670 IX. ON EMPIRICAL AND VERBAL STRUCTURES |
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the light signals propagate themselves to the right and to the left with equal velocities. Hence we can represent them by straight lines equally inclined to the
X axis. These lines are called 'light-lines'.The points
A', B' which are the intersection of the 'world-lines' of the points A and B with the light-lines-give us the 'times' at which the signals arrive. It follows from the drawing that A'B' is parallel to the X axis, which means that A' and B' are 'simultaneous' (equal 'times').Let us now take another case in which our points
A, B, C, move uniformly with an equal velocity (Fig. 3). Their world-lines will also be parallel to each other but inclined to the axis. In the drawing the light-lines will be represented by similar lines but their intersections with the world-lines of A and B will not be on a parallel to the X axis, and so they will not be simultaneous.We should notice that an observer who moved with the system in the direction
OX' would be perfectly entitled to claim that A' and B' are simultaneous to him. His co-ordinate system would be OX'V, in which the points A' and B' are on a parallel to his X' axis as he is at rest in his system OX'T'. The world-lines A, B, C, are parallel to the T' axis because the points are supposed to be at rest in this system and hence the x's have equal values for all t's. An important point should be noticed; namely, that we have only one space-time and that the indefinitely numerous ways different observers partition their 'space' and 'time' represent merely the indefinitely many ways in which it can be partitioned. If we keep the whole of it under consideration we see that we cannot divide it into 'space' and 'time', as any subdivision has both aspects.^{4}The Minkowski method of representation makes the change in our measurements of length, as given by the Lorentz-Einstein transformation, very obvious. A measuring rod is not purely a 'spatial' configuration, as in the actual world such a thing does not exist, but it is a space-time configuration.
Every point of the rod exists at each moment of 'time'. We see that in space-time we cannot represent our rod as a segment on the
X axis but must represent it structurally as a strip in the XT plane. We assume here for simplicity that the rod is one-dimensional (Fig. 4).A rod which is at rest in a system is represented by a strip parallel to the
T axis. If it is moving, its strip is inclined to the Taxis. The 'contraction' does |
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