The Local Validity of Special Relativity, Part 1: Geometry

In this two-part essay, we distinguish several senses in which general relativity has been regarded as "locally special relativistic". Here, in Part 1, we focus on senses in which a relativistic spacetime has been said to be "locally (approximately) Minkowskian". After critiquing several proposals in the literature, we present a result capturing a substantive sense in which every relativistic spacetime is locally approximately Minkowskian. We then show that Minkowski spacetime is not distinguished in this result: every relativistic spacetime is locally approximately every other spacetime in the same sense. In Part 2, we will consider "locally specially relativistic" matter theories.
Comment: 32 pages. Part 1 of 2