Multiple connected topology and gravitation

[humanino]

I can not figure out what is wrong with this : suppose one deals with a multiple connected universe, such as a torus. In order to make it simple, let us imagine we consider two very massive objects in this topology, say two well separated clusters of galaxies whose distances are large compared to their spatial extension. There are several paths from one to the other cluster in the multiple connected torus. So, does the gravitational influence of one to the other proceeds through all the different paths ?

I guess one first has the global topology given, with an average metric overall, and then individual movements can only locally and slightly affect the curvature. Especially, the time required for a signal to achieve the smallest closed path might correspond to the time-life of this universe. I thought it is exactly the case for a spherical universe, and maybe there are deeper reasons for it to hold true in a general, arbitrary configurations. I never heard of such a result, and have been unable to find more information by "googling" or "arXiving" it.

Let me go further to make it clear : suppose both clusters are located at largest distance possible on this torus, at antipodal points or "diametrically opposed". Independent of the expansion of this torus, this is an unstable equilibrium configuration. If for any reason this mutual position changes, it should result in a dramatically divergent collapse. Should it not ?

 

[hellfire]

Nice thoughts.

I guess one first has the global topology given, with an average metric overall, and then individual movements can only locally and slightly affect the curvature.

I think this is correct, if one assumes isotropy. I assume that otherwise the topology may evolve and this may happen depending on matter/energy flows and ‘local’ concentrations.

Let me go further to make it clear : suppose both clusters are located at largest distance possible on this torus, at antipodal points or "diametrically opposed". Independent of the expansion of this torus, this is an unstable equilibrium configuration. If for any reason this mutual position changes, it should result in a dramatically divergent collapse. Should it not ?

This is not necessarily correct if you do not assume that space is static. If space is dynamic and there is enough expansion rate (you may imagine an extreme situation in which the scale factor goes to infinity in a finite time -- some kind of big-rip), both clusters may not collapse, even the galaxies inside them will recede from each other after enough time.