I've seen questions and answers dealing with similar topics, but none that seem to provide what I'm looking for.
The Schwarzschild metric (and indeed any valid metric) should reduce to the Minkowski metric over a sufficiently small, linearized region. I am trying to do this mathematically by Taylor expanding the Schwarzschild metric terms, but struggling a bit with the math, specifically, what value of $r$ I should center it at, presumably not $0$ or $\inf$, maybe $1$? And what terms to neglect.
Can someone help me with this derivation, or at least tell me if I am on the right track?
Note: this is NOT the same as the Newtonian limit, where $r$ goes to infinity, because the locally Minkowski property should hold even at very high curvatures, including inside the horizon.