Compatibility equations, generalized Cesaro’s equations and expansion effects of spacetime

Abstract

A kinematic model of 4D continuum with different types of symmetry is introduced and a unified kinematic model of long-range electromagnetic and gravitational fields is proposed. We define the 4D vector potential, introduce 4D distortion tensor, and show that its antisymmetric part gives the classical definition of the electric field intensity vector and the pseudovector of magnetic induction. On the other hand, the symmetric part of the distortion tensor is interpreted as the gravitational field intensity. The structure of the gravitational field kinematic model is analyzed. For the 4D continuum, generalized compatibility equations are established including homogeneous Papkovich and Saint-Venant relations. It is shown that homogeneous Saint-Venant relations can be also integrated in quadrature’s respect to the 4D spherical deformation tensor, which can be defined through an integro-differential operator applied only to the components of the deviator tensor. We show that 4D Cesaro equations indicate the existence of two kinematic states in the spacetime in the absence of the strain deviator tensor. The first kinematic state proves the existence of an expansion metric effect of the 4D continuum since the speed of the observed point is always proportional to the distance to it. The second kinematic state indicates a purely geometric effect of uniformly accelerated expansion of event space.