I'm just starting to look into tensor analysis and general relativity. I have taken college level courses on Newtonian mechanics, electromagnetism, and special relativity that roughly follow the Feynman Lectures on Physics.
This Physics Stack Exchange answer states that
[spacetime] curvature allows and sometimes requires more and/or future curvature, just as a traveling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later. The vacuum allows curvature far from gravitational sources just as it allows electromagnetic waves far from electromagnetic sources. What electromagnetic sources allow is for electromagnetic fields to behave differently (namely to gain or lose energy as well as move in different ways and gain and lose momentum and stress). Similarly what gravitational sources do is allow curvature to react differently to itself than it otherwise would.
According to the answer:
what gravitational sources do is allow curvature to react differently to itself than it otherwise would
This raises my first question: How exactly does curvature normally react to itself / evolve without gravitational sources present? And how would we describe this mathematically?
My second question is a followup to the first:
When there are gravitational sources how do they change how curvature reacts to itself? And can this also be described mathematically?
The answer also says that:
curvature allows and sometimes requires more and/or future curvature, just as a traveling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later
My third question is: Is this describing the gravitational wave solution to the Einstein field equations? And how can I mathematically see how the gravitational wave propagates changes in spacetime curvature just like how electromagnetic waves propagate electric and magnetic fields by continuously producing them?