Geodesics in the closed accelerating University

Abstract

The need to introduce in the current cosmological models, dark energy and dark matter raises the question of revision of the basic concepts of the structure of the Universe. In particular, the accelerated expansion of the Universe can significantly affect the observed motion of free bodies. Conclusions about the dynamics of remote astrophysical systems may not be correct if they are based on an incorrect interpretation of the observed nature of the motion. In view of this, it seems relevant to study the general laws of motion in an accelerating expanding space. The present work is devoted to the study of the nature of the motion of test particles in a maximally symmetrical, closed, accelerating Universe. As a concrete model, we study the anti-de Sitter
space, that is, a space with positive scalar curvature. More specifically, as a model, a four-dimensional onesheeted hyperboloid is embedded in a flat five-dimensional enveloping space characterized by the Minkowski metric. The orientation of the hyperboloid is along the time axis. This choice of geometry allows you to maintain the straightness of 0-geodesics, both in the enclosed and in the enclosing spaces. The Minkowski metric of the ambient space induces a metric on the hypersurface of a hyperboloid that simulates an accelerated expanding spatially closed Universe. The pseudo-Cartesian coordinate system is introduced for saving formal equality of space dimensions of the space-time. The basic geometrical characteristics such as the metrics and
the connection are calculated in the coordinate system. The system of geodesic equations was solved for partial case of hyperbolic space-time. The choice supplies the possibility to investigate the common properties of spatial accelerating University. It is shown that number of faster-than-light particles (tachyons) has to be decreasing in the model due to annihilation processing.