After battling with the concept of time dilation for some time (no pun intended) I finally think I have a good grasp of the basics. I think part of my problem was how many books and websites presented the ideas and equations hence my want to write this.
Time dilation says that if a clock is moving quickly ( >40% of the speed of light) it will run slower than a clock that is stationary relative to it. This means that when 1 second passes on the moving clock more than one second will pass on the stationary clock.
An Example:
An astronaut takes a stopwatch up into space and travels 90% of the speed of light relative to his friend, on earth, who also has a stopwatch. They start their stopwatches simultaneously and then after the astronaut measures 6 years they stop their stopwatches simultaneously (don't ask how they manage to do this).
How much time has the friend measured?
Facts:
The speed of light (c) = 3 x 108 m/s
Speed of Astronaut (v) = 0.9c or 2.7 x 108 m/s
We need to know how much time stretches and for this we use the speed of light. The speed of light is not relative it is always the same in a vacuum. This makes it very useful. Scientists knew that the speed of light must be constant and the only way this could occur is if time stretched.
The gamma factor (ɣ) is how much time stretches and is given by:
This means that as the velocity of the clock (v) approaches the speed of light ɣ gets very large so time is stretched a lot. Alternatively when v is small ɣ is very close to 1 (i.e. no stretch) this is why most of the time we do not need to take into account time dilation.
We now get the equation:
t = ɣ τ
Where:
t = Time measure by a stationary clock
τ (tau) = Time measured by a moving clock
In the example:
v = 0.9c or 2.7 x 108 m/s
c = 3 x 108 m/s
ɣ = 1/√(0.19)
τ = 6 years
so the friend will have measured roughly 13.8 years. This does mean that the friend will have aged 7.8 years more the astronaut (13.8-6).
I hoped this has been helpful.
