Geometry in space-time
Now we have to talk about time. As you know from the special theory of relativity, measurements of space and measurements of time are interrelated. And it would be kind of crazy to have something happening to the space, without the time being involved in the same thing. You will remember that the measurement of time depends on the speed at which you move. For instance, if we watch a guy going by in a spaceship we see that things happen more slowly for him than for us. Let’s say he takes off on a trip and returns in $100$ seconds flat by our watches; his watch might say that he had been gone for only $95$ seconds. In comparison with ours, his watch—and all other processes, like his heart beat—have been running slow. Now let’s consider an interesting problem. Suppose you are the one in the spaceship. We ask you to start off at a given signal and return to your starting place just in time to catch a later signal—at, say, exactly $100$ seconds later according to our clock. And you are also asked to make the trip in such a way that your watch will show the longest possible elapsed time. How should you move? You should stand still. If you move at all your watch will read less than $100$ sec when you get back. Suppose, however, we change the problem a little. Suppose we ask you to start at point $A$ on a given signal and go to point $B$ (both fixed relative to us), and to do it in such a way that you arrive back just at the time of a second signal (say $100$ seconds later according to our fixed clock). Again you are asked to make the trip in the way that lets you arrive with the latest possible reading on your watch. How would you do it? For which path and schedule will your watch show the greatest elapsed time when you arrive? The answer is that you will spend the longest time from your point of view if you make the trip by going at a uniform speed along a straight line. Reason: Any extra motions and any extra-high speeds will make your clock go slower. (Since the time deviations depend on the square of the velocity, what you lose by going extra fast at one place you can never make up by going extra slowly in another place.) The point of all this is that we can use the idea to define “a straight line” in space-time. The analog of a straight line in space is for space-time a motion at uniform velocity in a constant direction. The curve of shortest distance in space corresponds in space-time not to the path of shortest time, but to the one of longest time, because of the funny things that happen to signs of the $t$-terms in relativity. “Straight-line” motion—the analog of “uniform velocity along a straight line”—is then that motion which takes a watch from one place at one time to another place at another time in the way that gives the longest time reading for the watch. This will be our definition for the analog of a straight line in space-time.