General relativity: is curvature of spacetime really required or just a convenient representation? I'm not really far into the general theory of relativity but already have an important question: are there formulations that can do without spacetime curvature and describe the general theory of relativity/all associated gravitational effects in global cartesian coordinates? Idea: Einstein chose spacetime curvature so that one had not to build gravitational effects into the rest of physics equations like Maxwell's equations. I assume that spacetime curvature provides a convenient way to apply physics laws not related to gravity unmodified locally because at small distances we may use special relativity as an approximation.
Please correct me everywhere I am wrong.
 A: The curvature of spacetime is a property of the spacetime manifold itself, it is not related to any particular choice of coordinates. The very essence of relativity lies in the Einstein field equations, which, colloquially, tell you that the energy-matter content of spacetime determines its curvature and metric. There is no formulation of general relativity known to me that avoids the idea of curved spacetime.
But this is not worrying - generally, such a geometric theory of physics is the kind we want. For classical mechanics, such a geometric picture is given by the Hamiltonian formalism and its symplectic manifolds. For gauge theories describing the other fundamental forces except for gravity, the geometric picture is given by the theory of principal bundles and the (e.g. electromagnetic) physical fields are amenable to be described as the curvature of such bundles (and wonderfully analogous to formulating GR with jet and frame bundles). For gravity, the geometric picture is that it is spacetime itself that has curvature. We don't even want to get rid of a description in terms of curvature, generally speaking.
Just because you can locally choose frames such that, at least at one point, the metric is flat, and hence you have SR, this is different from saying that the curvature is not required - a spacetime is flat only when there is a choice of coordinates such that the metric is flat everywhere.
A: There are formulations of classical general relativity that do not utilize the concept of the curvature of spacetime, which are equivalent to the traditional formulation that interprets gravity as the curvature of spacetime.
As with anything else, the framework that one chooses to do calculations in is a matter of circumstance, or philosophy, etc... sometimes both. But, history of science has shown time and time again, that it is well worth it to explore alternative formulations of theories that are known to be successful.
Quoted from the abstract of the paper above, "In these notes we discuss two alternative, though equivalent, formulations of General Relativity in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity, thus putting forward the existence of three seemingly unrelated representations of the same underlying theory."
This seems pretty clear that these operate in flat spacetime, i.e. as you say, "global cartesian coordinates." I can write a more thorough explanation if required by the OP.
A: In traditional GR, the possibility that spacetime can be curved is a fundamental requirement and moreover gravity is a fundamental force. However, in entropic gravity, it's argued that gravity is not fundamental and is an entropic effect and hence curvature is also not fundamental but an emergent property.
