Gravitation in the Space with Chimney Topology

Abstract

Searching for possible indicators of spatial topology of the Universe in the Cosmic Microwave Background data, one recognizes a quite promising interpretation which suggests that the shape of the space manifests itself in the form of anomalies in the large angular scale observations, such as the quadrupole and octopole alignment. Motivated by the presumptive existence of such a tempting connection, we study the chimney topology, T×T×R, which belongs to the class of toroidal topologies with a preferred direction. The infinite axis in this case may be attributed to the preferred axis of the aforementioned quadrupole and octopole alignment. We investigate the gravitational aspects of such a configuration. Namely, we reveal the form of the gravitational potential sourced by point-like massive bodies. Starting from the perturbed Einstein equations, which ensure the proper demonstration of relativistic effects, one can derive the Helmholtz equation for the scalar perturbation (gravitational potential). Through distinct alternative methods, we present the physically meaningful nontrivial exact solutions of this equation. Our approach excludes any presumptions regarding the spatial distribution of gravitating sources. We show that the particular solution that appears in the form of summed Yukawa potentials is indeed very convenient for the use in numerical calculations, in the sense that it provides the desired accuracy with fewer terms in the series.