Manuscript XI-G@1 · XI-G.tex · corollary · f029bbcd

Corollary: The Cauchy pairing on the pure-trace sector

G1: Cauchy reduction of the onset › The pure-trace sector of the cubic source · lines 739–761 · proof included or linked

XI-G@1#cor:Bsigtrace

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Let $γ_{a} = ψ_{a} η$ with $□ ψ_{a} = 0$ and Cauchy data $(ψ_{a}, ψ ˙_{a}) ∣_{Σ}$ . Then $B_{Σ} = \int_{R^{3}} [β_{00} J^{00} + 2 β_{0 i} J^{0 i} + β_{ij} J^{ij}] (0, x) d^{3} x$ with

J^{00} J^{0 i} J^{ij} = \frac{3}{2} ψ ˙_{1} ψ ˙_{2} + \frac{3}{2} \nabla ψ_{1} \cdot \nabla ψ_{2} + ψ_{1} Δ ψ_{2} + ψ_{2} Δ ψ_{1}, = - \frac{3}{2} (ψ ˙_{1} \partial_{i} ψ_{2} + ψ ˙_{2} \partial_{i} ψ_{1}) - (ψ_{1} \partial_{i} ψ ˙_{2} + ψ_{2} \partial_{i} ψ ˙_{1}), = \frac{3}{2} (\partial_{i} ψ_{1} \partial_{j} ψ_{2} + \partial_{i} ψ_{2} \partial_{j} ψ_{1}) + ψ_{1} \partial_{i} \partial_{j} ψ_{2} + ψ_{2} \partial_{i} \partial_{j} ψ_{1} + \frac{3}{2} δ^{ij} (ψ ˙_{1} ψ ˙_{2} - \nabla ψ_{1} \cdot \nabla ψ_{2}),

all evaluated at $t = 0$ . For the F1 data the $β_{ij}$ term is absent.

Proof. Insert $□ ψ_{a} = 0$ and $\partial_{t}^{2} ψ_{a} = Δ ψ_{a}$ into (eq:J3trace), with $\partial^{0} = \partial_{t}$ , $\partial^{i} = - \partial_{i}$ , $η^{ij} = - δ^{ij}$ .

Displayed formulas in this passage

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$J^{00} J^{0 i} J^{ij} = \frac{3}{2} ψ ˙_{1} ψ ˙_{2} + \frac{3}{2} \nabla ψ_{1} \cdot \nabla ψ_{2} + ψ_{1} Δ ψ_{2} + ψ_{2} Δ ψ_{1}, = - \frac{3}{2} (ψ ˙_{1} \partial_{i} ψ_{2} + ψ ˙_{2} \partial_{i} ψ_{1}) - (ψ_{1} \partial_{i} ψ ˙_{2} + ψ_{2} \partial_{i} ψ ˙_{1}), = \frac{3}{2} (\partial_{i} ψ_{1} \partial_{j} ψ_{2} + \partial_{i} ψ_{2} \partial_{j} ψ_{1}) + ψ_{1} \partial_{i} \partial_{j} ψ_{2} + ψ_{2} \partial_{i} \partial_{j} ψ_{1} + \frac{3}{2} δ^{ij} (ψ ˙_{1} ψ ˙_{2} - \nabla ψ_{1} \cdot \nabla ψ_{2}),$

Corollary: The Cauchy pairing on the pure-trace sector

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