Spacetime Pythagorean Theorem

The Pythagorean theorem of spacetime differs from the usual Pythagorean theorem in two ways. First, the vertical side of the triangle is multiplied by $c$. This is a trivial scale factor that gives time the same units as space. Second, the right side of equation (4.3) has a minus sign rather than a plus sign. This highlights a fundamental difference between spacetime and the ordinary $xyz$ space in which we live. Spacetime is said to have a non-Euclidean geometry -- in other words, the normal rules of geometry that we learn in high school don't always work for spacetime!

The main consequence of the minus sign in equation (4.3) is that $I^2$ can be negative and therefore $I$ can be imaginary. Furthermore, in the special case where $X = \pm cT$, we actually have $I = 0$ even though $X,~T \ne 0$ -- i. e., the ``distance'' between two well-separated events can be zero. Clearly, spacetime has some weird properties!

The quantity $I$ is usually called an interval in spacetime. Generally speaking, if $I^2$ is positive, the interval is called spacelike, while for a negative $I^2$, the interval is called timelike.

A concept related to the spacetime interval is the proper time $\tau$. The proper time between the two events A and C in figure 4.4 is defined by the equation

$$\tau^2 = T^2 - X^2 / c^2 .$$ (5.5)

Notice that $I$ and $\tau$ are related by

$$\tau^2 = - I^2 /c^2 ,$$ (5.6)

so the spacetime interval and the proper time are not independent concepts. However, $I$ has the dimensions of length and is real when the events defining the interval are spacelike relative to each other, whereas $\tau$ has the dimensions of time and is real when the events are timelike relative to each other. Both equation (4.3) and equation (4.5) express the spacetime Pythagorean theorem.

If two events defining the end points of an interval have the same $t$ value, then the interval is the ordinary space distance between the two events. On the other hand, if they have the same $x$ value, then the proper time is just the time interval between the events. If the interval between two events is spacelike, but the events are not simultaneous in the initial reference frame, they can always be made simultaneous by choosing a reference frame in which the events lie on the same line of simultaneity. Thus, the meaning of the interval in that case is just the distance between the events in the new reference frame. Similarly, for events separated by a timelike interval, the proper time is just the time between two events in a reference frame in which the two events are collocated.