The Role of the Apparent Horizon in the Evolution of Robinson-Trautman   %%EINSTEIN-Maxwell Spacetimes
  The `runaway solutions' of the Lorentz-Dirac equation of a charged particle interacting with its own field in classical electrodynamics are well-known. This type of self accelerated phenomena also exists in the solutions of the Einstein-Maxwell equations in general relativity. In particular, runaway solutions occur in a class of simple models known as the `Asymptotically Flat Robinson-Trautman Einstein-Maxwell' (AFRTEM) spacetimes. Consequently these spacetimes cannot evolve to their unique regular steady state, viz. a charged non-rotating black hole. This seems to contradict the established results that charged non-rotating black holes are stable under first order perturbations. We show that if an AFRTEM spacetime also possesses an apparent horizon, then it has a Lyapunov functional. This suggests that the evolution equations with additional constraints arising from the apparent horizon would evolve stably to a charged non-rotating black hole. We also demonstrate that the linearised equations of these restricted spacetimes are stable and the exponentially growing dipole modes, which give rise to self accelerated motions in classical electrodynamics are also eliminated. 
Anthony W.C.Lun and Ernest W.M.Chow