PART IIITHE GENERAL THEORY OF RELATIVITY
CHAPTER XXIIIPOTENTIALS AND FORCES
ACCORDING to classical science, force may be defined as the product of mass by acceleration. When no force is acting, the motion of a body is always Galilean; only when a force is acting will the motion of a body become accelerated.[76]
Now it must be noted that when a body is subjected to some definite muscular pull the magnitude of this pull or force depends solely on the muscular effort which we are willing or able to produce. On the other hand, when a body is situated in the gravitational field of the earth, the gravitational force which acts upon it and which is commonly known as the weight of the body not only depends on the mass of the earth but likewise varies with the mass of the body. In order to remedy this indeterminateness, it is usual to specify that the body on which the force is acting is one of unit mass; it is then called a test-body. Under these circumstances, we can explore a field of force by placing our test-body in successive regions of space and determining the magnitude and direction of the force which is acting on the body. When we have mapped out the magnitude and direction of the force for every point of space, we are in a position to state that we have determined the lay of thefield of force.
In physics we are often concerned with fields of force. The gravitational field surrounding the earth constitutes only one particular illustration. We may also consider the electric field surrounding an electron, or the magnetic field surrounding the poles of a magnet, or even combinations of these two fields, such as the electromagnetic fields produced by electric currents.
When the direction and magnitude of the force are the same throughout space and do not vary with time, the field of force is calleduniform. Uniform fields of force are very difficult to obtain; the simplest illustration we can think of would be that given by a river flowing with constant speed along a rectilinear course. The force exerted by the current would of course remain the same in magnitude and in direction at all points in the river; and for this reason the field of force produced by the current would constitute a uniform field.
Now, if we rest satisfied with this type of description of a field of force, we see that in the case of a uniform field, at any rate, there is no means of differentiating a region of the river or field that is upstream from one that is downstream. All parts of the river are identical. But it is obvious that this apparent identity between the different parts of the river is very superficial. It is a fact of common experience that whereas a boat will float downstream of its own accord, it will need the expenditure of a considerable amount of work to force it upstream again. Mathematicians were therefore compelled to take into consideration a new type of abstraction called apotential. Thanks to the introduction of this new concept, it became possible to differentiate between the various regions of the river (upstream or downstream) by stating that these regions differed in potential. The regions upstream were called the regions of higher potential, and the regions downstream were termed those of lower potential. Hence, it was possible to state, in the form of a general law, that bodies left to themselves would move of their own accord from regions of higher to regions of lower potential.[77]
So far we have been dealing with generalities, and it remains to be seen in what precise manner forces and potentials will be connected. The mathematical connection is expressed as follows:
Potentials are defined in such a way that the expression of the projection of the force along any given direction at any specific point of the field is given by the change in the magnitude of the potential when we pass from this point to the neighbouring point along this same direction. Just as an acceleration projected along a given direction is the time rate of change of the velocity projected along this same direction, so now a force is the space rate of change of the potential; and just as the acceleration is directed towards the increasing velocities, so a force is directed towards the lower potentials.
The existence of these potentials applies generally to all fields of force, regardless of whether these fields are of the uniform or non-uniform variety.[78]It is, as we have said, the variations in value of these potentials from point to point that are associated with the existence of forces. When this variation is uniform in one direction and no variation exists in directions at right angles, we are in the presence of a uniform field of force. When the variation is irregular, we have a non-uniform field of force. When the potential has the same constant value throughout space, so that it does not change in value from place to place, no forces can be present; there is then no field of force.
From this we see that a certain degree of indeterminateness surrounds the precise numerical value of a potential, since it is only its variation from place to place, and not its absolute value, that defines the force. This indeterminateness can be removed in the case of a non-uniform field of force by stipulating that in those regions of the field where the force vanishes (such as would be the case with the gravitational force at an infinite distance from matter), thepotential itself vanishes. A definite value having been attributed to the potential in one part of the field, we can determine its precise value from place to place in all other parts of the field.
Let us now give a few concrete illustrations of fields of force and of potentials in classical science. We have already mentioned the field of force surrounding matter. This was the gravitational or Newtonian field, and the potential at every point derived therefrom was called theNewtonian Potentialat the point. The distribution of the field of force was known in a perfectly definite manner when the potential distribution was known. In fact Newton’s law, which tells us how the field of force is distributed around matter, can also be expressed in an equivalent form byLaplace’s Equation, in which it is the potential distribution, and no longer the force distribution, that is described.
Let us now consider another type of field of force known as the inertial field. In perfectly free space, far from matter, there exists no field of force so long as we refer our observations to a Galilean frame. But if now we step into an accelerated or a rotating frame, we shall experience the effects of a field of force which we will ascribe to forces of inertia. Not only our own human bodies, but, in addition, all free bodies, will be subjected to the actions of these forces. Thus, consider a disk that is rotating, and a ball rolling without friction on the disk’s surface. If the ball is sent from the centre of the disk to its periphery it will of course follow a straight line at constant speed with respect to the earth or to any other Galilean frame. But then, with respect to the rotating disk itself, its course can no longer be rectilinear and uniform. Instead of following one of the radii of the rotating disk, it will follow a curve as though it had been pulled sideways by some force.
This result is general. Ifwe ourselvestried to advance along one of the radii of the rotating disk we should experience a real physical force pulling us sideways. To all intents and purposes, when we referred events to the rotating disk and not to the non-rotating earth (the rotation of the earth being so slow that we may neglect it as a first approximation), the physical existence of a field of force would have to be taken into consideration. The precise type of force we have mentioned is called theCoriolis force; in addition, there also exists another type of force, the better-knowncentrifugal force. Both these types of force are called forces of inertia. It is their ensemble which constitutes the field of inertial force existing in a rotating frame.
If, in place of a rotating frame, we had considered an accelerated railway compartment, the field of inertial force would have been disposed in a different manner. In the case of a train moving with constant acceleration along a straight line, the field of inertial force generated would have been of the uniform variety, disposed longitudinally through the train.
All these different illustrations show us that whereas, in a Galilean frame, no field of force exists, yet a field springs into existence automatically as soon as we place ourselves in any accelerated frame.It is for this reason that accelerated frames can be distinguished physically from Galilean frames; and it is owing to this generation of physical fields of force that accelerated motion must be regarded as absolute, whereas velocity, giving rise to no such fields, yields us no means of distinguishing one velocity from another, hence is relative.
Now we have seen that fields of force are accompanied by potential distributions. Hence it follows that, whereas, with respect to a Galilean frame, the inertial potential is zero at every point, or at least maintains some constant value, in the case of accelerated frames this inertial potential must vary from place to place in the frame. The magnitude of the force at any point of an accelerated frame is given, therefore, by a mathematical expression involving the variations in value of the potential in the neighbourhood of the point considered.
All we have explained so far pertains to classical science, but it appeared necessary to mention these results briefly before proceeding to a more systematic study of Einstein’s theory. We shall now see how all these results find a natural place in the space-time theory.
Consider a Galilean frame and an object moving freely in this frame. Its path will of course be rectilinear and its speed uniform, hence it will possess no acceleration, in accordance with Newton’s law of inertia. If we interpret these results in terms of space-time, we see that the world-line of the body is a straight line in space-time as referred to our mesh-system of equal four-dimensional cubes. If we now examine the motion of this same free body from some definite accelerated frame, all we have to do is to change our four-dimensional space-time mesh-system in an appropriate way. This change of mesh-system does not affect the world-line of the body, since this line remains a straight line or geodesic through flat space-time; but, on the other hand, it will certainly alter the appearance of the straight line when we refer our measurements to our curvilinear mesh-system. The erstwhile straight world-line will now appear curved in a definite way, and its precise mathematical equation as referred to our new curved mesh-system can be obtained without difficulty.
The physical significance of this apparent curvature of the world-line of the body will be that the body will now appear to possess acceleration with respect to our new non-Galilean frame. The precise magnitude of this acceleration will therefore be given by the apparent curvature of the world-line followed by the body, and this curvature is of course expressed implicitly by the equation of the world-line when referred to our curvilinear mesh-system. But for a given test-body the acceleration is related to the force acting on the body. From this it follows that the mathematical expression of the force can be obtained immediately.
This mathematical expression of the curved world-line, hence of the force of inertia at any particular point, is seen to be built up with the variations in value of the’s around this point. Were the’s to remain constant in value throughout, as would be the case in a Galilean frame, this mathematical expression would vanish invalue. It follows that the’s, of which this expression of the force is built, must correspond to the potentials of inertia. We have thus discovered the physical significance of the’s of space-time: they define potentials.
We see, indeed, that this identification is legitimate in every respect. Thus, in a Galilean frame, there are no forces of inertia, so that the potentials of inertia must be constants; and we know that in a mesh-system of equal four-dimensional cubes, which corresponds to a Galilean system, the’s are all constants and are given by,all other’s being zero. Again, in an accelerated frame, a field of inertial forces appears; hence the potential must vary from place to place; and we know that in a curvilinear mesh-system (corresponding to an accelerated frame) the’s lose their constant values and vary from place to place.
So far the reason for the existence of forces of inertia has been made apparent. They arise owing to the uneven spread ofnumbers which accompanies all curvilinear mesh-systems (accelerated frames).[79]When this occurs, the mathematical expression of the force of inertia assumes a definite numerical value at each point, whereas, when the’s are constants, as in a Cartesian mesh-system, this expression of the force maintains a zero value.
From this we see that forces of inertia arise from an attempt on our part to cut up space-time with curvilinear mesh-systems instead of Cartesian ones, just as they arise from our substitution of accelerated frames for Galilean ones. We cannot help but feel, however, that having proceeded thus far, a further generalisation is required. To be more explicit, it should be understood that the scheme of physics we have developed has compelled us to attribute a fundamental rôle to space-time. But Newton’s great law of universal attraction is expressed in terms of the separate space and time of classical science. If space-time is indeed as fundamental as Einstein has led us to believe, it appears incredible that Newton’s law should remain outside its scope. Yet this it certainly does, for if Newton’s law were a space-time law, it would preserve the same form in spite of any change in our space-time mesh-system. And this it fails to do.
Einstein’s next attempt was therefore to weld Newton’s law into the general fabric, a result he achieved about 1914. The mathematical generalisation which would allow this result to be obtained seems almost obvious to-day, since (as will be explained inChapter XXV) it reduces to assuming that in the neighbourhood of matter, four-dimensional space-time loses its flatness and becomes non-Euclidean or curved. However, it appears to have been through the medium of physical observation that Einstein was led to his superb generalisation, so we shall proceed to follow the historical order by explaining the significance of his postulate of equivalence.