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

Definition: Onset family, boundary pairing

G1: Cauchy reduction of the onset › The onset family and the boundary pairing · lines 634–649

XI-G@1#def:onset, XI-G@1#eq:Bsig

For smooth $γ_{1}, γ_{2}, ϕ$ on $M$ and $0 \leq s < T$ put

F_{s} (γ_{1}, γ_{2}; ϕ) := \int_{{s \leq t \leq T}} β [f, g, ϕ]_{μν} (y) J_{3}^{μν} (γ_{1}, γ_{2}) (y) d^{4} y, F_{0} = ⟨ u, γ_{1} \otimes γ_{2} \otimes ϕ ⟩,

and

B_{Σ} (γ_{1}, γ_{2}; ϕ) := - \frac{d}{d s}_{s = 0^{+}} F_{s} = \int_{R^{3}} β_{μν} (0, x) J_{3}^{μν} (γ_{1}, γ_{2}) (0, x) d^{3} x .

Displayed formulas in this passage

XI-G@1#eq:Bsig · line 643
$B_{Σ} (γ_{1}, γ_{2}; ϕ) := - \frac{d}{d s}_{s = 0^{+}} F_{s} = \int_{R^{3}} β_{μν} (0, x) J_{3}^{μν} (γ_{1}, γ_{2}) (0, x) d^{3} x .$
unlabeled · line 637
$F_{s} (γ_{1}, γ_{2}; ϕ) := \int_{{s \leq t \leq T}} β [f, g, ϕ]_{μν} (y) J_{3}^{μν} (γ_{1}, γ_{2}) (y) d^{4} y, F_{0} = ⟨ u, γ_{1} \otimes γ_{2} \otimes ϕ ⟩,$