The curvature of space-time
Now we want to relate what we have just been talking about to the idea of curved space-time. We have already pointed out that if the time goes at different rates in different places, it is analogous to the curved space of the hot plate. But it is more than an analogy; it means that space-time is curved. Let’s try to do some geometry in space-time. That may at first sound peculiar, but we have often made diagrams of space-time with distance plotted along one axis and time along the other. Suppose we try to make a rectangle in space-time. We begin by plotting a graph of height $H$ versus $t$ as in Fig. 42–18(a). To make the base of our rectangle we take an object which is at rest at the height $H_1$ and follow its world line for $100$ seconds. We get the line $BD$ in part (b) of the figure which is parallel to the $t$-axis. Now let’s take another object which is $100$ feet above the first one at $t=0$. It starts at the point $A$ in Fig. 42–18(c). Now we follow its world line for $100$ seconds as measured by a clock at $A$. The object goes from $A$ to $C$, as shown in part (d) of the figure. But notice that since time goes at a different rate at the two heights—we are assuming that there is a gravitational field—the two points $C$ and $D$ are not simultaneous. If we try to complete the square by drawing a line to the point $C'$ which is $100$ feet above $D$ at the same time, as in Fig. 42–18(e), the pieces don’t fit. And that’s what we mean when we say that space-time is curved.