Abstract

It is argued that, under the assumption that the strong principle of equivalence holds, the theoretical realization of the Mach principle (in the version of the Mach-Einstein doctrine) and of the principle of general relativity are alternative programs. That means only the former or the latter can be realized—at least as long as only field equations of second order are considered. To demonstrate this we discuss two sufficiently wide classes of theories (Einstein-Grossmann and Einstein-Mayer theories, respectively) both embracing Einstein's theory of general relativity (GRT). GRT is shown to be just that “degenerate case” of the two classes which satisfies the principle of general relativity but not the Mach-Einstein doctrine; in all the other cases one finds an opposite situation.These considerations lead to an interesting “complementarity” between general relativity and Mach-Einstein doctine. In GRT, via Einstein's equations, the covariant and Lorentz-invariant Riemann-Einstein structure of the space-time defines the dynamics of matter: The symmetric matter tensor Ttk is given by variation of the Lorentz-invariant scalar densityL mat, and the dynamical equations satisfied by Tik result as a consequence of the Bianchi identities valid for the left-hand side of Einstein's equations. Otherwise, in all other cases, i.e., for the “Mach-Einstein theories” here under consideration, the matter determines the coordinate or reference systems via gravity. In Einstein-Grossmann theories using a holonomic representation of the space-time structure, the coordinates are determined up to affine (i.e. linear) transformations, and in Einstein-Mayer theories based on an anholonomic representation the reference systems (the tetrads) are specified up to global Lorentz transformations. The corresponding conditions on the coordinate and reference systems result from the postulate that the gravitational field is compatible with the strong equivalence of inertial and gravitational masses