The General Relativity and Geometry
Third Assignment, due December 4 (Tue.)
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The modification to which the special theory of relativity has subjected the theory of space and time is indeed far-reaching, but one important point has remained unaffected. For the laws of geometry, even according to the special theory of relativity, are to be interpreted directly as laws relating to the possible relative positions of solid bodies at rest; and, in a more general way, the laws of kinematics are to be interpreted as laws which describe the relations of measuring bodies and clocks. To two selcted material points of a stationary rigid body there always corresponds a distance of quite definite length, which is independent of the locality and orientation of the body, and is also independent of the time. To two selected positions of the hands of a clock at rest relatively to the priviledged system of reference there always corresponds an interval of time of a definite length, which is independent of place and time. We shall soon see that the general theory of relativity cannot adhere to this simple physical interpretation of space and time. (Einstein 1916, 770)
In classical mechanics, as well as in the special theory of relativity, the coordinates of space and time have a direct physical meaning. To say that a point-event has the X_1 coordinate x_1 means that the projection of the point-event on the axis of X_1, determined by rigid rods and in accordance with the rules of Euclidean geometry, is obtained by measuring off a given rod (the unit length) x_1 times from the origin of coordinates along the axis of X_1. To say that a point-event has the X_4 coordinate x_4 = t, means that a standard clock, made to measure time in a definite unit period, and which is stationary relatively to the system of co-ordinates and practically coincident in space with the point-event, will have measured off x_4 = t periods at the occurrence of the event. (Einstein 1916, 773-4)
Einstein (1916), "Die Grundlage der allgemeinen Relativitaetstheorie", Annalen der Physik 49 (1916), 769-822. [English translation, Collected Papers vol. 6, Doc. 30, Princeton Univ. Press, 1997. ｭ�ｪwｰAｰCｰ羃Vｰｷｰ^ｰCｰ耒IｶW2ｪxﾑﾛｪAｶﾉﾚ棯B]
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Last modified December 6, 2001. (c) Soshichi Uchii