apply, what do the laws of physics say about <insert nonsense of your choice>".

Nothing travels faster than light and objects with mass cannot even travel AT the speed of light so positing that they CAN is magic and you can make up whatever answer you like.

 

[JEJoll]

I suppose that makes sense. Then I guess I'll ask the next closest question: What is the answer to the above if we substitute 2 times the speed of light for 99% the speed of light?

 

[HallsofIvy]

I think you mean the other way around! "If we substitute 99% the speed of light for 2 times the speed of light?". The time dilation formula says that if time t passes in your reference frame, you will observe $$t'= t\sqrt{1- v^2/c^2}$$ to pass in a reference frame moving at speed v relative to you.

With v= .99c, v/c= .99 so $$t'= t\sqrt{1- .99^2}= t\sqrt{1- 0.9801}= t\sqrt{0.0199}= 0.141t$$
or about 14% as fast as in your reference frame.

 

[JEJoll]

So, then, if I'm understanding correctly (and bare with me, I'm very much a laymen when it comes to physics), the traveler will experience time passage at only 14% of that of the observer (earth)?

So, based on what you've given me, if 15 years passed on earth, only 2.1 years would have passed for the traveler? 15 * .14 = 2.1

So then, if I'm interested in how much time passed on Earth for what seemed like 15 years to the traveler, I would solve 1/.14 * 15 to get my answer?

 

[Nugatory]

So then, if I'm interested in how much time passed on Earth for what seemed like 15 years to the traveler, I would solve 1/.14 * 15 to get my answer?

Somewhat surprisingly, the answer is no. As far as the traveller is concerned he is at rest while the Earth is moving away from him at .99c... so the traveller also finds that the moving Earth clock is running slow and only 1.14 years pass on Earth while the traveller experiences 15 years.

The key here is the relativity of simultaneity (to make sense of these problems you always have to consider time dilation, length contraction, and relativity of simultaneity because all three are always at work). According to the Earth guy, at the same time that his clock reads 15 years the traveller's clock reads 1.14 years - but because traveller has a different definition of at the same time it does not follow that according to traveller the two events "earth clock reads 15 years" and "traveller clock reads 1.14 years" happen at the same time. In fact, according to traveller at the same time that his clock read 15 the Earth clock read 1.14.

It gets more interesting if you do a round trip: Traveller leaves, goes to the distant star, turns around and returns to earth... And then the two of them compare their clocks to see which one experienced more time. This is the classic Twin Paradox, and we have many threads on it and http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html is an excellent reference.

 

[JEJoll]

All of this is so bizarre :D. Thanks for the input. I'll definitely check out that article, but I think that 30 mins before bed is not the best time to do so.