# Speed of light time ratio

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1. Apr 28, 2014

Hey guys,

I have what I'm sure is a very simple question that I can't seem to find a simple answer to. I'm wondering what the ratio of time passage is between someone traveling at X times the speed of light versus someone on the surface of a planet. Say for a scenario I was traveling at 10 times the speed of light for three weeks, how much time would pass on earth?

This is all purely science fiction and I'm not needing to get into the whole conversation of whether faster than light travel is possible. I'm merely interested in the ratio of time traveled.

I'd appreciate it if you kept this simple, I'm not much of a physicist, merely an artist/writer. So if equations are needed, please explain them.

2. Apr 28, 2014

### HallsofIvy

Staff Emeritus
I an bit sure what kind of answer you want! If you are going to assume that relativity is not correct (which you are doing when you have faster than light travel) you wouldn't want a calculation that takes relativity into account. So why not just say "three weeks"?

If, as you appear to be saying, you are not interested in being scientifically accurate, you can say pretty much whatever you want!

3. Apr 28, 2014

### chogg

Well, your scenario violates physics, so I can't answer your question directly. :-) You can't go 10x the speed of light. You can't even go 1x the speed of light, but any number less than 1 is possible in principle.

Still, I can explain a scenario where time passes more slowly.

There are two numbers you care about.
• $\beta$ is your speed, as a fraction of the speed of light. So if $\beta = 0.99$, you are going 99% the speed of light.
• $\gamma$ is the factor by which time is slowed down. So if $\gamma=10$, then 30 weeks pass on Earth if 3 weeks pass on the spaceship.

These two numbers are related:
$$\gamma = \frac{1}{\sqrt{1 - \beta^2}}.$$
Or, going the other way 'round,
$$\beta = \sqrt{1 - \frac{1}{\gamma^2}}.$$

So, suppose you wanted time to pass 10 times more slowly on the spaceship. Then you need to go $\sqrt{1 - \frac{1}{10^2}}$ times the speed of light (which works out to be about 99.5% the speed of light).

4. Apr 28, 2014

Well, I'm writing a book in my spare time for my own enjoyment. The major plot point is how relativity affects relationships between people. So, I need to know how much time passes back on earth to see how that would affect people. Is this group, gone for 3 weeks, going to come back to their friends 50 years older? Or only 1 year. That vastly changes the reunion.

I'm writing this while taking into account that time dilation does occur. So while you may be traveling for only 3 weeks of your time, I assume many years would be passing on earth.

There's being scientifically accurate to both a small and large degree. I want to be in the realm of realism, but there isn't much need to say a while equation to specify the time, down to seconds, that would pass on earth.

5. Apr 28, 2014

I'm basing my story about the theory of the Alcumbierre Warp Drive. Which, if you happened to have missed it, is the latest version of a possible faster than light ship. Using this theory, the passengers wouldn't be traveling the speed of light, only the space around them. This would allow the ship to go faster than light while avoiding all the nasty problems of increasing mass.

If you haven't heard of it, it's an interesting read.

6. Apr 28, 2014

### chogg

The idea that mass increases at high velocities is outmoded. Certainly the passengers on the ship would not feel any heavier! There are no "nasty problems of increasing mass".

For the plot point you want, warp drives don't matter. Just a regular, fast ship gets you the time difference you're looking for.

If I may suggest, you can have the spaceship accelerate at a rate of $g$ for some amount of time, then decelerate, and make a similar return voyage. The advantage is that it will make the trip very comfortable for your passengers: it will "feel" just like Earth's gravity. And you can still get quite a big difference in elapsed time.

I remember solving this problem in grad school. If you accelerate at $g$ for a year, then decelerate for a year -- and do the same for the return journey -- only four years pass for the people in the ship, but something like 200+ years pass on Earth.

7. Apr 28, 2014

The problem with eliminating faster than light travel is that it eliminates the possibility of traveling to other systems within a reasonable amount of time. At a speed less than that of light, it would take several lifetimes to reach a system like Sirius or IK Peg.

Outmoded? Really? I had always heard the the main problem with approaching the speed of light was the increase in mass of the objects. But that's beside the point.

Is there any way to get a hypothetical approximation of how much time would pass for both parties if it WERE possible to move through space at faster than the speed of light?

8. Apr 28, 2014

### chogg

Ah, but since time passes more slowly for the spaceship's occupants, they can indeed reach distant systems in a reasonable time -- for them!

From Earth's point of view, they are traveling near the speed of light for the amount of Earth-time which passes -- say, 50 years, or whatever you want.

From the spaceship's point of view, the distance becomes length contracted -- so they see themselves traveling only for a short time, but the distance is also shorter.

9. Apr 28, 2014

I can't believe that never occurred to me. For some reason it was stuck in my head that at the speed of light, it would take the occupants a year to travel a light year in their view of time. That's of course absurd now that I think about it. It would instead take a year for them to travel a light year from an observers standpoint. Thanks for your help and the equation, that'll help greatly.

Cheers

10. Apr 28, 2014

### phinds

There you go again. Your question is exactly equivalent to asking "if the laws of physics didn't apply, what would the laws of physics say about <fill in any nonsense you like>?"

11. Apr 28, 2014

### PAllen

[Ignoring very fundamental practical and theoretical problems with Alcubierre drive...]

It is wrong to think of someone in the interior of the bubble as having a 'speed'. In fact, using such a drive (if it could exist, which is extremely doubtful), you could go back in time as easily as to any event at any time/place in the universe. So, HallsofIvy's slightly facetious statement is literally true of Alcubierre drive - you can pick any answer you want.

12. Apr 28, 2014

### Staff: Mentor

But right now, even as we speak, you are traveling at 99.9999% of the speed of light relative to someone somewhere in this universe... Do you find yourself inconvenienced by increasing mass?

13. Apr 28, 2014

After my fundamental shift in my understanding of the universe a few hours ago in this very thread, I see what you all are saying. When you don't think about these things every day, it's easy to misunderstand the statement that "the universe is expanding faster than the speed of light". When I first read that, it seemed to say that the speed of light was no longer the universal speed limit.

However, I neglected to think of the relative speed of galaxies expanding. I see what you're saying about all of us constantly moving at 99.99...% of light constantly without any problems.

I'll have to go reread a lot of my research now. Should be fun.

14. Apr 28, 2014

### Staff: Mentor

Have you found the solution yet? The way I would approach this is to first decide on how much time you want to pass for each party, then pick a star a certain distance away and have the astronaut go there at very close to the speed of light and then stay there the appropriate amount of time. For example, if you want 20 years and two weeks to pass on earth and two weeks to pass for the astronaut, you could have the astronaut travel to a star 10 light years away at very nearly the speed of light, then stay there for 2 weeks, then travel back to earth.