If we now reflect that the force of gravitation introduced by Newton does not exist—such action at a distance is very problematical—and that in empty space there are only objects freely left to themselves, we are driven to the following conclusion, which unites in a simple way the previously separated sisters, inertia and weight: Every moving body freely left to itself in the universe describes a geodetic.
Far from the massive stars this geodetic is a straight line, because there the universe is almost Euclidean. Near the stars it is a curved line, because there the universe is not Euclidean. A fine conception, combining in a single rule the principle of inertia and the law of gravitation! A brilliant synthesis of mechanics and gravitation, putting an end to the schism which so long kept them separate and non-corresponding sciences!
In this bold and simple theory gravitation is not a force. The planets have curved paths because near the sun, just as in the neighbourhood of every concentration of matter the universe is curved or warped. The shortest path from one point to another is a line that only seems straight to us—poor pygmies that we are—because we measure it with very small rods and over small distances. If we could follow the line over millions of miles, and during a sufficient period, we should find it curved.
In a word—to use an illustration that must be regarded only as an analogy—the planets describe curved paths because they follow the shortest path in a curved universe, just as at a sports ground cyclists have no need to turn the handles when they reach the corner, but pedal straight on, because the slope of the ground compels them of itself to turn. In the sports ground, as in the solar system, the curvature is greater in proportion as the machine is nearer to the inner edge of the track.
All that now remains is to assign to the universe, to space-time, such a curvature at its various points that the geodetics will exactly represent the paths of the planets and of falling bodies, admitting that the curvature of the universe is caused at each point by the presence or vicinity of material masses.
In this calculation we have to take into account the fact that the “Interval”—that is to say, the part of the geodetic between two points that are very near each other—must be an invariant whoever may be the observer. In this way the same geodetic will be a curved or even wavy line for the drunken man we introduced and a straight line for a stationary observer. The length of the line is the same, whether it appears straight or curved.
Taking all this into account, and doing prodigies of mathematical skill of which we have sufficiently indicated the object, Einstein has succeeded in expressing the law of gravitation in a completely invariant form.
In calculating, on the ground of Newton’s law, the “Interval” of two astronomical events—for instance, the successive falls of two meteorites into the sun—we should find that the “Interval” has not precisely the same value for observers who are moving at different velocities.
With the new form given to the law by Einstein the difference disappears. The two laws, however, differ little from each other, as was to be expected in view of the accuracy with which astronomers found Newton’s law verified during a couple of centuries. The improvement made in Newton’s law by Einstein means, in a word (and to use the old language of the Euclidean universe), that we consider the law accurate with the reserve that the distances of the planets from the sun are measured by a scale which decreases slightly in length as the sun is approached.