The first thing on our list of strange observations from Einstein’s Special Theory of Relativity is that time slows down in a reference frame that is moving rapidly relative to an observer’s frame—time dilation. By moving rapidly, we mean that it is moving at a significant fraction of the speed of light (670 million miles per hour). Let’s discuss how we will test this in our thought experiment. Our experiment consists of two spacecraft, one with human observers and an alien spaceship moving at half the speed of light. We will compare two identical clocks, one on the alien spacecraft and one on ours, and see if they run at the same rate. But what kind of clocks can we use to illustrate this effect?
Einstein’s first postulate assumes Galilean relativity. That means the aliens cannot perform any experiment inside their spacecraft to determine if they are moving, such as comparing the rates of different types of clocks. Therefore, all types of clocks in the spaceship will run at the same rate, even biological processes and our perception of time. Thus, we can choose any type of clock for our thought experiment. We will choose the light clock because it can provide a clear illustration of time dilation.
In the light clock, a light pulse bounces between two mirrors, separated by a distance L, as shown at the top of the figure. Since we know this distance and the speed of light, we can determine the time it takes for a light pulse to go from the bottom, reflect off the top mirror, and return to the bottom. The light pulse is detected and simultaneously reflected upward from the bottom for the next cycle. The number of cycles that make this round trip is our measure of time.
First, we will carefully calibrate the light clocks while the two spacecraft are in the same reference frame, before the alien ship speeds away. This is like the phrase “synchronize your watches,” which we hear in movies to ensure all clocks display the same time and remain in sync. As shown in part a of the figure, the calibration process verifies that the stationary spacecraft clocks run at the same rate on both ships
Let’s see what happens when the alien spaceship moves past us, illustrated in part b of the figure. The ship is traveling at speed v relative to our spaceship frame. On our ship, it takes a certain time (t seconds) for the light in the clock to move vertically from the bottom to the top mirror. During that time, the spaceship moves a horizontal distance equal to its speed multiplied by the time, or vt.
In our reference frame, we observe that the light in the alien spacecraft clock travels along a slanted path rather than vertically. The slanted path is longer than the vertical path seen when the spaceship was stationary. The length of the longer slanted path can be calculated using the Pythagorean theorem, shown in part c of the figure.
It seems like magic, or at least a significant paradigm shift, when we apply Einstein’s second postulate—the speed of light does not change regardless of the reference frame. As we watch the clocks, it takes longer for the light to travel the slanted path between the mirrors in the moving spaceship clock than it takes to travel the vertical path between the mirrors in our clock. We conclude that the clock in the alien spaceship is running slower and, because of Galilean relativity, time itself appears to be passing more slowly in the alien spaceship as viewed from the reference frame of our spaceship.
Would an observer traveling in the alien spaceship notice their clock running slower and everything slowing down? The answer is no. An observer on the alien spaceship would see the light pulses on their clock continue to travel vertically as normal. But they would observe our spacecraft moving in the opposite direction at high speed. They would perceive that the light in the clock on our spaceship is traveling a longer, slanted path, and therefore our light clock would appear to be running more slowly. The alien spacecraft observer would see time passing normally in their spaceship, but time slowing down in our spacecraft.
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