What is the greatest interest in studying a flat metric of spacetime rather than one with a curved metric?
closed as unclear what you're asking by AccidentalFourierTransform, probably_someone, Chris♦, John Rennie, Jon Custer Jul 15 '18 at 1:06
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A flat metric is an asymptotic local representation of the curved metric (this fact in itself already offers an insight on some important qualities of spacetime). This representation is of a paramount importance, as it reveals fundamental properties of reality, such as the existence of anti-matter, the constancy of the local speed of light, and more. A flat metric is the basis for the quantumm field theory. Quantum electrodynamics is the most precise theory ever created by the humankind that matches observations to better than one part in a trillion. In contrast, the math of a curved metric is very complex and thus less productive. Even a two-body problem has no analytical solution in General Relativity. Thus the best approach is to take advantage of the insight offered by both, flat and curved metrics to the extent possible afforded by the analytical and numeric methods we have.
You could ask why is Newtonian gravity and flat geometry good? Why not GR and 4D geometry?
On the surface of Earth, there are a lot of problems that have a mathematical solution, that has solutions that are very good, and have little error in 2D or 3D in flat space. They are much easier to calculate.
Why would we use curved spacetime GR measurements, on these simple problems when the more complex math adds only very little to the precise answers?