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When I started learning about GR I wondered if it emerged from SR (which the name suggests) or if the connection between the two is mere technical. GR describes the behaviour of the metric of space-time, which is locally Minkowskian and therefore SR applies.

But is a curvature-based theory of gravity possible where the metric is locally Euclidean, i.e. the speed of light is not constant and space and time are essentially uncoupled?

Or is there a closer relation between GR and SR that I am missing?