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 𝒵=∫ BabBab√γ d3x. Its transverse radiativ e part carries an additional frequency-squared weight relative to gravitational-wave energy: δ𝒵k≃ωk2Wk in the radiation zone, with W=8πG EGW. 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↔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 AdS4, 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.
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BibTeX
@misc{lehner2026gravitational,
title = {{Gravitational Enstrophy: Local Geometric Origin and Energy-Cascade Constraints}},
author = {Luis Lehner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.03697},
eprint = {2608.03697},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03697},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Gravitational Enstrophy: Local Geometric Origin and Energy-Cascade Constraints
AU - Luis Lehner
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.03697
UR - https://arxiv.org/abs/2608.03697
AB - 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 𝒵=∫ B_abB^ab√γ d^3x. Its transverse radiativ e part carries an additional frequency-squared weight relative to gravitational-wave energy: δ𝒵_k≃ω_k^2W_k in the radiation zone, with W=8πG E_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↔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.
ER -