Is there a formulation or theory of classical relativistic gravity yielding the same predictions as the standard General Relativity (when the predictions are expressed in GR-free language which presumably always can be done) but formulated or written following standard non-gravitational interactions, e.g., such as based on mathematical structures built on top of basic causal structure such as Minkowski space (besides the obvious embedding of the pseudo-Riemannian manifold in the space)?
Or, any set of assertions compatible with the prediction of GR as we understand them today (again, when expressed in a GR-free and purely physical language including background independence) necessarily leads to GR and the identification of gravity with spacetime?
Best Answer
This issue is discussed somewhat in Steven Weinstein's essay "Naïve Quantum Gravity", published in Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity (eds. Callender & Huggett, 2001). He begins by noting that
So it's perhaps more elegant to view GR as a theory of curved spacetime; but could we get away with thinking about it as a weird non-linear field theory on Minkowski spacetime anyhow? It's certainly possible in some circumstances; famously, Steven Weinberg's 1972 book on general relativity tries to eschew geometric thinking as much as possible, viewing GR as a field theory that has the Equivalence Principle as a fundamental principle and showing how object such as the metric and curvature tensors can be thought of as arising from this principle. It seems to me that this is not far off from the OP's idea in the second paragraph that "any set of assertions compatible with the prediction of GR as we understand them today (again, when expressed in a GR-free and purely physical language including background independence) necessarily leads to GR and the identification of gravity with spacetime", though Weinberg (in 1972) might have disputed the last part.
However, Weinstein notes that there are still a few problems with this approach:
* Weinstein includes a footnote here which refers the reader to an "interesting philosophical analysis of this line of thinking" in Reichenbach's The Philosophy of Space and Time (Eng. translation), (1958 [1927]).