I am having trouble understanding how to obtain a spacelike slicing of the Schwarchild black hole. I understand there is not a globally well defined timelike killing vector, so we can define t=cte slices outside the horizon and r=cte slices inside the horizon.
In the literature people define connector slices that join these two spacelike surfaces.
What is the formal definition of a spacelike connector slice? What is the most practical way to go about finding its mathematical expression?