Page:Eddington A. Space Time and Gravitation. 1920.djvu/187

XI] depends on the route, and no definite meaning can be attached to the interval between them without specifying a route, yet in the limit there is a definite small interval between $$P$$ and $$Q$$ when they are sufficiently close together.

To understand the meaning of these new coefficients $$\kappa$$ let us briefly recapitulate what we understand by the $$g$$'s. Primarily they are quantities derived from experimental measurements of intervals, and describe the geometry of the space and time partitions which the observer has chosen. As consequential properties they describe the field of force, gravitational, centrifugal, etc., with which he perceives himself surrounded. They relate to the particular mesh-system of the observer; and by altering his mesh-system, he can alter their values, though not entirely at will. From their values can be deduced intrinsic properties of the world—the kind of space-time in which the phenomena occur. Further they satisfy a definite condition—the law of gravitation—so that not all mathematically possible space-times and not all arbitrary values of the $$g$$'s are such as can occur in nature.

All this applies equally to the $$\kappa$$'s, if we substitute gauge-system for mesh-system, and some at present unknown force for gravitation. They can theoretically be determined by interval-measurement; but they will be more conspicuously manifested to the observer through their consequential property of describing some kind of field of force surrounding him. The $$\kappa$$'s refer to the arbitrary gauge-system of the observer; but he cannot by altering his gauge-system alter their values entirely at will. Intrinsic properties of the world are contained in their values, unaffected by any change of gauge-system. Further we may expect that they will have to satisfy some law corresponding to the law of gravitation, so that not all arbitrary values of the $$\kappa$$'s are such as can occur in nature.

It is evident that the $$\kappa$$'s must refer to some type of phenomenon which has not hitherto appeared in our discussion; and the obvious suggestion is that they refer to the electromagnetic field. This hypothesis is strengthened when we recall that the electromagnetic field is, in fact, specified at every point by the values of four quantities, viz. the three components of electromagnetic vector potential, and the scalar potential of