The Dynamics of a Particle in General Relativity

Some applications of the Gaussian integral transformation to the conservation law of the matter tensor yield the complete dynamical law for a material particle in a Riemannian space, in the form of two relations which correspond completely to Newton's two fundamental laws. The "moving force" of the motion law comes out as a volume integral which can be transformed into a surface integral. The inertial mass and the coordinates of the center of mass, on the other hand, are genuine volume integrals which cannot be transformed into boundary integrals. For weak and static matter the principle of the geodesic line can be established, irrespective of spherical symmetry. But even for static and infinitesimal matter an interesting exception occurs if the total mass of the particle happens to drop down to a quantity of second order, as is indeed the case if the scalar condition $T=0\mathrm{}$ is satisfied inside of the particle. Then the acceleration is smaller than it should be according to the geodesic principle which has the consequence that the gravitational mass of the sun, determined in the customary manner by the acceleration of its planets, may be underestimated. This gives a new clue for the understanding of the anomalous value of the light deflection on the limb of the sun, discovered by Freundlich and his collaborators. However, the theoretical prediction of an increase of 83 percent, deduced for a static and spherically symmetric particle, is too large in comparison with the 25-40 percent increase, indicated by Freundlich's measurements.