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By P. W Bridgman

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2) where in the second version we have introduced components parallel and perpendicular to the relative velocity between the inertial systems. 9 Velocity Addition In considering the possibility that the velocities are not proportional we enter the domain where all three space dimensions become important. We already handled this situation before, using ordinary Euclidean vector algebra. But now some warnings will be appropriate. If one wants to interpret the formal scalar product v w geometrically, one has to observe that the component triple v resp.

5. 1) Consider the world line of an arbitrarily (but subluminally, as it can be used for signalling and may be at rest in some system) moving mass point (Fig. 11). According to sect. 3 it has to remain inside the light cone of each of its points. For each such point there is an inertial system I that is instantaneously comoving with the mass, its instantaneous rest system. (With three space dimensions, this does not yet specify a frame since the directions of the spatial axes would remain unspecified, but this will not concern us here.

The figure shows two clocks, 1 and 2, at rest at the origins of I and I, so that their world lines are just the t- and f-axis, respectively. x ~__________________~X Fig. 12. Time dilation The unit point on the f-axis corresponds to the event 'clock 2 shows f = 1'. This event obviously occurs at a time t > 1 in I, so that the moving clock goes slow as judged from I. But this is also the case for clock 1 as judged from I, where we have, according to eq. , fordx = O. 4) Time dilation is a reciprocal effect: judged from every inertial system, the clocks of any other go slow.

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A sophisticate's primer of relativity by P. W Bridgman

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