PeterDonis
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lucas_ said:Can you really assume SR, GR as approximately described by Newtonian mechanics, and then include higher-order non-Newtonian correction terms in a power series in ##\frac{1}{c^2}##?
For many purposes, yes. This is called the post-Newtonian approximation:
https://en.wikipedia.org/wiki/Post-Newtonian_expansion
https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism
A good discussion of this approach, and its "unreasonable effectiveness", is in this paper by Will:
https://arxiv.org/pdf/1102.5192.pdf
The "unreasonable effectiveness" part includes the fact that this method works pretty well for predicting the gravitational wave signatures from black hole mergers, even though a black hole is a case which, intuitively, should not be treatable by this method since the region at and inside the horizon cannot be viewed as simply being a perturbation of flat spacetime (which is the underlying rationale of the approach in the first place).
However, this method does not work for cosmology, which suggests that the real limitation of the approach is not being able to view gravity as a perturbation of flat spacetime, but being able to view it as associated with isolated massive objects separated by large regions of empty space. The universe, globally, cannot be described that way, because the universe as a whole is not an "isolated object" in this sense. So there is at least one domain in which viewing gravity as just Newtonian plus correction terms does not appear to work and a different approach is needed.