Time-dependent, spherically symmetric background in Kaluza-Klein compactified Horndeski theory and the speed of gravity waves

We revisit the models recently derived from a Kaluza-Klein compactification of higher dimensional Horndeski theory, where the resulting electromagnetic sector features non-trivial couplings to the Horndeski scalar. In particular, this class of theories admits the electromagnetic waves propagating at non-unit speed, which in turn allows to relax the constraints on Horndeski theories following from the multi-messenger speed test. In this work, we prove that both the gravitational wave and its electromagnetic counterpart propagate at the same, though non-unit, speed above an arbitrarily time-dependent, spherically symmetric background within the theories in question. Hence, we support the statement that several subclasses of Horndeski theories are not necessarily ruled out after the GW170817 event provided the photon-Galileon couplings are allowed. We also formulate a set of stability conditions for an arbitrary solution within the discussed theoretical setting.