CHAPTER XXIXTHE GENERAL LAWS OF MOTION
IN classical science, when we wished to describe a body’s motion, it was of course insufficient to describe its spatial course in our system of reference. It was necessary, also, to define its motion along this course. As soon, however, as we reason in terms of space-time instead of in terms of separate space and time, all the particularities of motion are fixed by the single space-time world track of the body. This opens up the possibility of describing the laws of motion of free bodies in a purely geometrical manner in four-dimensional space-time, so that kinematics becomes a four-dimensional statics. Let us now pass to a more precise determination of the problem.
According to Newtonian mechanics, free bodies in interstellar space far from matter, moving under no constraint, always described rectilinear courses with constant speeds. This law was known as Newton’s law of inertia. We remember, of course, that a law of motion of this type remained completely indeterminate until we had defined the frame of reference to which this motion would be referred, and we saw that any Galilean frame answered our requirements. Now it was assumed that Newton’s law of inertia was universally valid in the case of free bodies moving under no constraint; when, therefore, the law was found to be at fault in the neighbourhood of large masses of matter, the discrepancies were ascribed to a real force of gravitation emanating from matter and compelling the erstwhile free bodies to move under constraint. In other words, in a field of gravitation, bodies were no longer free; so the validity of the law of inertia, applying to perfectly free bodies, was not overthrown thereby. The motions actually followed by bodies in a gravitational field would thus result from a compromise between the courses the bodies would have followed had no gravitational field been present, and the perturbating influence due to the field. These laws of motion were expressed by Kepler’s three laws of planetary motion, according to which planets described ellipses round the sun (Kepler’s first law), with variable velocities determined by the law of areas (Kepler’s second law) and by the dimensions of the ellipses described (Kepler’s third law).
As in the case of the law of inertia, Kepler’s laws of planetary motion held only for a Galilean or inertial frame and, in particular, for the Galilean frame attached to the sun’s centre. Thus there existed a complete duality between the two types of law and motion, those for free bodies and those for bodies moving under gravitational constraint.
In Einstein’s space-time theory this duality is removed. There is now but one law of motion, namely:All free bodies pursue geodesics, or the straightest of paths, through space-time, regardless of whether we are considering regions in the neighbourhood of matter or remote regions in interstellar space. This law is in full accord with thePrinciple of Least Action, and may be regarded as a direct consequence of this fundamental principle.[97]
True, owing to the curvature of space-time around matter and its flatness in distant regions, the actual courses and motions will vary according to the more or less great proximity of matter; and we might be tempted to say that we had succeeded in removing a duality in the laws of motion only by introducing a new duality referring to the structure of space-time—curved around matter and flat for distant regions. But this line of argument would be incorrect. There is but one law of space-time curvature or gravitation; for this law, which ascribes a definite curvature to space-time in the presence of matter, tells us also how this curvature dies away to perfect flatness at infinity. The existence of flat space-time far from matter is thus comprised in the one statement of the law of space-time curvature or gravitation. Thus, a precise determination of the motions of bodies, whether near matter or far from matter, is given by the one law of space-time curvature together with the law of least action, which states that the space-time courses or world lines of the bodies will be geodesics.
Let us now examine another aspect of the question. We have seen that according to Einstein’s ideas it was no longer necessary to differentiate between free bodies (those moving in interstellar space) and bodies moving under constraint (those moving in the vicinity of attracting matter). Henceforth, a planet moving round the sun is just as much a free body as one moving in interstellar space. If it moves in a different way, in these two cases, it is simply because the structure of space-time is curved around matter, instead of being flat as in interstellar space. There is, therefore, no justification for invoking a physical pull or force of gravitation in the proximity of matter; this so-called physical pull is inexistent so long as the planet moves without any additional interference. In all cases the planets and bodies in interstellar space are guided through space-time by the intrinsic structure of the space-time wherein they are situated.
Now structure implies geometry, and space-time, like any other mathematical extension, is amorphousper se; hence thisstructure of space-time must be attributed to a foreign cause, to the presence of the four-dimensional metrical field of space-time similar to a magnetic field. It is this four-dimensional field which acts as a guiding influence on bodies, constraining them along the geodesics of space-time. When, therefore, the observer partitions space-time into his space and time mesh-system, the effect of the guiding field is to direct bodies along certain paths in space and to determine their motions along these paths. Were there no guiding field, or metrical field, were space-time completely amorphous, there would be no space and time partitions, motion would be meaningless and inconceivable; even were it conceivable, the bodies would not know how to move.
It would appear, then, to be more correct to state that there can exist no such thing as freely moving bodies, since all bodies are subjected to the influence of the guiding field. We can, however, retain the usual terminology by remembering that by a freely moving body we mean one submitted to the action of the guiding field and to nothing else. The important point to understand is that as regards freedom of motion, all bodies in empty space, whether situated in a gravitational field or far from matter, are on the same footing. In all cases their actions are controlled by the guiding field, and by the guiding field alone. The sole difference is that in the neighbourhood of matter, the lay of the guiding field is curved slightly. It is incorrect, therefore, to invoke a real physical pull acting on the body when it moves near matter, and to deny the existence of such a pull when the body is moving in interstellar space.
And yet it might be said: “We experience a real physical pull when we stand on the earth, whereas we should experience no such pull were we floating in interstellar space.” Let us see why it is that this real physical pull appears to arise in one case and not in the other.
A freely moving body follows, as we have explained, a geodesic of space-time, regardless of whether space-time be curved by the proximity of matter or whether it be flat. So long as the body is not interfered with, it will continue to follow its geodesic, allowing itself to be guided by the space-time structure or field. But suppose now that the motion is interfered with, so that the body is torn away from the geodesic which it would normally have followed. The guiding field will immediately react and oppose this foreign interference. It is as though the geodesic were a groove from which the body could not easily be diverted.
Consider then a body moving uniformly in interstellar space, according to the law of inertia. If we interfere with this body’s motion, accelerate it along its straight line or compel it to describe a curve, we shall, in effect, be tearing it away from its space-time geodesic. The guiding field will react and we shall experience a physical pull which we call a force of inertia or a centrifugal pull. In exactly the same way we may explain the appearance of the physical pull which we ascribe to gravitation when we stand on the earth’s surface.
Thus, a body which is falling freely towards the earth is following ageodesic in the curved space-time that surrounds the earth. But if, for some reason or other, the free fall of the body is interfered with, if, for instance, the rigidity of the earth’s crust prevents the body from falling farther along the geodesic, the geodesic will react and will strive to force the body through the earth’s crust. It was this force which we formerly ascribed to the pull of gravitation. But we see once again that this force has exactly the same origin as the forces of inertia we discussed previously, so that there exists no essential difference between a force of inertia and one of gravitation. They are both real physical forces in that they will produce physical effects; and they will both disappear in exactly the same way when we cease to interfere with the natural motion of the body. In short, they are both of them but manifestations of the four-dimensional metrical field or guiding field of space-time.