  We consider the operator product expansion (OPE) structure of scalar primary
operators in a generic Lorentzian CFT and its dual description in a
gravitational theory with one extra dimension. The OPE can be decomposed into
certain bi-local operators transforming as the irreducible representations
under conformal group, called the OPE blocks. We show the OPE block is given by
integrating a higher spin field along a geodesic in the Lorentzian AdS
space-time when the two operators are space-like separated. When the two
operators are time-like separated however, we find the OPE block has a peculiar
representation where the dual gravitational theory is not defined on the AdS
space-time but on a hyperboloid with an additional time coordinate and
Minkowski space-time on its boundary. This differs from the surface Witten
diagram proposal for the time-like OPE block, but in two dimensions we
reproduce it consistently using a kinematical duality between a pair of
time-like separated points and space-like ones.
