I understand that gravity in GR is a manifestation spacetime curvature dictated by the field equations by the principle that objects follow the geodesic path in spacetime.
And, I get how gravitational radiation travels like waves because there are wave equations in the field equations.
But, I have trouble reconciling gravitational waves with larger scale spacetime curvature.
If the earth suddenly disappeared, then the metric field would begin to flatten out and become a Minkowski Spacetime and this effect would travel away from where the earth is, presumably by gravitational waves.
Likewise, if a planet suddenly sprang into existence, a Schwarzchild Spacetime would emerge.
What I have trouble understanding and visualizing is how this effect is propagated and sustained by gravitational waves.
Should I picture the gravitational waves as constantly acting in the field to produce curvature of spacetime? If so, how do the gravitational waves around us in Schwarzchild spacetime differ from the gravitational waves in other spacetimes to produce another metric?
Or am I thinking about this the wrong way?
Would it be fruitful to attempt to understand this by analogy with the propagation of Electromagnetic fields?