Chapter 3: The Invariant Interval

By Mark Lawrence

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In Chapter 2, we saw that the speed of light, c, is the same number for everyone, everywhere. The speed of light does not depend on how fast you are going, nor on how fast the source of the light is going, you always measure the same number. We learned that two different observers moving at different speeds cannot synchronize their clocks with each other. We learned that moving clocks appear to be slow, and moving objects appear to contract in the direction of their motion. The factor by which clocks slow down and objects contract is Ö(1 - v2 / c2). We also learned that the space concept of position had to be replaced by the space-time concept of event, which is a particular position at a particular time.

We're used to using the Pythagorean theorem to calculate distances. So, in Figure 3.1 below, A2 = B2 + C2. We need to be able to calculate distances in special relativity, so we need to know how this formula should look in space-time.

Figure 3.1
Here's what we know: the speed of light is always c. Speed is distance divided by time, so this means sqrt(  X2 + Y2 + Z 2 ) / T = c. We can multiply both sides by T then square both sides to get X2 + Y2 + Z2 = c2 T2. Or, c2T2 - X2 - Y 2 - Z2  = 0. We call this quantity c2T 2 - X2 - Y2 - Z2 the interval. We're going to see that in special relativity the interval takes the place of distance. But, there's a big difference - the interval is how far you went minus the time is took to get there. For a ray of light, the interval is always zero, because at light speed, 1 meter of distance takes 1 meter of time, and 1 - 1 = 0. This fact, that all people see the same speed of light, will be elevated from a curiosity to a fundamental axiom. The equation c2T2 - X2 - Y2 - Z2 will similarly be elevated from a special equation about light to a fundamental equation about the distance between any two events.

We're used to distance2 =  X2 + Y2 + Z2. This formula is called a metric, and this particular type of metric is called positive-definite. Positive because the sum of three squares is always positive. Definite because for any X,Y,Z we can always calculate a unique number. So, in 3-space, we use the 3-distance sqrt( X2 + Y2 + Z2). But, we're working in 4-space now, so we need to figure out what the 4-distance is. In Special Relativity, the idea of distance will be replaced by the interval c2T2 - X2 - Y2 - Z2, which is not positive definite. We can see that if something moves a short distance in a long time, the interval is positive. If something moves at precisely the speed of light, the interval is zero. And if something moves faster than light, or more reasonably if we consider two points which are far apart in space but not in time, the interval is negative.

This is very different from flat Euclidean space. This is why we use a new word, interval: to help remind us that this is a very strange kind of distance that can be positive, zero, or negative. For example, the interval between the Earth and the Sun is about -7 minutes if we consider where the Earth and Sun are at the same time; the interval is zero if we consider the path that a ray of light would take from the Sun to the Earth; and the interval is about 6 months if we consider the path that a typical NASA satellite would take. We're not used to a type of distance where how long you take to go somewhere counts as part of the distance. Similarly the interval from Los Angeles to New York is not the 3-distance of 3,000 miles. The interval from Los Angeles to New York is zero for a ray of light, it's about six hours for a traveler on a 747, and it's about five days for someone driving a car. If we consider where Los Angeles and New York are at exactly the same instant, so that T2 = 0, then the interval is -3,000 miles, but now it's a negative number.

Right away we can see that these factors of c are going to be popping up all over the place. Also, we see that there's some confusion on whether the interval is measured in meters or seconds or hours or feet or light years or whatever. Actually, we're used to this for distance. If we asked someone, "How far is it from Los Angeles to New York," we would not be surprised to hear 3,000 miles, or 5,000 kilometers, or maybe even six million feet, or 50 million centimeters. But the idea that it could be two seconds or six hours or -3,000 miles from Los Angeles to New York seems very strange. How can a distance be the same as a time? Why is the speed of light this strange number, 186,000 miles per second? Is there something special about this number, 186,000, that God particularly liked?

Let's think about horses for a minute. A horse's height is measured in hands, where a hand is four inches (don't ask me why, I've never owned a horse). So, the horse below stands about 15 hands high at the shoulders, and is about 7 feet long. When the horse rears up on its hind legs, if we were to measure the dumb way we might find that the horse is now 22 hands high but only 5 feet long.

A horse standing The same horse rearing up

The confusion here is because we're measuring height in hands, and length in feet. It would make a lot more sense if we were using the same units in both directions, like hands for both length and height. As it is, if we want to know the distance from the horse's rear hoof to his nose, we can't use Pythagoris' theorem, we can't say (15 hands)2 + (7 feet)2 = distance2, because hands are not in the same units as feet. We could say something like (15 hands * 4 inches per hand / 12 inches per foot)2 + (7 feet)2 = distance 2 . You can see this is a real pain - it's really not very convenient to use different units for different dimensions.

Similarly, we have a built-in confusion about space and time: we measure time in seconds and distance in feet or meters. However, knowing that the speed of light is the same for everyone, we can use the speed of light to convert seconds into feet or meters. From now on, we'll agree that we're going to use the same units for time and space. For example, as we've already seen, one foot of distance equals one nanosecond at the speed of light, so if we say a foot of time we mean the same thing as if we say a nanosecond. A meter of time is about 3 nanoseconds. Velocity, distance per time, is now meters per meter, so velocity has no dimensions. The speed of light is now just 1 with no units.  The speed limit on most freeways is 65 miles per hour which equals about c / 10,000,000. If physicists were running the highways, apparently highway signs would say "Speed limit 10-7." A traffic ticket for going 85 would read "excessive speed: 1.3*10-7 in a 10-7 zone." That's it, no units. So, if you were talking to God, I think He would tell you that the speed of light was one, and that 186,000 is some strange number that we humans made up.



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