Toward a Theory of Gravitational Wave Turbulence

General relativity describes the dynamics of gravitational waves, which can feature nonlinear interactions, such as those underlying turbulent processes. Theoretical and numerical explorations have demonstrated the existence of gravitational wave turbulence, of which a full and general mathematical description is currently not known. Here, we take essential steps towards such a theory. Leveraging a formulation exactly recasting general relativity as a set of nonlinear electrodynamics equations, we demonstrate that general relativity admits an Elsasser formulation -- the same type of equation underpinning magnetohydrodynamic turbulence. We further show that nonlinear interactions described by this equation are in part Alfvénic, linking gravitational wave turbulence to Alfvénic turbulence. Our work paves the way for a new understanding of nonlinear gravitational wave dynamics through insights from magnetohydrodynamics.