  We initiate a comprehensive study of a set of solutions of topologically
massive gravity known as null warped anti-de Sitter spacetimes. These are
pp-wave extensions of three-dimensional anti-de Sitter space. We first perform
a careful analysis of the linearized stability of black holes in these
spacetimes. We find two qualitatively different types of solutions to the
linearized equations of motion: the first set has an exponential time
dependence, the second - a polynomial time dependence. The solutions polynomial
in time induce severe pathologies and moreover survive at the non-linear level.
In order to make sense of these geometries, it is thus crucial to impose
appropriate boundary conditions. We argue that there exists a consistent set of
boundary conditions that allows us to reject the above pathological modes from
the physical spectrum. The asymptotic symmetry group associated to these
boundary conditions consists of a centrally-extended Virasoro algebra. Using
this central charge we can account for the entropy of the black holes via
Cardy's formula. Finally, we note that the black hole spectrum is chiral and
prove a Birkoff theorem showing that there are no other stationary axisymmetric
black holes with the specified asymptotics. We extend most of the analysis to a
larger family of pp-wave black holes which are related to Schr\"odinger
spacetimes with critical exponent z.
