Abstract
We investigate the influence of the chimney topology T×T×R of the Universe on the gravitational potential and force that are generated by point-like massive bodies. We obtain three distinct expressions for the solutions. One follows from Fourier expansion of delta functions into series using periodicity in two toroidal dimensions. The second one is the summation of solutions of the Helmholtz equation, for a source mass and its infinitely many images, which are in the form of Yukawa potentials. The third alternative solution for the potential is formulated via the Ewald sums method applied to Yukawa-type potentials. We show that, for the present Universe, the formulas involving plain summation of Yukawa potentials are preferable for computational purposes, as they require a smaller number of terms in the series to reach adequate precision.
Highlights
The shape of the space, whether it is positively curved, negatively curved, or flat, and whether there is a limit to the size of the Universe are all among essential topics of contemporary debate in theoretical physics and cosmology
We study the chimney topology T × T × R in terms of the gravitational characteristics of the Universe, manifested in the shape of the gravitational potential and force
As we show in the present work for the chimney topology, it becomes possible to obtain exact solutions of this equation that are nontrivial and physically meaningful
Summary
The shape of the space, whether it is positively curved, negatively curved, or flat, and whether there is a limit to the size of the Universe are all among essential topics of contemporary debate in theoretical physics and cosmology. Non- connected spacetimes are timelike geodesically incomplete, since they have singularities [1]. If the Universe is multiply connected, it may have a finite volume and yet be negatively curved or flat [2]. Spaces with toroidal topology in one to three dimensions may be presented as common examples of multiply connected spaces.
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