Lesson 2 - Time Dilation

Clocks speed is relative to observer's velocity

page1 In lesson 1 we saw that different observers do not agree on which events happen at the "same time". In this lesson we are going to see that different observers see time passing at a different rate. You should see the discussion page for information about deriving the scaling factor γ. For this lesson, I just want to illustrate that different observer's clocks to not move at the same rate.

page2 Let's put a person in our diagram called Alice, and a spot with a mirror. Alice is going to flash a beam of light, and when it hits the mirror it will bounce back. I'm going to add some green events when the beam leaves Alice, $P_1$ , when it hits the mirror, $P_2$ , and returns to Alice, $P_3$ .

page3 If I turn Alice's clock on, I can move time forward and see that her clock reads 4 at event $P_3$ . That makes sense: the speed of light is 1, and the the mirror was 2 units away, so the light had to travel 4 units.

page4 Now let's view these events from a velocity relative to Alice and her mirror. I'll pick -0.7, and notice that the scaling factor γ is about 1.4. I'll add a second observer, Ben, off to the side at $x=6$ . At this velocity we see that the first beam of light has farther to travel and takes more time. The second beam of light has less distance to travel and takes less time. The first key point is that the beams of light still travel at speed 1 in Ben's reference frame — this is the fundamental assumption of Special Relativity. But from Ben's point of view, the difference in velocity between Alice and the light is not 1. It is smaller than 1 when they are moving in the same direction and larger than 1 when they are moving toward each other.

page5 The second key point is that the the total time for both beams of light is more for Ben than for Alice. Alice said the time between $P_1$ to $P_2$ is 2 and between $P_2$ to $P_3$ is also 2. But Ben says the time between $P_1$ to $P_2$ is almost 5 and between $P_2$ to $P_3$ is about 1. The total time Ben sees is between 5 and 6.

page6 See the the discussion page for an explanation why velocities don't seem to average the way we expect.

page7 I'm going to advance the time now. Watch Alice's clock and Ben's clock. Since Ben is standing still, our clock matches his. As we advance forward, we see that when we get to event $P_3$ , Alice's clock reads about 4 and Ben's reads about five and half.

page8 So now we see that Ben thinks Alice's clock moves more slowly, relative to his own. However, by symetry, Alice should see that Ben's clock moves more slowly. This seems like a paradox. To resolve the paradox. Let's look at the two points in time when we do the measurements. I'm adding an instant when Ben and Alice's clocks are 0, at $P_1$ . Then I'm adding an instant at $P_3$ . Let's color one red and the other purple.

page9 I said these were instants in time. However, in special relativity, an instant in time is only meaningful if you've picked an observer. Since Ben is standing still, these two instants in time are relative to Ben.

page10 When I switch back to Alice's reference frame, these two lines that represent "instants" to Ben, are no longer horizontal lines. That means that from Alice's point of view, the events that occur on the red line do not happen at the same time. So Alice does not think that Ben's clock reads 0 at the same time that Alice's clock reads 0. She thinks that Ben's clock reads a little more than 4, and when her clock reads 4, she thinks his reads about 7.

page11 That's the end of lesson 2. In the next lesson I'll talk about length contraction. And then in lesson 4, we'll come back to the twin paradox, which looks at how both Alice and Bob think the other's clock is advancing slowly.