The Vector Meson Field and Projective Relativity

In the projective theory of relativity, if, instead of the projective metric $G_{\alpha \beta }$ of index $2N$, one uses the restricted metric ${\gamma }_{\alpha \beta }\left(\equiv \frac{G_{\alpha \beta }}{G_{00}}\right)$ of index zero, one is led to a formal unification of the gravitational and electromagnetic fields of the general theory of relativity. The present paper discusses the case of the general metric $G_{\alpha \beta }$. It is shown that a natural four-dimensional, gauge invariant variational principle, involving the curvature scalar of $G_{\alpha \beta }$, yields field equations unifying the gravitational and vector meson fields, the range of the meson force being determined by the inverse length ${\mathrm{}\left(12\right)\mathrm{}}^{\frac{1}{2}}N$. Reasons are given for supposing that an extension to conformal geometry could prove of interest.