Einstein’s theory of gravitation
Einstein’s form of the equations of motion—that the proper time should be a maximum in curved space-time—gives the same results as Newton’s laws for low velocities. As he was circling around the earth, Gordon Cooper’s watch was reading later than it would have in any other path you could have imagined for his satellite.6 So the law of gravitation can be stated in terms of the ideas of the geometry of space-time in this remarkable way. The particles always take the longest proper time—in space-time a quantity analogous to the “shortest distance.” That’s the law of motion in a gravitational field. The great advantage of putting it this way is that the law doesn’t depend on any coordinates, or any other way of defining the situation. Now let’s summarize what we have done. We have given you two laws for gravity: Those two laws correspond to similar pairs of laws we have seen earlier. We originally described motion in a gravitational field in terms of Newton’s inverse square law of gravitation and his laws of motion. Now laws (1) and (2) take their places. Our new pair of laws also correspond to what we have seen in electrodynamics. There we had our law—the set of Maxwell’s equations—which determines the fields produced by charges. It tells how the character of “space” is changed by the presence of charged matter, which is what law (1) does for gravity. In addition, we had a law about how particles move in the given fields—$d(m\FLPv)/dt=q(\FLPE+\FLPv\times\FLPB)$. This, for gravity, is done by law (2).  In the laws (1) and (2) you have a precise statement of Einstein’s theory of gravitation—although you will usually find it stated in a more complicated mathematical form. We should, however, make one further addition. Just as time scales change from place to place in a gravitational field, so do also the length scales. Rulers change lengths as you move around. It is impossible with space and time so intimately mixed to have something happen with time that isn’t in some way reflected in space. Take even the simplest example: You are riding past the earth. What is “time” from your point of view is partly space from our point of view. So there must also be changes in space. It is the entire space-time which is distorted by the presence of matter, and this is more complicated than a change only in time scale. However, the rule that we gave in Eq. (42.3) is enough to determine completely all the laws of gravitation, provided that it is understood that this rule about the curvature of space applies not only from one man’s point of view but is true for everybody. Somebody riding by a mass of material sees a different mass content because of the kinetic energy he calculates for its motion past him, and he must include the mass corresponding to that energy. The theory must be arranged so that everybody—no matter how he moves—will, when he draws a sphere, find that the excess radius is $G/3c^2$ times the total mass (or, better, $G/3c^4$ times the total energy content) inside the sphere. That this law—law (1)—should be true in any moving system is one of the great laws of gravitation, called Einstein’s field equation. The other great law is (2)—that things must move so that the proper time is a maximum—and is called Einstein’s equation of motion. To write these laws in a complete algebraic form, to compare them with Newton’s laws, or to relate them to electrodynamics is difficult mathematically. But it is the way our most complete laws of the physics of gravity look today. Although they gave a result in agreement with Newton’s mechanics for the simple example we considered, they do not always do so. The three discrepancies first derived by Einstein have been experimentally confirmed: The orbit of Mercury is not a fixed ellipse; starlight passing near the sun is deflected twice as much as you would think; and the rates of clocks depend on their location in a gravitational field. Whenever the predictions of Einstein have been found to differ from the ideas of Newtonian mechanics Nature has chosen Einstein’s. Let’s summarize everything that we have said in the following way. First, time and distance rates depend on the place in space you measure them and on the time. This is equivalent to the statement that space-time is curved. From the measured area of a sphere we can define a predicted radius, $\sqrt{A/4\pi}$, but the actual measured radius will have an excess over this which is proportional (the constant is $G/3c^2$) to the total mass contained inside the sphere. This fixes the exact degree of the curvature of space-time. And the curvature must be the same no matter who is looking at the matter or how it is moving. Second, particles move on “straight lines” (trajectories of maximum proper time) in this curved space-time. This is the content of Einstein’s formulation of the laws of gravitation.