Shock-waves in the gravitational wave compatible Horndeski theories with linear kinetic term

Propagating shock-waves can be discussed in terms of junction conditions between space-time regions separated by a hypersurface. Recent observations of gravitational waves and their electromagnetic counterparts established that the former also propagate with the speed of light. Hence energetic gravitational waves could be perceived as shock-waves on null hypersurfaces. The most generic scalar-tensor theories with at most second order dynamics, the Horndeski-theories were severely constrained. We derive junction conditions across a null hypersurface for the subclass of allowed Horndeski-theories with linear kinetic term dependence, exploring a formalism based on a transverse null vector. We obtain a 2+1 decomposed generalised Lanczos equation, with the jump of the transverse curvature induced by both the distributional energy-momentum tensor of the wavefront of the shock-wave, and by the jump in the transverse derivative of the scalar. The surface density, current and pressure of the distributional light-like shock-wave and the transverse derivative of the scalar are also constrained by a scalar junction equation.