Dynamical space and gravitation within a purely geometrical framework: A new approach to gravitation

We state a purely geometrical framework apparently implementing Machian ideas on inertia. Only coupling constants dimensionless in natural units have been introduced in the theory. In anynonvacuum cosmos the field equations describing the gravitational phenomena in cosmological units turn out to be identical to Einstein's equations, with the Einstein gravitational coupling expressed in terms of the parameters defining the cosmological structure. This dependence, however, is not detectable. Indeed, such equations do not need to incorporate the standard Machian requirements (apart from the requirement that they are not conceivable in the total absence of matter) in order to be Machian, since, just on the basis of Mach's principle, one cannot expect to be able to detect Machian effects in Nature by using a system of units based on gravitational phenomena. On the contrary, the equations describing the gravitational field in local atomic units are Machian in the standard sense and, in particular, they incorporate the ideas that the frame has to be fixeddirectly in connection with the observed distribution and motion of matter and that there does not exist any kind of space-time in the total absence of matter. Finally, to reconcile, at least in the weak-field approximation, Einstein's equations (considered as equations describing the gravitational phenomena in local atomic units) with Mach's principle and to be in agreement with cosmological observations, we suggest that our cosmos be identified with a superuniverse model in which the background structure is homogeneous (in space and in time) and isotropic, while our universe is represented by one of the local inhomogeneities of the background. Then we prove that in any region of our universe in which the gravitational field issufficiently weak and smooth the equations, describing the gravitational field in local atomic units, are expected to approximate Einstein's equations all the better, the more the dimensions of our universe are negligible with respect to the dimensions of the background and the background curvature is small. As regards the experimental predictions of the present approach, any prediction for experiments involving only purely gravitational measurements is identical to that of Einstein's theory and the above result also guaranteesa fortiori the agreement with the available experimental data, also asnonpurely gravitational experiments are concerned.This paper appeared as Istituto Matematico L. Tonelli, preprint 78–10 (April 1978) (unpublished).