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Pick an arbitrary timelike geodesic in an arbitrary spacetime. We demonstratethat there is a particular limiting process, an "ultra-local limit", in whichthe immediate neighborhood of the timelike geodesic can be "blown up" to yielda general (typically non-diagonal) Bianchi type I spacetime. This processshares some (but definitely not all) of the features of the Penrose limit,whereby the immediate neighborhood of an arbitrary null geodesic is "blown up"to yield a pp-wave as a limit.

Any spacetime has a Bianchi type I spacetime as a limit