Causal structure of general relativistic spacetimes
  We present some of the recent results and open questions on the causality problem in General Relativity. The concept of singularity is intimately connected with future trapped surface and inner event horizon formation. We offer a brief overview of the Hawking Penrose singularity theorems and discuss a few open problems concerning the future Cauchy development (domain of dependence), breakdown criteria and energy conditions for the horizon stability. A key question is whether causality violating regions, generating a Cauchy horizon are allowed.   We raise several questions concerning the invisibility and stability of closed trapped surfaces from future null infinity and derive the imprisonment conditions. We provide an new perspective of the causal boundaries and spacelike conformal boundary extensions for time oriented Lorentzian manifolds and more exotic settings. 
E. M. Howard
gr-qc