On Birkhoff's New Theory of Gravitation

It is pointed out in the first place: (1) in Birkhoff's gravitational theory based on "flat" space-time, the "red shift" is accounted for by the energy change of the photon as it travels from the emitting body, whereas the photon plays no especial role in the Einstein theory; (2) the solution of the problem of two or more bodies is feasible in the new theory because of its simpler character. Four comments of H. Weyl concerning the Birkhoff theory are discussed, and it is concluded that these are to be taken with much reserve. In regard to the third of these comments it is pointed out that the "perfect fluid" used by Birkhoff as the ultimate carrier of mass and electric charge is to be characterized as the simplest fluid with disturbance velocity that of light ($c$). It is affirmed to be a glaring defect of earlier relativistic theories that the disturbance velocity in matter has been taken as arbitrary, although that of gravitation and of the electromagnetic field have been equal to $c$. The differential equations of the theory are then set up. An additional cosmological term in the gravitational potentials $h_{\mathrm{ij}}$ is suggested, namely, $h_{\mathrm{ij}}^{*}=\left(\frac{K}{8}\right)(t^2-x^2-y^2-z^2)g_{\mathrm{ij}},$ where $x$, $y$, $z$, $t$ are Lorentz coordinates and $K$ is the (small) cosmological constant. The explicit formula for the rate of advance of periastron $P$ of two bodies (mass points) of masses $m$ and $m_2$ is given, as obtained from the solution of the two-body problem in the theory, and its possible application to double stars is referred to. The authors propose to give a detailed development of the theory and its applications in papers to be published shortly elsewhere.