On Apr 24, 2:11 pm, Dan <dan.ms.ch...@gmail.com> wrote:
> Space can be 'closed' , and also , 'locally flat' , in the sense that > the Riemann tensor vanishes ,or there exists ,for any point of the > space, a non-infinitesimal spherical section around that point that's > indistinguishable from flat space .
Let?s look at spacetime. According to GR, it is the curvature of spacetime that causes gravity. So, spacetime is curved around a gravitating mass. In free space, the Einstein tensor vanishes which means the Ricci tensor also vanishes which mean the Riemann tensor also vanishes. So, you have vanished Riemann tensor in curved spacetime. That means the curvature tensors really do not address the curvature thing. The field equations are merely differential equations that allow you solve the local geometry and nothing more. <shrug>
> The only metric I can think of where the traveler in a straight line > *must always* return to his starting point for every possible starting > point is some sort of n-sphere, though it would have to have > exceedingly small curvature to match experiments. It would be > interesting to find another metric where this is valid, though I doubt > another one exists .
Write down the metric for n-sphere please. <shrug>
> I for one think of the universe as infinite in extent, and mostly > flat, though my reasons tend to be philosophical : > Our world should be the most perfect, that is "the one which is at the > same time the simplest in hypothesis and the richest in > phenomena" (Leibniz) . And , like it or not, an infinite (and 'roughly > isotropic') universe matches these criteria.
At least, you admit your own version of cosmology is purely speculation. The so-called experts believe in their speculated ?reality? more whole heartedly. <shrug>