Causal structures of pp-waves
  We discuss the causal structure of pp-wave spacetimes using the ideal point construction outlined by Geroch, Kronheimer, and Penrose. This generalizes the recent work of Marolf and Ross, who considered similar issues for plane wave spacetimes. We address the question regarding the dimension of the causal boundary for certain specific pp-wave backgrounds. In particular, we demonstrate that the pp-wave spacetime which gives rise to the N = 2 sine-Gordon string world-sheet theory is geodesically complete and has a one-dimensional causal boundary. 
Veronika E. Hubeny and Mukund Rangamani