Exact Kerr–Schild Spacetimes from Linearized Kinetic Gravity Braiding

We generalize our recent work on k-essence sourcing Kerr–Schild spacetimes to the kinetic gravity braiding scalar field. For k-essence, in order for a perturbative Kerr–Schild-type solution to become exact, the k-essence Lagrangian must either be linear in the kinetic term (with the Kerr–Schild congruence autoparallel) or unrestricted, provided the scalar gradient along the congruence vanishes. A similar reasoning for the pure kinetic braiding contribution leads to either a vanishing Lagrangian or a scalar that is constant along the congruence. From the scalar dynamics we also derive an accompanying constraint. Finally, we discuss pp-waves, an example of Kerr–Schild spacetime generated by a constant k-essence along the Kerr–Schild congruence with a vanishing Lagrangian. This allows for the construction of a Fock-type space, consisting of a tower of Kerr–Schild maps first yielding a vacuum pp-wave from flat spacetime; next a k-essence-generated pp-wave from the vacuum pp-wave; and finally an arbitrary number of k-essence pp-waves with different retarded time-dependent metric functions.