The ABC of Relativity by Bertrand Russell
Author:Bertrand Russell
Language: eng
Format: mobi, epub
ISBN: 9781329946170
Publisher: Lulu.com
Published: 2016-03-03T03:00:00+00:00
Each of these postulates requires some explanation. Our first postulate requires that, if two events are close together (but not necessarily otherwise), there is an interval between them which can be calculated from the differences between their co-ordinates by some such formula as we considered in the preceding chapter. That is to say, we take the squares and products of the differences of co-ordinates, we multiply them by suitable amounts (which in general will vary from place to place), and we add the results together. The sum obtained is the square of the interval. We do not assume in advance that we know the amounts by which the squares and products must be multiplied; this is going to be discovered by observing physical phenomena. But we do know, because mathematics shows it to be so, that within any small region of space-time we can choose the co-ordinates so that the interval has almost exactly the special form which we found in the special theory of relativity. It is not necessary for the application of the special theory to a limited region that there should be no gravitation in the region; it is enough if the intensity of gravitation is practically the same throughout the region. This enables us to apply the special theory within any small region. How small it will have to be, depends upon the neighbourhood. On the surface of the earth, it would have to be small enough for the curvature of the earth to be negligible. In the spaces between the planets, it need only be small enough for the attraction of the sun and the planets to be sensibly constant throughout the region.
In the spaces between the stars it might be enormous – say half the distance from one star to the next – without introducing measurable inaccuracies.
Thus, at a great distance from gravitating matter, we can so choose our co-ordinates as to obtain very nearly a Euclidean space; this is really only another way of saying that the special theory of relativity applies. In the neighbourhood of matter, although we can still make our space very nearly Euclidean in a very small region, we cannot do so throughout any region within which gravitation varies sensibly - at least, if we do, we shall have to abandon the view expressed in the second postulate, that bodies moving under gravitational forces only move on geodesies.
We saw that a geodesic on a surface is the shortest line that can be drawn on the surface from one point to another; for example, on the earth the geodesies are great circles. When we come to space-time, the mathematics is the same, but the verbal explanations have to be rather different. In the general theory of relativity, it is only neighbouring events that have a definite interval, independently of the route by which we travel from one to the other. The interval between distant events depends upon the route pursued, and has to be calculated by dividing the route into a number of little bits and adding up the intervals for the various little bits.
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