Ill-posedness of a double-null free-evolution scheme for black hole spacetimes

We suggest that `free-evolution' integration schemes for the Einstein equations (that do not enforce constraints) may contain exponentially growing modes that render them useless in numerical integrations of black hole spacetimes, independently of how the equations are differenced. As an example we consider the evolution of Schwarzschild and Reissner - Nordström spacetimes in double-null coordinates.