. Theory of the relativity of motion . r be noted, moreover, that many im-portant results of the theory of relativity can be more easily obtainedif we do not try to employ this four-dimensional geometry. Thereader should also be on his guard against the fallacy of thinking thatextension in time is of the same nature as extension in space merelybecause intervals of space and time can both be represented byplotting along axes drawn on the same piece of paper. 174. Idea of a Time Axis. In order to grasp the method let usconsider a particle constrained to move along a single axis, say OX,and let u
. Theory of the relativity of motion . r be noted, moreover, that many im-portant results of the theory of relativity can be more easily obtainedif we do not try to employ this four-dimensional geometry. Thereader should also be on his guard against the fallacy of thinking thatextension in time is of the same nature as extension in space merelybecause intervals of space and time can both be represented byplotting along axes drawn on the same piece of paper. 174. Idea of a Time Axis. In order to grasp the method let usconsider a particle constrained to move along a single axis, say OX,and let us consider a time axis OT perpendicular to OX. Then theposition of the particle at any instant of time can be represented by apoint in the XT plane, and its motion as time progresses by a line inthe plane. If, for example, the particle were stationary, its behavior 188 Four Dimensional Analysis. 189 in time and space could be represented by a line parallel to the timeaxis OT as shown for example by the line ab in figure 16. A particle. Fig. 16. dx moving with the uniform velocity u — -jr could be represented by a straight line ac making an angle with the time axes, and the kine-matical behavior of an accelerated particle could be represented by acurved line. By conceiving of a /owr-dimensional space we can extend thismethod which we have just outlined to include motion parallel toall three space axes, and in accordance with the nomenclature ofMinkowski might call such a geometrical representation of the space-time manifold the world, and speak of the points and lines whichrepresent the instantaneous positions and the motions of particles as world-points and world-lines. 175. Non-Euclidean Character of the Space. It will be at onceevident that the graphical method of representing kinematical eventswhich is shown by Figure 16 still leaves something to be desired. Oneof the most important conclusions drawn from the theory of relativitywas the fact that it is impossible for a par
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