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We consider some classes of Horndeski theories in four dimensions for which acertain combination of the Einstein equations within a spherical ansatz splitsinto two distinct branches. Recently, for these theories, some integrabilityand compatibility conditions have been established which have made it possibleto obtain black hole solutions depending on a single integration constantidentified as the mass. Here, we will show that these compatibility conditionscan be generalized to accommodate a time dependence by promoting the constantmass to an arbitrary function of the retarded (advanced) time. As a directconsequence, we prove that all the static black hole solutions can be naturallypromoted to non static Vaidya-like solutions. We extend this study in arbitraryhigher dimensions where the pure gravity part is now described by the Lovelocktheory and, where the scalar field action enjoyed the conformal invariance. Forthese theories, the splitting in two branches is also effective, and we showthat their known static black hole solutions can as well be promoted toVaidya-like solutions.

From static to Vaidya solutions in scalar tensor theories