Page:CarmichealPostulates.djvu/22

From these results it follows that the law of the parallelogram of velocities is only approximate. This conclusion of the theory of relativity has given rise, in the minds of some persons, to the most serious objections to the entire theory.

Suppose that both the velocities considered above are in the line of relative motion of S and $S'$. Then we have

$u={\frac {v+u}{1+{\frac {vu'}{c^{2}}}}}$

This equation gives rise to the following theorem:

Theorem VIII. If two velocities, each of which is less than c, are combined the resultant velocity is also less than c (MVLR).

To prove this we substitute in the preceding equation for v and $u'$ the values

$v=c-k,\ u'=c-l,$

where each of the numbers k and l is positive and less than c. Then the equation becomes

$u=c{\frac {2c-k-l}{2c-k-l+{\frac {kl}{c}}}}.$

The second member is evidently less than c. Hence the theorem.

If, however, either one (or both) of the velocities v and $u'$ is equal to c — and hence k or l (or both) is equal to zero — we see at once from the last equation that u = c. Hence, we have the following result:

Theorem IX. If a velocity c is compounded with a velocity equal to or less than c, the resultant velocity is c (MVLR).

Remark. — A conclusion of importance is implicitly involved in the results obtained in §§ 7-11. It can probably be seen in the simplest way by reference to the first two equations (A), these being nothing more nor less than an analytic formulation of theorems IV. and VI. If β is in absolute value greater than 1 — whence $1-\beta ^{2}$ is negative — the transformation of time coordinates from one system to the other gives an imaginary result for the time in one system if the time in the other system is real. Likewise, measurement of length in the direction of motion is imaginary in one system if it is real in the other. Both of these conclusions are absurd and hence the absolute value of β is equal to or less than 1. If it is one, then any length in one system, however short, would be measured in the other as infinite; and a like result holds for time. Hence β is less than 1. But $\beta =v/c$, the ratio of the relative velocity of the two systems to that of light. Hence, the velocity of light is a maximum