# Non-contracting closed space GR?

1. Feb 16, 2010

### Xenoman

I have an idea regarding general relativity in closed space, my claim is that general relativity in closed space (such as a 3-sphere) is non-contracting. I here post three ways to arrive at the same conclusion, what is the error in it? (Or how do I prove it rigorously)

Poissons equation for gravity in classical mechanics (Nabla^2 \Phi = 4 \pi G \rho) does not hold if the universe is a sphere (although perhaps not reasonable in Newtonian Mechanics), instead one has to take in to account the potential \Phi from the "other direction radially on the sphere" which amounts to a revised Poisson's equation in the spheric situation. If one builds GR on this one gets a different GR where terms cancel and the Friedmann solution of such a space is stationary.

From the Einstein-Hilbert action it is possible to derive Einstein field equations given that certain terms vanish at infinity (Take "Symmetry Transformation, the Einstein-Hilbert Action, and Gauge Invariance" p. 16 eqn (33) as example if you wish: http://web.mit.edu/edbert/GR/gr5.pdf). In a closed spheric homogeneus space this does not seem true, but rather some terms should remain because Gauss's law does not apply here (due it is spheric). Possibly the extra terms could be akin to \Lambda g_{\mu \nu}?

When deriving the Einstein field equations from energy conservation laws (http://en.wikipedia.org/wiki/Einstein_field_equations) then one gets the Einstein tensor covariantly differentiated and set to zero. To get the complete field equations they are integrated (and integration constants are \Lambda g_{\mu \nu} and T_{\mu \nu}). But if the space over which it is integrated is a closed sphere, then is there not an integration constant present due to that? (one akin to \Lambda g_{\mu \nu}?)

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