### The Relativity of Distance and Time

Einstein’s special theory of relativity says that distances and times are relative quantities—that is, they depend on reference frames. Roughly speaking, the length of an object gets shorter and its internal processes slow down relative to reference frames in which it is moving. (This is similar, but not identical, to Lorentz’s length contraction and time dilation hypotheses, as I’ll explain in a moment.) More precisely, distances and times depend on reference frames in the following ways.

1. The relativity of distance: When an object or set of objects is moving relative to a given frame of reference, the lengths of those objects and the distances between them are shorter (in their direction of motion) according to that reference frame, compared to their lengths and distances according to a reference frame in which they are at rest.
2. The relativity of time: When an object is moving relative to a given frame of reference, its internal processes (e.g. the internal motions of its molecules) occur more slowly in that reference frame, compared to reference frames in which the object is at rest.

To better understand these ideas, let’s return to the example of the alien spaceship. From the farmer’s perspective on Earth, the beam of light appears to be going only 100,000 km/s faster than the alien (since its speed is 300,000 km/s and the spacecraft is going 200,000 km/s in the same direction). But when the alien measures the speed of the beam as it passes her, she finds that it is moving 300,000 km/s relative to her spaceship. This sounds paradoxical, but it makes sense if we accept the above two claims.

How, you ask? Well, suppose the alien measures the speed of light by dividing the length of her spaceship by the time it takes for the beam to travel from one end of the ship to the other, and she gets the correct answer: 300,000 km/s. What happens if the farmer performs the same calculation, dividing the length of the moving spaceship (according to his perspective on Earth) by the time it takes (according to his own wristwatch) for the beam of light to traverse the length of the ship?

The relativity of distance (1) implies that the spaceship is shorter from the farmer’s perspective, and the relativity of time (2) implies that the alien’s clock—or whatever aliens use to measure time—must tick more slowly from the farmer’s perspective. For both of these reasons, the farmer will reckon that the beam’s speed relative to the spaceship is much slower than it seems from the alien’s perspective. From the farmer’s perspective, the length of the spaceship is shorter, and the light takes more time to go that short distance (because his wristwatch is running faster). If he does the calculations, he’ll find that the beam of light is moving only 100,000 km/s relative to the ship—or so it seems in his frame of reference, which regards the earth (not the spaceship) as being at rest. From the farmer’s point of view, the beam of light is moving 300,000 km/s relative to Earth, not relative to the spacecraft. But from the alien’s point of view, it’s the other way around.

The above suggestions sound an awful lot like Lorentz’s length contraction and time dilation hypotheses, and indeed Einstein’s special theory of relativity is similar to the Lorentzian theory—so similar, in fact, that it uses the very same equations Lorentz had formulated to describe the length contraction and time dilation effects for his aether theory! (These equations, called the Lorentz transformations, are explained in appendix E [link coming soon].) However, there are two important differences in how Einstein interpreted the ideas of length contraction and time dilation.

First, Lorentz had suggested that length contraction and time dilation are physical effects caused by an object’s motion through aether. But according to Einstein, there is no aether, and hence nothing to cause such effects. On Einstein’s view, length contraction and time dilation are not effects at all. Times and distances are simply relative quantities which differ from one reference frame to another, in the same way that the speed of an object differs from one reference frame to another. The train example discussed at the beginning of this chapter provides a helpful analogy. When I regarded the train as being at rest, the conductor’s speed was only 0.5 m/s; but when I regarded the earth, or the sun, or the galaxy as being at rest instead, his speed was much faster. Nothing caused the train conductor’s speed to increase. The only thing that changed was my own perspective. The conductor’s actual speed didn’t change—in fact, there’s no such thing as his actual speed. Speed is just a relative quantity, and there’s no fact of the matter how fast the train conductor is really moving. In the same way, distances and times are relative quantities. There’s no fact of the matter how long the alien spaceship really is, or how much time it really takes for the beam of light to get from one end to the other. These are relative quantities, so they differ depending on one’s perspective.

Secondly, Lorentzian relativity says that the aether’s reference frame is the only frame in which measurements of distance and time can be objectively correct. If an observer is moving through aether, her measurements of time and distance will be distorted, and therefore incorrect. But according to Einstein, all inertial reference frames are equally valid. No frame of reference is uniquely privileged.

For example, length contraction and time dilation apply to the farmer from the alien’s perspective, just as they apply to the alien from the farmer’s perspective. From the alien’s perspective, the farmer is the one who is moving at ⅔ the speed of light, so the farmer is the one experiencing length contraction and time dilation. In other words, the alien will consider the farmer’s clock to be ticking more slowly, just as the farmer will consider the alien’s clock to be ticking more slowly. Similarly, the farmer and the alien will disagree about which objects are shorter. Suppose the alien spaceship is exactly the same length as the farmer’s barn when they are not moving relative to each other (for example, when the spacecraft is hovering above the barn to abduct the farmer’s cows). Since the farmer and the alien share the same frame of reference at that time, they’ll agree that the barn and spaceship are the same length. Later, when the spaceship is moving at ⅔ the speed of light relative to the barn, the alien spaceship will be shorter than the barn according to the farmer’s reference frame. But according to the alien, the barn will be shorter than the spaceship. In fact, the entire earth will be shorter in the alien’s reference frame. In her view, our planet is shaped like a pancake rather than a sphere, “squished” in the direction of our motion relative to her spaceship!

The alien’s clock runs slower in the farmer’s reference frame, but the farmer’s clock runs slower in the alien’s reference frame. Can both be correct? What if the alien and the farmer synchronize their clocks initially, and meet up again later to see whose clock really elapsed more time? This apparent problem is essentially the same as a famous puzzle called the “twin paradox,” which we’ll discuss (and solve) later in this chapter.

And there’s no fact of the matter who’s right and who’s wrong. Provided the alien spaceship is moving inertially (not accelerating), the alien is entitled to regard her spaceship as being at rest, just as the farmer regards the earth as being at rest. Technically, the farmer’s reference frame isn’t inertial, since the earth is rotating on its axis and also orbiting the sun. But our planet’s acceleration is mild enough to be ignored for most farming purposes. There’s no fact of the matter how long an object really is, or how much time really elapses from one moment to another. Distances and times depend on frames of reference, and there is no uniquely correct (or “privileged”) frame of reference.

Does this mean everything is relative, there are no absolutes? Absolutely not! As discussed at the beginning of this chapter, all theories of relativity—including Galileo’s, Lorentz’s, and Einstein’s—hold that some quantities are relative while others are absolute. The theories merely disagree about which quantities are absolute and which aren’t.

Galilean relativity and Lorentzian relativity regarded distances and times as absolute quantities; Einstein disagrees. On the other hand, the speed of light was a relative quantity according to Galileo and Lorentz, but Einstein says it is absolute. And there are other absolute quantities in Einstein’s theory as well—for example, spacetime intervals, which will be introduced shortly.