Analytic continuation of Spacetime Metrics

Compacted conformal infinity
by choosing advanced and retarded null coordinates v, w defined
by v = t + r, w = t — r , the second (spherical) metric becomes:
Minkowski spacetime
with its conformal infinity
structure imbedded in the
Einstein static universe
with two spatial
dimensions suppressed.
define p and q by tan p = v and tan q = w, introduce a
conformal factor and extend ds 2 to cover the whole manifold ->
Einstein static universe as the cylinder x 2 +y 2 = 1 imbedded in a
3-d Minkowski space.
conformally map Minkowski spacetime into a patch of the
cylinder; the boundary = conformal structure of infinity
two null surfaces
any future-directed timelike geodesic will originate from i - and terminate at i +
all null-like geodesics begin at past null infinity and end on future null infinity
all spacelike geodesics both begin and end at spacelike infinity i 0
the shaded region = conformal to the whole of Minkowski space-time