Time dilation formulas

[pwagen]

I was thinking of the hypothetical situation of someone going away from Earth on a spaceship travelling close to the speed of light. Are there any simple formulas for calculating the time dilation of their trip?

 

For example, say the ship is going away at 1 c for 2 days (ship time), then comes back. Is it possible to calculate how much time has passed on Earth?

 

If we stay on Earth and watch the ship go away, and it comes back after exactly five years, is it possible to know how much time has passed aboard the ship?

 

If such formulas (intelligible for a layman not too well versed in heavy physics) exist, I'm assuming they'd include the speed of light? If so, what would happen aboard the ship at speeds beyond the speed of light? Purely hypothetical, of course.

[swansont]

The factor by which time is dilated is

 

[math]\gamma = \frac {1}{ \sqrt{1-\frac{v^2}{c^2}}}[/math]

 

So if gamma is 2, a trip that the spaceship thinks takes 1 day, the earth observer says takes 2 days (i.e. the space clock is slowed by a factor of 2)

 

We can't say what happens at v > c. The theory doesn't cover that.

[pwagen]

Pretty much exactly what I was looking for, much appreciated!

[pwagen]

I've looked about a bit, but can't find anything about distance being a factor (apart from gravitational time dilation). Am I then correct in thinking that it doesn't matter if my spaceship goes to Sirius and back, or simply orbits the Earth - given the same velocity, the time dilation would be the same?

 

Or is the velocity relative to Earth, and only a moving away or moving to movement counts?

 

Furthermore, would it be correct saying that if you could somehow teleport to a place far, far away (say in this galaxy, so the expansion of the universe isn't that noticable), time wouldn't be dilated?

[swansont]

Distance matters because more time difference will accumulate for longer trips, and distance and time are related.

 

Satellites moving in nominally circular orbits exhibit time dilation. The kinematic time dilation will be the same if the speed is the same.

[pwagen]

I know about GPS having to take into account time dilation when calculating positions, but I wasn't sure whether it was because of their speed or their distance.

 

[...]and distance and time are related.

Any more so than what you said about longer trips? In my last example of being able to teleport to a place far away (or use a wormhole, whatever suits best), wouldn't that take velocity out of the equation and just deal with distance? Or is that thinking so far away from any current understanding of physics that it's just silly?

 

Reason I'm asking these questions is because I was thinking of the science fiction techs available in books, like in the Foundation universe, and what they would be like if they existed here, in one form or another. Naturally, Asimov would have to write his books without taking time dilation into account, or the stories would have been very different from what they were.

[swansont]

I know about GPS having to take into account time dilation when calculating positions, but I wasn't sure whether it was because of their speed or their distance.

 

Any more so than what you said about longer trips? In my last example of being able to teleport to a place far away (or use a wormhole, whatever suits best), wouldn't that take velocity out of the equation and just deal with distance? Or is that thinking so far away from any current understanding of physics that it's just silly?

 

Reason I'm asking these questions is because I was thinking of the science fiction techs available in books, like in the Foundation universe, and what they would be like if they existed here, in one form or another. Naturally, Asimov would have to write his books without taking time dilation into account, or the stories would have been very different from what they were.

 

I can't really decipher all the issues you'd have with a wormhole

[pwagen]

Because of my sucky explanation? I'd be happy to try and clarify my question. Might even try to use that formula you gave me above!

[elfmotat]

I've looked about a bit, but can't find anything about distance being a factor (apart from gravitational time dilation). Am I then correct in thinking that it doesn't matter if my spaceship goes to Sirius and back, or simply orbits the Earth - given the same velocity, the time dilation would be the same?

 

Or is the velocity relative to Earth, and only a moving away or moving to movement counts?

 

If you take a longer trip, the time dilation accumulates. For example say you take a 1 year long trip in your spaceship at 99% the speed of light then ~7 years will pass on Earth. A 2 year long trip will for you is about a 14 year long period for Earth, etc.

 

Furthermore, would it be correct saying that if you could somehow teleport to a place far, far away (say in this galaxy, so the expansion of the universe isn't that noticable), time wouldn't be dilated?

 

I suppose it really depends on what you mean by "teleport." We can't use our theories to describe scenarios which violate those very theories. The closest thing to "teleporting" that is allowed by GR (allowed so far as we can tell thus far) I suppose would be a wormhole.

 

If you move one mouth (call it end #1) of a wormhole at high velocity and then bring it back, the end 1 will be time-dilated with respect to end 2. This can result in some wacky phenomena. For example if you entered at end 1, then you would exit at end 2 before you even entered at end 1.

 

This is all assuming that stable wormholes are possible in the first place. If I had to guess I would say that they aren't, though this is by no means proven and certainly isn't an area of expertise for me. My perspective is a practical one: time travel to the past opens up a slew of causality paradoxes which need to be resolved (Grandfather Paradox, etc.).

 

I know about GPS having to take into account time dilation when calculating positions, but I wasn't sure whether it was because of their speed or their distance.

 

Well in this case you're mixing gravitational time dilation (time "slows" when you're near massive bodies) with kinetic time dilation (time "slows" as your velocity increases). In general, if you want to calculate the time that someone on Earth experiences relative to an orbiting observer, you use:

 

[math]T=T_\oplus k\sqrt{1-\frac{r_s}{r}-\frac{v^2}{c^2}}[/math]

 

where [math]T[/math] is the time experienced by the satellite, [math]T_\oplus[/math] is the time experienced on the surface of the Earth, [math]r_s=2GM_\oplus/c^2\approx 0.008869~ meters[/math] is the Schwarzschild radius of Earth, [math]r[/math] is the radial coordinate (which is approximately equal to the distance from the center of the Earth to the satellite), [math]v[/math] is the tangential velocity of the satellite, and [math]k=1/\sqrt{1-r_s/r_\oplus}\approx 1.000000000695[/math] is a constant.

 

If you want to incorporate radial velocity then things get a lot more complicated.

Edited January 22, 2013 by elfmotat

[pwagen]

I've thought about my question during the day, and let me rephrase the wormhole issue. I realized you don't need a wormhole for the example anyway. So let's say we take elfmotat's speed of 99% of the speed of light, and go away for a year (ship time), then return home. So two years will have passed on the ship, and 14 back on Earth. I'm in not totally mistaken, I'm with you so far.

 

But let's say you go away for a year, then stay where you arrive for another year, then go back home. So, you've been away on a trip for 3 years, but how long has passed on Earth? That's what I was trying to get at with my wormhole example. Nowhere in the formulas do I see distance (apart from the "orbiting observer", but I'm guessing that wouldn't apply here). So in short, 15 years would have passed on Earth? That is, 14 for the journey and one for the time you stayed. Is that somewhat accurate?

 

 

Thanks to both of you for your replies, it's very informative. And sorry if I'm being a bit slow, but I really want to wrap my head around it properly.