Second-Order Effective Geometry of de Broglie Gravitational Waves

We investigate the non-linear dynamics of de Broglie gravitational waves within a post-Minkowskian framework. Starting from the perturbative solution proposed by Feoli and Scarpetta, we derive the second-order gravitational field generated by the wave’s self-interaction. Owing to the extremely high oscillation frequency of de Broglie gravitational waves, we employ the Isaacson/Misner–Thorne–Wheeler averaging procedure to obtain an effective coarse-grained geometry. The resulting averaged theory takes the form of an effective linear perturbation theory at second order in the wave amplitude. Near the symmetry axis, the resulting spacetime exhibits a non-vanishing static curvature generated entirely by non-linear effects. We show that geodesics in this effective geometry undergo bounded transverse oscillations governed by a harmonic equation. Possible implications for de Broglie’s wave–particle framework are also discussed.