I Solving General Relativity's Schwarchzild & FRW Metrics

[ChrisVer]

I had the following question
how are the Schwarchzild metric that describes a spherically symmetric matter distribution (such as a star) be compatible with the FRW metric that describes the 'overall universe' that the star resides in/is part of its matter distribution?
Then we say that FRW solution is not applicable in small distances because the Universe is no longer homogeneous and isotropic. How did this happen? we started from an homogeneous and isotropic universe and we ended up (because of the very small perturbations or inner interactions?) with structure formation and the "destruction" of what we started with?

 

[Nugatory]

I had the following question
how are the Schwarchzild metric that describes a spherically symmetric matter distribution (such as a star) be compatible with the FRW metric that describes the 'overall universe' that the star resides in/is part of its matter distribution?

"Compatible" just means that their domains of validity don't overlap, or if they do overlap they don't disagree.

There's only one spacetime for the entire universe and the metric that describes it is not any of the known exact solutions of the EFE. However, if we look at one little region around a star, we find that the Schwarzschild metric is a pretty good approximation in that region. It's not perfect because it predicts that the spacetime curvature goes to zero as we move further away from the star... but that's OK, we just don't use the solution there. For example, the Schwarzschild solution around our sun works pretty well for predicting the orbits of the planets, but we wouldn't expect it to describe conditions at a distance of 100,000,000 light-years away from the sun. Likewise, the cosmological metrics give good results at cosmological scales where local fluctuations average out, but aren't so good at local scales - modeling the Earth as a sphere works for astronomers and even pilots, but for ants, not so much.