. Theory of the relativity of motion . ificance of the lines we draw, conventions which are funda-mentally not so very unlike the conventions by which we interpret assolid, a figure drawn in ordinary perspective. Consider for example the diagram shown in figure 18, where wehave drawn a pair of perpendicular axes, OXh and OX4, and thetwo unit hyperbolae given by the equations Xi2 — X42 = 1, (303) Xi2 — x2 = — 1, together with their asymptotes, OA and OB, given by the equation xx2 - x42 = 0. (304) This purely Euclidean figure permits, as a matter of fact, a fairlysatisfactory representation of t
. Theory of the relativity of motion . ificance of the lines we draw, conventions which are funda-mentally not so very unlike the conventions by which we interpret assolid, a figure drawn in ordinary perspective. Consider for example the diagram shown in figure 18, where wehave drawn a pair of perpendicular axes, OXh and OX4, and thetwo unit hyperbolae given by the equations Xi2 — X42 = 1, (303) Xi2 — x2 = — 1, together with their asymptotes, OA and OB, given by the equation xx2 - x42 = 0. (304) This purely Euclidean figure permits, as a matter of fact, a fairlysatisfactory representation of the non-Euclidean properties of themanifold with which we have been dealing. Four Dimensional Analysis. 209 OXi and OX4 may be considered as perpendicular axes in thenon-Euclidean XiOX4 plane. Radius vectors lying in the quadrantAOB, will have a greater component along the X4 than along the X\axis and hence will be 5-vectors with the magnitude s = \£42 — x{2,where X\ and #4 are the coordinates of the terminal of the vector. X4. Fig. 18. 7-radius-vectors will lie in the quadrant BOC and will have the mag-nitude s = Vxi2 — #42. Radius vectors lying along the asymptotes OA and OB will have zero magnitudes (s = yxx2 — x±2hence will be singular vectors. Since the two hyperbolae have the equations Xi2 — x42 0) and ■- 1 and Xi2 — x42 = - 1, rays such as Oa, Oa, Ob, etc., starting from theorigin and terminating on the hyperbolae, will all have unit we may consider the hyperbolae as representing unit pseudo-circles in our non-Euclidean plane and consider the rays as repre-senting the radii of these pseudo-circles. A non-Euclidean rotation of axes will then be represented bychanging from the axes OXi and OX4 to OXi and OX4r, and takingOa and Ob as unit distances along the axes instead of Oa and Ob. 15 210 Chapter Thirteen. It is easy to show, as a matter of fact, that such a change of axesand units does correspond to the Lorentz transformation. Let X\an
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