correspond to the time directions of the two different observers. Classical science, assuming time to be absolute, considered the two lines coincident. Relativity has proved the error of this belief.
PART III
THE GENERAL THEORY OF RELATIVITY
CHAPTER XXIII
POTENTIALS 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 the field 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 called uniform. 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 a potential. 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.