One thing I have read in various popular science magazines is that time goes slower as you approach the speed of light. My question is lets say in a craft moving away from the eath at the speed of light to a point 1 light year away, and the returning to earth, would the occupants of this craft discover that more time than two years has elapsed on earth when the return?

Well, anything with mass cannot move at the speed of light. That’s one problem with the question. But if we assume that it was moving at close to the speed of light, then the answer is yes.

By how much.? Say a craft the size of the Apollo command module traveling near the speed of light to a point 1 light year away and then back…

Yes, and the closer to the speed of light you get the greater the time discrepancy between the traveler and the folks back on earth.

I remember hearing of experiments where a highly accurate clock was taken on something like a long, fast plane ride and compared to an identical sort of clock upon return, and there was a measureable difference between the two consistent with relativistic principles (an extremely tiny fraction of a second, given the relatively low speed).

Oh, god. Please don’t ask me to dig out my relativity notes. In any case, the difference is highly dependent on how close you are to c, the difference between taking the journey at .9c and .99c is astronomical.

It depends very much on how close to the speed of light. At 86% of c, two years of travel as seen from earth would be measured as 1 year by the ship. But .999c, two years from earth would be about a month ship-time, and at .999999c, two years from earth would be about 1 day ship time

See here http://www.1728.com/reltivty.htm

Travelling at 90% of the speed of light, one year on the spacecraft would be about 2.3 years on earth. Consequently, 2 years on the ship would be about 4.6 years on earth.

Increase the speed to 95% and 3.2 years would pass on earth for each year on the ship.

At 99%, 7 years would pass on earth.

At 99.9% 22 years would pass on earth.

At 99.99% 70 years would pass on earth.

And so on.

For reference, the formula for this calculation is the Lorentz transformation: 1 / sqrt( 1 - ( v[sup]2[/sup] / c[sup]2[/sup] ) )

Where v is the relative velocity and c is the speed of light.

This transformation has the convenient property of working not just for time dilation, but length contraction as well. (The ship gets smaller along its direction of travel from the perspective of Earth.)

And mass also. Mass increases by the same factor.

I knew I was forgetting one.

Assuming 1g acceleration to 0.9c, how long would it take to accelerate to 0.9c? (I’m too lazy to dig out the formula.)

Well, you would have to ignore relativistic effects. If you do that and simply use newtonian physics, about 318 days.

The formula relating velocity and acceleration is v = a*t. one gravity is 9.8 cm/sec^2. The speed of light is appoximately 3x10^10 cm/s so to get to 0.9 c would require 2.7x10^9 /9.8 = 2.8 x10^8 seconds = 318 days.

Now I’m not positive if I have to do a relativistic correct there so that it would feel like 1 gravity to astronauts on the rocket. But that should be approximately right in any case

Yes. But strictly speaking, this effect would NOT be the result of *velocity* time dilation, but rather *gravitational* time dilation.

Velocity time dilation means that time passes more slowly for all objects that are moving relative to me. So from the point of view of an observer on Earth, time is passing more slowly on the spaceship. But from the point of view of an observer on the spaceship, time is passing more slowly on Earth!

Gravitational time dilation means that time passes more slowly for objects that are undergoing acceleration. This can be the result of being in a gravitational field, or being acted upon by another force. In order to leave Earth and come back the spaceship has to accelerate to nearly lightspeed, decelerate to turn around, accelerate on the way back, and decelerate to arrive. Taken collectively these accelerations and decelerations result in less time having passed on the spaceship when it finally arrives home and the clocks are compared.

It’s these inertial frames of reference that seem like the truly difficult part of the problem.

At least, they’re the part that I’ve never understood.

So it essentially means that deep space travel is impossible, mainly because no one wants to return home and find they are 150 years in the future?

No, it’s math. It says nothing about deep space travel. That’s a sociological and economic question.

Acceleration can be thought of as a succession of inertial frames of reference, each at a somewhat higher speed than the previous one (infinitesimally, if you want to really get it down); hence, you don’t need to appeal to GR, as has already been stated.

In a simplified way, you can view the whole thing as a system consisting of three inertial reference frames: that of the Earth, that of someone leaving Earth (at fixed speed – picture him just getting beamed up, or something), and that of someone returning the Earth (again, being beamed over to another spaceship going in the other direction, again at a fixed speed). This frame-changing accounts for the asymmetry of the problem, not any General Relativity effect; the paradox only arises if you consider the problem to only comprise two reference frames, which then should be interchangeable.

I think you could certainly find qualified astronauts willing to do such a mission and return to find it’s 150 years into the future.

And technical. As mentioned upthread, as velocity increases, so does mass, so you have to continue to increase thrust to maintain the same acceleration, and at some point it’s impossible to provide enough thrust. So those speeds we’re talking about are infeasible given today’s technology (and maybe infeasible due to the laws of physics).