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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?

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I think the analogy with the EM field is a good one with some provisos e.g. gravitational waves require quadrapolar or higher oscillation (gravitational dipoles don't radiate) and gravitational waves are self interacting. The electric potential round a point charge doesn't need continual EM waves to sustain it, and likewise the curvature round the Earth doesn't need continual gravity waves to sustain it. For for both gravity and electromagnetism waves are only required to propagate changes in the fields, not static fields.

Planets, or any form of mass/energy, can't spring in and out of existance any more than electric charges can spring in and out of existance, so it's not useful to use this as an analogy. However if you take any randomly shaped electric field you can decompose it as a Fourier series to get the combination of EM waves needed to (momentarily) create it. I guess the same would be true of gravitational waves, at least in the weak field approximation.

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Thank you. Then I suppose our spacetime consists of a Schwarzschild part and an almost undetectable part due to volatile events at great distances and the two aren't related. –  MadScientist Jun 4 '12 at 16:21
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That's certainly a good description of the solar system at present. If you consider two black holes merging then the separate gravity wells merge into a single Schwarzschild well and radiate intense gravity waves as they do so. The gravity waves carry off all the multipoles leaving behind a Kerr geometry. Is this the sort of thing you were thinking of? –  John Rennie Jun 4 '12 at 18:35
    
That process is an example of what I was thinking of. Mine were artificial examples. But, then again, a large comet could theoretically destroy earth and the Schwarzchild Metric would be disturbed. My example is a Schwarzchild Spacetime flattening out into a Minkowski one which is much less interesting than your example. I'll research the process you've described. –  MadScientist Jun 5 '12 at 14:02

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