  We consider superconformal and supersymmetric field theories on
four-dimensional Lorentzian curved space-times, and their five-dimensional
holographic duals. As in the Euclidean signature case, preserved supersymmetry
for a superconformal theory is equivalent to the existence of a charged
conformal Killing spinor. Differently from the Euclidean case, we show that the
existence of such spinors is equivalent to the existence of a null conformal
Killing vector. For a supersymmetric field theory with an R-symmetry, this
vector field is further restricted to be Killing. We demonstrate how these
results agree with the existing classification of supersymmetric solutions of
minimal gauged supergravity in five dimensions.
