  We investigate the global structure of a recently discovered simple exact,
non-stationary solution of topologically massive and new massive gravity with
the asymptotic charges of an undeformed BTZ black hole. We establish the
existence of a timelike singularity in the causal structure of the spacetime
even in the absence of angular momentum. The dynamical trapping and event
horizons are determined and we investigate the evolution of the outer horizon
showing that it may increase or decrease with time, depending on the value of
the mass parameter. Finally, we test two proposals for dynamical entropy on
this solution, one of them depending on the Kodama vector. In addition we show
that the Kodama vector leads to the correct entropy for all stationary black
holes in 2+1 dimensions.
