## The impossible in mathematical form

While the condition that the speed of a light signal is the same relative to all observers may be difficult to accept from the point of view of physics and even more difficult on the basis of 'common sense', there is no problem in expressing it in mathematical form. We simply define two frames, S and S', and express the speed of light in each frame in terms of its respective coordinates.

Whether or not the idea of relative time agrees with our intuition, each frame of reference has not only its own space coordinates but also its own time coordinates as illustrated in Figure 15.8.

Speed of light in frame S (= c) Speed of light in frame S' (also = c)

fi X

If the speed of light is the same in the two frames of reference, then x2 + y2 + z2 - c2t2 = x'2 + y '2 + z'2 - c2t'2 (15.3)

Note that space and time are treated on the same footing. However, since its square is negative, t is imaginary and also has different units. (See Chapter 15.7.4.)

The observer on earth uses the frame of reference S, while the astronaut uses S'. Assume that the origins of the two frames in Figure 15.8 coincide at t = t' = 0. A light source at the common origin sends out a light signal at that instant, which spreads out in all directions (like an enormous, rapidly growing, spherical balloon) from the origin of each frame of reference.

This is where Einstein's second postulate comes in, and things become counter-intuitive. Each observer has his own opinion and his own point of view. Even when the frames become separated, each says that the light is spreading out from the origin of his frame. Each says that the radius of the 'sphere of light' equals the speed c multiplied by the time elapsed in his frame of reference. The counter-intuitive part is that we cannot use a 'universal clock', but must assume a different rate of time flow in the two frames. Another part, not easy to accept, is that the two are equally right.

The astronaut, using the clock in his spacecraft (frame of reference S'), obtains the same result for the speed of light as the observer on earth using his own clock in frame S. The earth observer says that the photon has travelled a distance

^x2 + y2 + z2 in a time t while the astronaut measures the distance as ^x/2 + y'2 + z'2 and the time as t'. They agree that the

speed of the photon is equal to c, but they do not agree on the values of the distances or of the times. To the earth observer, the marked distance over which the astronaut is timing the light appears to have 'shrunk', and the astronaut's clock appears to be slower than his own. However, the ratio of the distance travelled by the light to the time taken remains the same and equals c.

Time dilation is the key to the resolution of the paradox. To 'solve the impossible' we must wipe out any preconceived ideas about time, and introduce a dramatic new concept. We must assume that time is not the same when measured in the two frames of reference. Newton's concept of absolute time, 'which, by its nature, flows without reference to anything external', has to be abandoned.

Time is relative: The logical consequence of the relativity postulate

The timing apparatus used by the earthman and by the astronaut may be the same, and both may be working correctly, but each is located in a different 'time frame'. Time flows at a different rate in their respective frames of reference. There is nothing to say that one is 'right' and the other is 'wrong'. There is no central arbiter, and no universal clock. The astronaut's biological clock, his pulse, and his sense of time, will be related to the passage of time in his own frame of reference.

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