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.
