ON THE 'WORLD* OF MINKOWSKI 669 |
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The expression
ct, where c is the velocity of light, 300,000 kilometres per second, gives us the distance travelled by light in the 'time' t. It is natural to regard the velocity of light, which is a constant and translates easily into the language of length, as a unit of 'time'. In the Minkowski world it is customary, because of its convenience, to regard 1 second as the equivalent of 300,000 kilometres and measure lengths or 'times' in seconds or kilometres indiscriminately.Let us imagine a scale graduated in kilometres, and clocks whose faces are also graduated in kilometres (1/300,000 of a second). If the clocks are set correctly and we look at them from A the sum of the reading of any clock and the scale division beside it is one for all because the scale reading gives the correction for the 'time' taken by light, travelling with unit velocity, to reach A.
If we lay the scale in line with the two events and note the clock and scale readings,
h and xi, of the first event, and the corresponding readings, h and *2, of the second event, thenwhere 5 represents the 'interval'mentioned above.
If we set the scale moving in the direction AB then the divisions would have advanced to meet the second event and the difference
(xwould be smaller. But this is _{t} xi) compensated, because (h h) also becomes altered. When A is advancing to meet the light coming from any of the clocks on the scale the light arrives too quickly, and the reading of the clock appears smaller.The net result is, roughly, that it does not matter what uniform motion is given to the scale, the final results for the interval s are always equal.
^{3}We can now understand the vital importance of the minus sign with the 'time' co-ordinate. In fact, if in our equations all the signs were plus, using the 'space' and 'time' of one observer, one value of s would be obtained; but using the 'space' and 'time' of another observer, a different value would be obtained. With the minus sign for the 'time' co-ordinate, we see that we can have values of
s which are equal for all observers. If the distances increase, the 'time' element increases also, and so the difference may not be changed, but with the positive sign this would not be the case.We see that the interval s represents something which concerns only the events under consideration. The corresponding entity in ordinary geometry is
distance, which is independent of the accidental choice of co-ordinates. The minus sign makes the geometry of space-time non-euclidean.To familiarize ourselves with what has been already explained about simultaneity and the geometry of space-time, we will work it out once more, but now by the Minkowski method.
It will be enough to use two dimensions, one represented on the X axis, the other on the T axis. Let us consider three points A, B, C, at rest in our system O on the X axis (Fig. 2). In our space-time they will be represented by three parallels to the T axis. Let C be midway between A and B so that A C = CB. Let us assume that light signals are sent in both directions from C at the moment t =0. We assume that the system is 'at rest', which means that |
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