Cartesian Coordinates
Section Two
Physics makes use of the Cartesian coordinates:
... we see that it will be advantageous if, in the description of position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates. (p. 8)
Einstein says that "it will be advantageous if... it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions"; what exactly does he mean by this? The German original is as follows:
Aus dieser Ueberlegung sieht man, dass es fuer die Beschreibung von Orten vorteilhaft sein wird, wenn es gelingt, sich durch Verwendung von Messzahlen von der Existenz mit Namen versehener Merkpunkte auf dem staren Koerper, auf den sich die Ortsangabe bezieht, unabhaengig zu machen. Dies erreicht die messende Physik durch Anwendung des Kartesischen Koordinatensystems. (The Collected Papers of Albert Einstein, Vol. 6, doc. 42, 429)
He cannot mean that, in each Cartesian system of coordinates, we can specify the position of any point without depending on the existence of marked poisitions; for, you have to specify the origin of such a system, and otherwise you cannnot obtain the coordinate values! Presumably he is trying to say that the choice of such Cartesian coordinates does not affect the distance between any two points; the distance (and change of distance) is one of the essential quantities in mechanics. Notice that the (Cartesian) coordinate system can be chosen arbitrarily, since the resulting distance between any two points is not affected by this choice. For instance, the distance between the red and the blue point in the preceding figure is independent, no matter which coordinate system you may choose; <x, y, z> and <x', y', z'> may change, but not the distance between the two.
Now, the problem is: Can we really do physics, classical or relativistic, only in terms of relative distance (thus definable) and its derivatives? Although Einstein does not mention this basic problem, this is a rather hard question. A number of physicists, mathematicians, and philosophers attacked this problem. Huygens, Leibniz, Mach, Poincare, and most recently, Julian Barbour. See, for instance, The Machian Problem.
Last modified, April 18, 2002. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp
