Bound states in the continuum of gravitational waves

Abstract: Bound states in the continuum (BICs) are ubiquitous wave phenomena, but have not yet been demonstrated for gravitational waves (GWs). Here, in-plane periodic perturbations, exponentially localized at the plane $z = 0$, are shown to lead to distributional surface energy tensors at this plane and to be regular vacuum solutions ($T_{μν} = 0$) of the linearized Einstein field equations outside of it. These are achieved by explicitly calculating the Ricci tensor components and the Ricci scalar from the metric perturbation tensor. To fulfill each vacuum solution ($R_{{σν}<em>{(+, \times)}} = 0$ and $R</em>{{}<em>{(+, \times)}} = 0$), different surface polariton-like dispersions are required. These bound perturbations decay exponentially to a flat metric ($$h</em>{{BIC}_{(+,\times)}} \propto e^{- k_z |z|}$$), and each localized metric has a correspondence to a different planar GW polarization ($+,\times$). The Lorenz gauge-fulfilling solutions exist at the $Γ$ point in momentum space, dwelling within the continuum of wavevectors of propagating GWs. The strains in the unit cell of the periodic perturbations have opposite parities relative to the corresponding planar GWs under a $C_2$ in-plane rotation, making them incompatible by symmetry with their propagating counterpart, indicating localization via symmetry protection.