Eddington on Kinds of Space and Field Equations
Chapter V of Space, Time and Gravitation (1920)
Einstein's explanation of field equations, and of the spacetime of general relativity may be too short to be intelligible to ordinary people. Arthur Eddington is known as a good writer, and his exposition in this regard seems better than Einstein's own, although Eddington's is a bit more technical. Let us begin with Eddington's explanation of coordinate systems, which he calls "mesh-systems".
Consider the following four systems of coordinates:
As you already know, the choice of a coordinate system is free in general relativity. Since in any of the four systems two dimensional space is the subject matter, we may use x and y for two axes; in some of them, these variables may be understood in terms of degrees, rather than in terms of distances.
Now, what is important in any of them is the expression for the line element, dsds; this expression may be called a metric, for the sake of brevity:
Coefficients in front of variables, otherwise expressed as g_ik, are a metric. Here, Eddington's point is that although (1)-(4) look all different, there is an essential difference between the first three and the last. The first three all express a flat space, and this fact can be expressed in terms of a differential equation; it gives the criterion of identity of space (or of gravitational field, Einstein may say) in question. The metric of (4) on the other hand cannot be reduced to any of them. And the point of general covariance is to keep this distinction intact, through coordinate transformations.
Now, when Einstein talks about classical mechanics and special relativity, he almost exclusively uses a coordinate system of type (1); but the fact is, even in classical mechanics and special relativity, it is quite all right to use other types of coordinate system, and different expressions of metric appears in them. The reader should not be misled by this; the apparent difference of metric does not necessarily mean the difference of geometry. The geometries of (1) and (4) are indeed different, but this requires another criterion, other than the difference of their metrics! We can indeed provide such a criterion, in terms of a differential equation containing g's, as Eddington points out. Einstein of course knew this, but he seldom mentioned this in his popular expositions of general relativity (see, e.g., p. 175 of Relativity: the special and the general theory, where he refers to "Riemann condition").
The same point can be seen more intuitively, if you ask "Can this triangle--on an Euclidean plane--be transferred to another coordinate sytem, without changing its size?" If the answer is "Yes", then that coordinate system expresses the same kind of space as the Euclidean space; if "No", as is the case with a spherical surface, then the kind of space is different.
With this preparation, it may become easier to go to the field equations. Eddington now poses the question: Can every possible kind of space-time occur in an empty region in nature? (86). The answer is "No". Only certain kinds of space-time can occur in an empty region in nature, and the law which determines what kinds can occur is the law of gravitation (86). This amounts to reducing the choice of metric g's, and Eddington says that we need a three-fold process.
(1) many sets of values can be dismissed because they can never occur in nature,
(2) others, while possible, do not relate to the kind of space-time present in the problem considered,
(3) of those which remain, one set of values relates to the particular mesh-system that has been chosen. (86)
Thus the first problem is to inquire the criterion for (1).
In solving this problem Einstein had only two clues to guide him.
(1) Since it is a question of whether the kind of space-time is possible, the criterion must refer to those properties of the g's which distinguish different kinds of space-time, not to those which distinguish different kinds of mesh-system in the same space-time. [This is the condition of general covariance]
(2) We know that flat space-time can occur in nature (at great distances from all gravitating matter). Hence the criterion must be satisfied by any values of the g's belonging to flat space-time.
It is remarkable that these slender clues are sufficient to indicate almost uniquely a particular law. (86)
For more specific arguments for arriving at field equations, go through the rest of Eddington's chapter. Elementary portion of tensor calculus is used, but it is readable, and the reader can grasp the intuitive significance of tensor and conditions expressed in terms of tensor.
I wish to add a note here, for uninitiated readers. Eddington asked "what kinds of spacetime in an empty region in nature?" A beginner may wonder, "Why, of course a flat spacetime!" No! An empty region of spacetime may be affected by an adjacent region. Suppose there is a spherical, heavy star. Its exterior space is certainly an "empty region" in nature; but since the star generates a gravitational field, this empty region (the exterior of the star) is not a flat spacetime at all, and its geometry is curved. As you may have already heard of, this is the subject of Schwarzschild solution of the field equations, and its geometry is called Schwarzschild Geometry. Thus the law of gravity, expressed by field equations, must allow such possibilities.
For instance, consider the equatorial plane of the sun, a plane cutting across the sun at the equator.
This is a two-dimensional surface, but it is curved, and its curvature may be visualized by embedding it in a 3-dimensional (flat) space. This sort of diagram is called an embedding diagram (polar coordinates--which, by themselves, have no physical meaning--are drawn in order to help your imagination). The white bowl at bottom is the interior of the sun, and it should be neglected here. The point is merely to show (visualize) the curvature of the exterior equatorial plane, according to the Schwarzschild geometry, and the curvature is exaggerated. It should be noted that only the surface of this 2-dimensional curved space has physical meaning (you may see a third dimension, but it has no physical meaning). What is essential in this diagram is the metrical property of the curved surface.
For more on this, see Schwarzschild Geometry.
Last modified, August 8, 2002. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp
