Proof of Universe Expansion

First you need to make some assumptions - that the universe is homogeneous and isotropic.

Then the solution of Einstein's equations is a FRW metric, see for instance http://en.wikipedia.org/wiki/Friedmann-Lemaître-Robertson-Walker_metric

The expansion or contraction is governed by Friedman's equations

http://en.wikipedia.org/wiki/Friedmann_equations


If you add a cosmological constant to these equations, there is a static solution (the Einstein static universe), but it is unstable, and also requires some very special parameter values.

So you can't quite "prove" that the universe must be expanding or contracting (as there is the static Einstein solution), but it would require a lot of coincidences to have this happen if GR is correct.

Of course, observation strongly suggests that the universe is expanding.