Time in Relativity

By Steve Humphrey
Illustrations by Andrea Hutchinson


From the Scholium in Newton’s “Principia Mathematica,” it says “Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.” In this, he contrasts true, or absolute, time with common, or sensible, time. The latter is the way we experience time, through motion and clocks, but the former is actual time. This would imply that time passes at the same rate at every time and place in the universe, both at the largest scales and the smallest. Newton’s view reflects our commonsense, intuitive notion of time, as a kind of giant clock in the sky that dictates how things occur.

Henri Poincaré posed a puzzle related to absolute time. How can we tell that two distinct time intervals are the same length? Newton says time flows “equably,” but how can we know that? We can’t take two different time intervals and hold the endpoints up against one another. All we have are clocks, which are said to “measure” time intervals, which is puzzling. How do clocks “measure” time? And we can’t even know that clocks are isochronous. All we can do is compare one clock with another, and if they both “tick” together, we can say that the clocks are in sync, but we still don’t know that each successive hour is of the same length.

Einstein’s Special Theory of Relativity holds that there is no absolute state of motion. This means that whether something is at rest or in uniform motion is relative. Think about being on an airplane. Without looking out of the window, you can’t tell that you are moving. No experiment can be performed that would tell you that the airplane is in motion and the Earth below is at rest. We don’t find ourselves pushed up against the back wall. When we pour bourbon into our glass, it doesn’t spill all over us (unless we’ve had too many). If motion is relative, what does this tell us about time?

Consider a light clock, consisting of two mirrors, one above the other, one meter apart, and let a light ray bounce back and forth between the mirrors, and let one round trip constitute a “tick.” The yellow dotted line represents a light ray.

Now, put this clock in motion, moving from left to right.

These are the successive positions of the upper mirror.

These are the positions of the lower mirror.

Now, we know the velocity of light is constant, i.e., the same in all reference frames. But, as we can see from the above, the light ray must travel farther when the mirror is moving. Thus, each “tick” must take longer in the moving frame than in the rest frame. However, there is no fact in the matter of which reference frame is actually in motion, there is only relative motion. So, from the point of view of the “moving” frame, it is the rest frame’s clock that is slow. This phenomenon of “Time Dilation” has been tested and found to be real, and is independent of any particular choice of clock.

K mesons (kaons) are unstable subatomic particles that are created when high energy cosmic rays from space collide with air molecules in the upper atmosphere. In the lab, they decay into two pions in an extremely short period of time. So short, in fact, that we should not expect to see any at the Earth’s surface, because they would have decayed before they arrived. But, in fact, we do observe kaons on Earth. Why? Because the “internal clock” of the kaon runs slow relative to us because of its great velocity relative to the Earth.

Time dilation leads to the famous “Twin Paradox.” Consider two twins, Adventurous Alice and Boring Bob. Alice jumps into a rocket and zooms away. From Bob’s point of view, Alice’s clock is running slow, but from Alice’s POV, it is Bob’s that is running slow. What is true of the clocks is also true of the twins’ aging process. So, Bob thinks Alice is aging more slowly than he is, while Alice believes the same is true of Bob. While they are widely separated, there is no paradox, but what happens when Alice comes home? On the face of it, Bob would be older than Alice and Alice would be older than Bob. Paradox! Now, I really need some bourbon!

The point is, whether two temporal intervals are of the same duration, depends upon the relative state of motion. Contrary to Newton and our ordinary intuitions, there is no absolute duration. Meaning, absolute time does not “flow equably.” Think about that and console yourself with a nice glass of your favorite bourbon.

Steve Humphrey has a Ph.D in the history and philosophy of science, with a specialty in philosophy of physics. He teaches courses in these subjects at the University of California, Santa Barbara, and has taught them at the University of Louisville.