Hi! I don’t know this stuff well. Does this sound right? In Newtonian Mechanics, an infinite, homogeneous universe should produce a net force of zero on every stellar body. So nothing moves. So a static universe is possible. It’s still an unstable situation (like “balancing the pencil on its point”), but it’s possible.
But in GR, gravity doesn’t push things around. It warps space, the same way that velocity does. Now if you’re accelerating in the +x direction, then length contracts along that direction. Space between massive bodies experiences the same effect as acceleration, so it contracts. So after a little while, all the stellar bodies find themselves closer together, even though it would seem that nothing was accelerated & nothing moved. If that’s correct, then the end result would be that the universe wants to contract, and the only way for the universe to avoid such a contradiction is escape velocity. Thus the line that GR space must be expanding or contracting. It could of course be static, but just for an instant.
In other words, in GR, the universe must tend toward contraction for the same reason that the rope breaks between the spaceships in the famous example. (Which I have never fully understood!)
I think ‘static’ here means static in reference to the metric of space. In big bang cosmology it’s assumed that the matter in the Universe behaves as a perfect fluid, but within this fluid individual particles are still moving around relative to each other (and separate from the expansion of space).
Newtonian physics implicitly assumes a static flat space, so there can’t be any metric expansion anyway whether particles are moving around and whether the Universe is homogeneous or not.
In general relativity there isn’t this assumption and if you assume that the Universe is homogeneous and isotropic that allows only 3 different kinds of geometries whose evolution is governed by an arbitrary positive function of time (which in turn is governed by the Einstein field equations).
The reason for the non-static nature of isotropic and homogeneous cosmology in GR is that the form of the Einstein field equations mean that the rate at which the Universe is expanding must always be changing if the density of the Universe is non-zero and the isotropic pressure of the matter in the Universe is non-negative. This means that for the Universe to be static in i. and h. cosmology the Universe must either be devoid of matter or there must be a dark energy-like negative pressure which exactly in the right proportion to the matter content of the Universe.
Space is not empty so we can rule that out and even if there were a negative pressure term to create a static Universe it is unstable, so small departures from isotropic and homogeneous Universe would cause a runaway-effect which caused the Universe to expand or contract.