Page:The principle of relativity (1920).djvu/252

 (observing) system S´ have velocity +v with respect to the system S.

Then velocity of signal with respect to system S´ is given by W´ = (W - v)/(1 - Wv/c^2)

Thus "time" from A to B as measured in S´, is given by l/W´ = l(1 - Wv/c^2)/(W - v) = t´     (1)

Now if v is less than c, then W being greater than c (by hypothesis) W is greater than v, i.e., W > v.

Let W = c + μ and v = c - λ.

Then Wv = (c + μ)(c - λ) = c^2 + (μ + λ)c - μλ.

Now we can always choose v in such a way that Wv is greater than c^2, since Wv is >c^2 if (μ + λ)c - μλ is >0. that is, if μ + λ > μλ/c; which can always be satisfied by a suitable choice of λ.

Thus for W > c we can always choose λ in such a way as to make Wv > c^2, i.e., λ - Wv/c^2 negative. But W - v is always positive. Hence with W > c, we can always make t´, the time from A to B in equation (1) "negative." That is, the signal starting from A will reach B (as observed in system S´) in less than no time. Thus the effect will be perceived before the cause commences to act, i.e., the future will precede the past. Which is absurd. Hence we conclude that W > c is an impossibility, there can be no velocity greater than that of light.

It is conceptually possible to imagine velocities greater than that of light, but such velocities cannot occur in reality. Velocities greater than c, will not produce any effect. Causal effect of any physical type can never travel with a velocity greater than that of light.

[P. C. M.]