Page:Eddington A. Space Time and Gravitation. 1920.djvu/226

by halving the gauge. Therefore for action-density we must take an expression which will be diminished 16-fold by halving the gauge. Now $G$ is proportional to $1/^{2}$, where $R$ is the radius of curvature, and so is diminished 4-fold. The invariant $B_{\mu \nu \sigma }^{\rho }B_{\rho }^{\mu \nu \sigma }$ has the same gauge-dimensions as $G^{2}$; and hence when integrated through a volume gives a pure number independent of the gauge. In Weyl's theory this is only the gravitational part of the complete invariant $\left(B_{\mu \nu \sigma }^{\rho }-{\tfrac {1}{2}}g_{\mu }^{\rho }F_{\nu \sigma }\right)\left(B_{\rho }^{\mu \nu \sigma }-{\tfrac {1}{2}}g_{\rho }^{\mu }F^{\nu \sigma }\right)$, which reduces to $B_{\mu \nu \sigma }^{\rho }B_{\rho }^{\mu \nu \sigma }+F_{\nu \sigma }F^{\nu \sigma }$.

The second term gives actually the well-known expression for the action-density of the electromagnetic field, and this evidently strengthens the identification of this invariant with action-density.

Einstein's theory, on the other hand, creates a difficulty here, because although there may be action in an electromagnetic field without electrons, the curvature is zero.

Historical Note

Before the Michelson-Morley experiment the question had been widely discussed whether the aether in and near the earth was carried along by the earth in its motion, or whether it slipped through the interstices between the atoms. Astronomical aberration pointed decidedly to a stagnant aether; but the experiments of Arago and Fizeau on the effect of motion of transparent media on the velocity of light in those media, suggested a partial convection of the aether in such cases. These experiments were first-order experiments, i.e they depended on the ratio of the velocity of the transparent body to the velocity of light. The Michelson-Morley experiment is the first example of an experiment delicate enough to detect second-order effects, depending on the square of the above ratio; the result, that no current of aether past terrestrial objects could be detected, appeared favourable to the view that the aether must be convected by the earth. The difficulty of reconciling this with astronomical aberration was recognised.