  We show that the equivalence problem for three-dimensional Lorentzian
manifolds requires at most the fifth covariant derivative of the curvature
tensor. We prove that this bound is sharp by exhibiting a class of 3D
Lorentzian manifolds which realize this bound. The analysis is based on a
three-dimensional analogue of the Newman-Pen-rose formalism, and spinorial
classification of the three-dimensional Ricci tensor.
