On the definition of distance in curved space, and the displacement of the spectral lines of distant sources

Abstract
The displacement of lines in the spectrum of a distant star, on the assumption that space-time is of constant curvature (the “de Sitter world”) has been extensively studied. The question, however, is not yet settled, different investigators reaching conclusions which are not apparently concordant. The origin of some of the discordances may be traced to the ambiguities which are involved in the use of the terms “ time,” “spatial distance” and “velocity,” when applied by an observer to an object which is remote from him in curved space-time. The “interval” which is defined by ds 2 = ∑ p, q g pq dx p dx q involves space and time blended together; and although any particular observer at any instant perceives in his immediate neighbourhood an “instantaneous three-dimensional space,” consisting of world-points which he regards as simultaneous, and within which the formulæ of the “restricted relativity theory” are valid, yet this space cannot be defined beyond his immediate neighbourhood; for with a general Riemannian metric, it is not possible to define simultaneity (with respect to a particular observer) over any finite extent of space-time. The concept of “spatial distance between two material particles” is, however, not really dependent on the concept of “simultaneity.” When the astronomer asserts that “the distance of the Andromeda nebula is a million light-years,” he is stating a relation between the world-point occupied by ourselves at the present instant and the world-point occupied by the Andromeda nebula at the instant when the light left it which arrives here now; that is, he is asserting a relation between two world-points such that a light-pulse, emitted by one, arrives at the other; or in geometrical language, between two world-points which lie on the same null geodesic. The spatial distance of two material particles in a general Riemannian space-time may, then, be thought of as a relation between two world-points which are on the same null geodesic . It is obviously right that “spatial distance” should exist only between two world-points which are on the same null geodesic; for it is only then that the particles at these points are in direct physical relation with each other. This statement brings out into sharp relief the contrast between “spatial distance” and the “interval” defined by ds 2 = ∑ p, q g pq dx p dx q ; for between two points on the same null geodesic the “interval” is always zero. Thus “ spatial distance ” exists when, and only when, the “interval” is zero.

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