Manuscript XI-E@1 · XI-E.tex · section · 46e55514

Variation from tensor-density covariance

Route 3: the cubic Einstein tree and its computed variation › Variation from tensor-density covariance · lines 1511–1521

For a contravariant rank-two density define

(L_{ξ}^{dens} v)^{μν} = ξ^{ρ} \partial_{ρ} v^{μν} - v^{ρ ν} \partial_{ρ} ξ^{μ} - v^{μ ρ} \partial_{ρ} ξ^{ν} + (\partial_{ρ} ξ^{ρ}) v^{μν} .

Displayed formulas in this passage

unlabeled · line 1514
$(L_{ξ}^{dens} v)^{μν} = ξ^{ρ} \partial_{ρ} v^{μν} - v^{ρ ν} \partial_{ρ} ξ^{μ} - v^{μ ρ} \partial_{ρ} ξ^{ν} + (\partial_{ρ} ξ^{ρ}) v^{μν} .$