Eddington on Natural Motion
Chapter IV of Space, Time and Gravitation (1920)
What do you think is the point when Eddington discusses the problem of fields of force? The beginner may be confused, but the crucial discussion begins at page 69. In both the Newtonian mechanics and the special relativity, the natural motion was a uniform rectilinear motion, following the law of inertia (no force acting on it). You can find the essential link here, for understanding what is going on in the general relativity.
In the general relativity, we have no priviledged frame. Thus a natural motion may be linear or curved, depending on which frame you are taking for your description; hence we have to give up saying that "gravitational field (in the extended sense) exists when a test particle deviates from a straight track". But then, what is the criterion for natural motion? The answer is, in Wheeler's language, the principle of extremal aging; for this, see Principle of Extremal Aging. What Eddington is trying to get at is essentially the same. And if a test particle deviates from such a natural motion, then there is a gravitational field; the description of such a field is of course relative to a chosen frame, but the criterion for natural motion is independent from such a choice of frame.
Our attention is thus directed to the natural tracks of unconstrained bodies, which appear to be marked out in some absolute way in the four-dimensional world. There is no question of an observer here; the body takes the same course in the world whoever is watching it. Different observers will describe the track as straight, parabolical, or sinuous, but it is the same absolute locus. (70)
We are now able to state formally our proposed law of motion---Every particle moves so as to take the track of greatest interval-length between two events, except in so far as it is disturbed by impacts of other particles or electrical forces. (72)
Determining a longest path (or a shortest path, in some cases) is the task of geometry, but this geometry is not necessarily Euclidean. Thus the quest for an adequate description of gravitational field came this way:
presence of field ª¬ reference to natural motion ª¬ principle of extremal aging ª¬ geometry
The final expression for gravitational fields, or the curvature of spacetime, will be given in terms of Riemann curvature tensor (Riemann curvature is invariant, independent of a choice of a frame), and this plays an essential role for deriving Einstein's field equations. For a brief review of this, see The Parable of the Apple.
Last modified, June 21, 2002. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp