Green's function for Klein-Gordno equation in curved spacetime

[paweld]

Is it possible to define unambiguously retarded and advanced Green's function
in spacetime without timelike Killing vector. Most often e.g. retarded Green
function G_R(t,\vec{x},t',\vec{x}') is defined to be 0 unless t'<t
but maybe one can express this condition using only casual structure
(without time coordinate)?

[Stingray]

Is it possible to define unambiguously retarded and advanced Green's function
in spacetime without timelike Killing vector. Most often e.g. retarded Green
function G_R(t,\vec{x},t',\vec{x}') is defined to be 0 unless t'<t
but maybe one can express this condition using only casual structure
(without time coordinate)?

Yes, this can be done using the causal structure. You don't need a Killing vector to decide if two points are in the past or future of each other (or are spacelike-separated). A good practical introduction to Green's functions in curved spacetimes may be found in http://relativity.livingreviews.org/Articles/lrr-2004-6/.