Regarding Einstein's static universe John Baez explains in The Meaning of Einstein's Equation(adsbygoogle = window.adsbygoogle || []).push({});

To see this, consider a small ball of test particles, initially at rest relative to each other, that is moving with respect to the matter in the universe. In the 13 local rest frame of such a ball, the right-hand side of equation (2) is nonzero. For one thing, the pressure due to the matter no longer vanishes. Remember that pressure is the flux of momentum. In the frame of our moving sphere, matter is flowing by. Also, the energy density goes up, both because the matter has kinetic energy in this frame and because of Lorentz contraction. The end result, as the reader can verify, is that the right-hand side of equation (2) is negative for such a moving sphere. In short, although a stationary ball of test particles remains unchanged in the Einstein static universe, our moving ball shrinks.

So in this universe the ball of test particles shrinks perpendicular to the direction of motion, which I think means that geodesics converge accordingly. Now, in this spacetime the spatial geometry is spherical.

Can someone explain what happens to said ball and how works geodesic deviation in case of a static spacetime with flat spatial geometry instead, like the 3-torus?

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# I Geodesic deviation in static spacetime

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