Gravitational Enstrophy: Local Geometric Origin and Energy-Cascade Constraints

Abstract

Two-dimensional incompressible fluids conserve energy and enstrophy in the inviscid limit. Fjørtoft's argument uses this pair to constrain nonlinear energy transfer. We examine the corresponding question in General Relativity using the magnetic Weyl functional $\mathcal{Z}=\int B_{ab}{B}^{ab}\sqrt{\gamma }{d}^{3}x$. Its transverse radiativ e part carries an additional frequency-squared weight relative to gravitational-wave energy: $\delta {\mathcal{Z}}_k\simeq {\omega }_k^2{W}_k$ in the radiation zone, with $W=8\pi G{E}_{\mathrm{GW}}$. We derive its balance law from the Bianchi identities, obtaining the algebraic cancellations, bulk sources and boundary fluxes. Conservation of positive e nergy and curvature norm through nonlinear transfer gives a Fjørtoft constraint when their weights are ordered. Resonant weak-wave interactions generally change the c urvature norm; $2\leftrightarrow 2$ dynamics instead conserves energy and wave action. We examine the competition between transfer and linear losses in Kerr and AdS backg rounds, where long-lived or confined modes provide time for nonlinear evolution. In the hydrodynamic regime of AdS${}_4$, a fluid-weighted magnetic Weyl functional inherit s fluid enstrophy conservation in the inviscid limit. We compute its proportionality coefficient to boundary enstrophy in closed form at leading gradient order, giving a bulk diagnostic for holographic turbulence.

Updated discussion includes an additional relevant quantity mediating energy-cascade behavior; added references and sharpen some estimates. 38 pages (1/2 are appendices for specific details), 3 figures