General covariance and the passive equations of physics (Q2892578)

After \textit{A. Einstein} had recognized in a 1927 paper with \textit{J. Grommer} [Sitzungsberichte Akad. Berlin 1927, 2--13 (1927; JFM 53.0817.03)] that the equation of motion for a point-like test particle, i.e., the geodesic equation, does need not be postulated independently of the gravitational field equations but can be derived from them, in 1938 he went a step further. Together with \textit{L. Infeld} and \textit{B. Hoffmann} he showed, in the so-called EIH paper [Ann. Math. (2) 39, 65--100 (1938; Zbl 0018.28103)], that the gravitational equations for empty spaces are sufficient to determine the motions of bodies by virtue of the gravitational field generated by them, where the bodies are represented as point singularities of the field.NEWLINENEWLINE In this paper, it is clarified that the general coordinate covariance of the field equations leads to their overdetermination and, thus, to the equations of motion for their sources, where the nonlinearity of the field equations is responsible for the existence of terms expressing the interaction of the moving bodies. The author of the present paper considers the covariance (but not the nonlinearity) aspect as the key idea of the EIH paper. Using Souriau's method, which extracted this aspect in modern mathematical language given by the theory of generalized functions, he shows that there is a fundamental connection between general covariance and that what he calls passive equations of physics. These passive equations describe the motion of a small object in the presence of a force field, where one ignores the effect induced by this small object. (However, regarding the nonlinearity of Einstein's equations, this effect was considered in the EIH paper for \(N\) gravitationally interacting particles.) In particular, the author derives the geodesic equation from the covariance with respect to the group of diffeomorphisms of compact support on a given manifold (acting on a metric) and the Schrödinger equation from the covariance with respect to the group of all invertible linear transformations of a finite-dimensional vector space (acting on a rank 1 operator).NEWLINENEWLINEFor the entire collection see [Zbl 1237.00021].