Time Dilation
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We will now look at the first - and easiest to derive - consequence of the two postulates of special relativity. It is called time dilation. The word dilate means to expand, spread out or become wider. The pupils in your eyes dilate when you are anxious or in the dark. What we will soon find out is that time in a moving reference frame dilates. I will wait until after the derivation to discuss the meaning of this.
Consider a train car like the rectangle in the diagram below. The point S is the source of a pulse of light, and D is the detector. I drew them spaced apart, but think of them as one and the same point. The ceiling of the train car is a mirror off of which the light will reflect. If you run the animation the path of the light is traced out as observed from the platform. If you change the value of v/c to zero you will see the path as observed in the proper frame in which the train car is at rest.
Derivation of Time Dilation
We want to compare the duration of time that it takes the light pulse to go from source to detector in the proper frame, to other frames in which the train is observed to move at speed v. The elapsed time in the proper frame is denoted with a zero subscript to indicate that the source and detector are at v=0 with respect to the observer. Keep in mind that the proper time is measurable by a single clock in the proper frame... one that could sit right at the location of the source and detector. This elapsed time will simply be the distance of travel divided by the speed that the light travels at, which is in every reference frame. So we get if the height of the train car is .
In comparison, when viewed from the platform (as you see in the animation above) the light is seen to travel in a triangular shaped path as the train travels past. Clearly the light travels along a longer path length. The speed the light travels at, however, will still be the same as in the other frame since that is the second postulate of special relativity! The elapsed time will certainly be longer than in the proper frame. We want to find out how much longer, and on what it depends. All we need to do is find the length of the triangular path. Since the distance to the ceiling is the same as the distance back down, we can just double that length. The base of that larger triangle will be and the height of the triangle will be The hypotenuse of the triangle is then The time will then be that distance divided by the speed of light, or This leads to Lastly, we can relate the duration in the proper frame to this result and write
This is often written for short, where This is the Greek letter gamma, and is called the Lorentz factor. Our result indicates that a duration of time (like one second) will last longer in a frame observed in motion. This is called time dilation. In other words, moving clocks - and all things governed by time - will run slower when observed in moving frames. Conversely, clocks always run fastest in the proper frame. Another thing to point out is that in the frame of the platform, to measure the interval of time between the source emitting a light pulse and the detector receiving it, we'd need two clocks. This is because the clocks need to be at rest with respect to the platform at all times, and as such the two events (emission and detection of light) will occur at two different locations.
It is worth noting that we are always in our own proper frames. I bring this up because students often wonder if this result somehow implies that we could live much longer if we were to board a high speed rocket. Since we are in our own proper frame, the perceived duration of our lifetime will not be altered by boarding a fast-moving rocket or something of the sort. On the other hand, everybody else's lifetime - all those you left behind - will be altered as far as you are concerned. To them, however, time will pass as it always did. So not only will time seem to pass at the same rate on the rocket as back on earth, but you will likely be bored out of your mind looking out the windows at the darkness of space year after year.
Implications of Time Dilation
The most obvious implication is that time is not a universal quantity. It varies depending on reference frame, and as Einstein later found, it is also affected by gravitation and acceleration. When we observe particles traveling large fractions of light speed in particle physics experiments, we readily see these effects. We measure the half-lives of unstable particles to be longer when the particles are traveling faster. We must expect that these same effects are not limited to fundamental particles or imagined clocks, but rather to whatever metronome dictates the rate at which all processes proceed according to natural law.
While we can't pretend that we are light and ask what the universe would look like from light's reference frame since light always travels at the speed of light, we can still look at interesting limiting scenarios. We can see that traveling astronauts come home younger than those they left behind. In the limit going to light speed they wouldn't have aged at all upon return, while the rest of the world ages and passes away. Perhaps this is what light would see if it could stop and take a look around.
If you have watched movies about time travel, clearly there is a mechanism in place (time dilation) in nature to allow travel to the future. The past, however, is not something we can travel to via time dilation.
Lastly I should mention that GPS navigation depends upon time dilation and also on the gravitational effects on time. Without carefully accounting for both of these effects, the most accurate GPS readings would not be possible. After all, GPS relies on a 3D version of triangulation where synchronized signals are sent from satellites to a receiver unit that you have in phones these days. The receiver needs to calculate how much time has elapsed since the signal was sent, but time up high where gravity is weaker ticks faster than down where the receiver is. Also, the synchronized ticking of the clocks on the satellites depends on their velocities of orbit. In this sense relativity is a part of everyday life these days, and not just for particle physicists.