What equations do I use to do basic time dilation problems?

Question

I understand the concepts of relativity, but I don't know where to go to set up equations to do any actual calculations. For example, what equations would I use to find out the following:

1. A ship leaves earth and accelerates up to .5c over the course of 1,000 years. it travels for another 1,000 years, and then decelerates for 1,000 years to 0c. How many years have passed e.g.

   (a) on the ship, relative to the ship?

   (b) on the ship, relative to the earth?

   (c) on the earth relative to the earth?

   (d) on the earth relative to the ship?

   (e) at the arrival spot, relative to the arrival spot?

   (f) in the radio frequencies transmitted by earth that are currently being received by the ship (e.g. if they were listening to a radio station from earth, what would be the year of its broadcast?)

2. The ship in question 1 was moving in a straight line away from earth. If it then started a second journey in a random direction (say, perhaps, at a 90 degree angle) how would you calculate everything from question 1 again?

3. What if the ship accelerated to .9c instead?

Answer 1

I would recommend Einstein(1905) On the electrodynamics of moving bodies. It is pretty accessible and you will find the equations you are interested in and give you a clear conceptual foundation of the subject, so that you can answer your own questions.

The basic question that you use is

$ t_e = \gamma t_r$

Where $\gamma = \sqrt\frac{1}{1-v^2/c^2}$

where $t_e$ is the time measured on earth and $t_r$ is the time measured on the rocket. Now when you say 1000 years it is either earth reference frame or the rocket's.

Now this does not take into account the acceleration of the rocket. If you are interested in acceleration draw the accelerated trajectory. Calculate its proper time.($\int ds$) That would be the time as measured on the rocket.

Answer 2

For non-accelerated motion you use the Lorentz transformations. These tell you what the event $(t, x)$ looks like when viewed by a moving observer. Be vary catious about just multiplying or dividing by factors of $\gamma$ as it's very easy to fall into apparent paradoxes like the pole in a barn. If you use the Lorentz transformations you won't make this kind of mistake.

Treating accelerated motion is a bit more involved, but fortunately John Baez has done all the work for you in his article on the relativistic rocket. As an example let's take the first leg of your journey. Suppose the rocket starts accelerating at (0, 0) in both frames, and you want to compare the time passed on the rocket with the time passed on Earth. A quick look at John's article will tell you:

$$ t_{Earth} = \frac{c}{a} \sinh\left( \frac{a \space t_{rocket}}{c} \right) $$

where $a$ is the acceleration of the rocket.