III. The General Theory of Relativity
We have seen that the first or "mechanical" theory of relativity was built up on the notion of inertial systems as deduced from the law of inertia; the "special" theory of relativity was built up on the universal significance and invariance of
, the velocity of light in vacuo; the third or general form of relativity is to be founded on the principle of the equality of inertial and gravitational mass and in contradistinction to the other two is to hold not only for systems moving uniformly and rectilinearly with respect to one another, but for all systems whatever their motion; i.e. physical laws are to preserve their form for any arbitrary transformation of the variables from one system to another.
Mass enters into the formulæ of the older physics in two forms: (1) Force = inertial mass multiplied by the acceleration. (2) Force = gravitational mass multiplied by the intensity of the field of gravitation; or,
Observation tells us that for a given field of gravitation the acceleration is independent of the nature and state of a body; this means that the proportionality between the two characteristic masses (inertial and gravitational) must be the same for all bodies. By a suitable choice of units we can make the factor of proportionality unity, i.e.
.