  Pick an arbitrary timelike geodesic in an arbitrary spacetime. We demonstrate
that there is a particular limiting process, an "ultra-local limit", in which
the immediate neighborhood of the timelike geodesic can be "blown up" to yield
a general (typically non-diagonal) Bianchi type I spacetime. This process
shares 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.
