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§ 4. Equivalents of the Postulates.

The problem of determining sets of postulates logically equivalent to those which have been used as a basis of the theory of relativity is obviously important. So far as a justification of the theory is concerned it is, however, unnecessary to have complete logical equivalents; all that is essential is to find a set of postulates, experimentally demonstrable, by means of which it is possible to demonstrate the characteristic conclusions of relativity concerning the relations of units of time and units of length in two systems of reference.^[1] Such a set of postulates we shall call essential equivalents of the postulates of relativity. The object of this section is to determine essential equivalents of postulate R, that is, such postulates as may be taken in connection with postulates M, V, L, so that the new set shall be essentially equivalent to M, V, L, R.

For this purpose let us first consider the relation between the transverse mass of a moving body and its mass at rest as given in theorem I. Let us suppose that this theorem is true^[2] (whether proved experimentally or otherwise); and let us seek its consequences. Suppose that the experiment by means of which we proved theorem I. is now repeated. If we again assume the law of conservation of momentum and equate the two observed changes in momenta, it is clear that we shall have a relation between measurements of time as carried out in the two systems of reference, and that this relation will be precisely the same as in the usual theory of relativity. Having this relation concerning time units we can then proceed as in the first paragraph in § 3 to derive the usual relations between units of length. Hence we have the following result:

Theorem V. If $m_{0}$ and $t\left(m_{v}\right)$ have the same meaning as in theorem I. and if for any particular kind of matter whatever we have the relation

$t\left(m_{v}\right)={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}$

then this fact and postulates $\scriptstyle \left(MVLC_{1}\right)$ form an essential equivalent of postulates $\scriptstyle \left(MVLRC_{1}\right)$.

Next, let us suppose that for some particular kind of matter we have the relation

$t\left(m_{v}\right)=\left(1-\beta ^{2}\right)l\left(m_{v}\right)$

where $t\left(m_{v}\right)$ and $l\left(m_{v}\right)$ denote the transverse mass and the longitudinal mass, respectively, as above. Then repeat the experiments by means of

1. It is obvious that we should then be able to demonstrate theorems I. and II. concerning the mass of a moving body.
2. All that is essential to the argument is the truth of theorem I. for a particle of matter of some one kind; it need not be assumed to be true universally.