Use of the Flat-Space Metric in Einstein's Curved Universe, and the "Swiss-Cheese" Model of Space

Using a second metric tensor ${\gamma }_{\mu \nu }$ as proposed by Rosen, Gupta's supplementary condition for the gravitational field (which has the form of De Donder's coordinate condition) is written in general-covariant form. This supplementary condition appears to be of physical importance because of the use made of it by Gupta in the quantization of Einstein's gravitational field. This physical significance of the supplementary condition singles out a manifold of coordinate systems, which contains as sub-manifolds infinitely many metric spaces each allowing only "Lorentz transformations" leaving the ${\gamma }_{\mu \nu }$ constant. A particle is called "at rest" if it is not accelerated with respect to some of these "Lorentz" frames. Although the ${\gamma }_{\mu \nu }$-metric may be important in the formulation of the quantum theory of gravitons, it does not enter in the line element describing the results of physical measurements of time or distance, which are described by a line element containing as metric the gravitational tensor $g_{\mu \nu }$, so that space, flat with respect to hypothetical measurements by unrealistic rods keeping their $\gamma$-metric length on displacement, is actually found to be curved by physical measurements by realistic rods keeping their $g$-metric length on parallel displacement.