A classification of two-dimensional spacetimes with non-compact Cauchy surfaces

A new scheme for classifying spacetimes is given in two-dimensional cases. To be precise, we classify two-dimensional spacetimes with non-compact Cauchy surfaces up to causal isomorphisms. This presents a new ideal boundary in such a way that two spacetimes have the same ideal boundary if and only if two spacetimes are causally isomorphic. In other words, ideal boundaries determine completely its interior causal structure. In this process, we also show that the group of all causal automorphisms of two-dimensional spacetimes with non-compact Cauchy surfaces is a subgroup of the group of all causal automorphisms on R12.