  Real time holography is studied in the context of the embedding space
formalism. In the first part of this paper, we present matching conditions for
on-shell integer spin fields when going from Euclidean to Lorentzian signature
on AdS background. Using the BTZ black hole as an example, we discuss various
ways of lifting the physical solution from the AdS surface to the whole
embedding space. The BTZ propagator for higher spin field is expressed
elegantly in terms of the embedding coordinates. In the second part of the
paper, we develop the proposed duality between higher spin theory and vector
models. We obtain a specific map between the field configurations of these two
theories in real time, so called Lorentzian AdS/CFT map. We conclude by
exploring the matching conditions for higher spin fields satisfying the
proposed bulk quadratic action. The physical and ghost modes can be treated
independently during the Wick rotation; only physical modes are considered to
be external modes.
