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arXiv · 2608.03697

Gravitational Enstrophy: Local Geometric Origin and Inverse-Cascade Constraints

Luis Lehner
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Abstract

Two-dimensional fluids conserve energy and enstrophy, driving inverse energy cascades via Fjørtoft's argument. We show General Relativity admits an analogous structure: for linear radiative perturbations of Petrov type D backgrounds (Kerr, Kerr--AdS), the gravitational-wave energy $W = \sum_k W_k$ and magnetic Weyl enstrophy $\mathcal{Z} = \int B_{ab} B^{ab} \sqrtγ \, d^3x \approx \sum_k ω_k^2 W_k$ are approximately conserved in the zero-angular momentum frame, where vorticity coupling vanishes identically and curl exchange cancels mode-by-mode. This yields a gravitational Fjørtoft constraint implying nonlinear energy transfer proceeds preferentially toward lower frequencies. The constraint is dynamically active in near-extremal Kerr ($τ_{\text{damp}} \gg τ_{\text{nl}}$) and confined geometries (AdS), but suppressed in generic ringdown. In AdS, $\mathcal{Z}$ maps holographically to the boundary fluid enstrophy.

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Luis Lehner. 2026-08-05. Gravitational Enstrophy: Local Geometric Origin and Inverse-Cascade Constraints. https://arxiv.org/abs/2608.03697

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