Manuscript III@2 · III.tex · corollary · 82dad3e5

Corollary: Prescribed-metric scalar symplectic form

The coupled first-order jet and retarded harmonic section › A scalar symplectic consequence with metric fixed · lines 1424–1469 · proof included or linked

III@2#cor:symplectic

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For coefficientwise spacelike-compact solutions, define

J_{k}^{μ} (A, B) = a_{k}^{μν} (A \partial_{ν} B - B \partial_{ν} A),

σ_{k, t} (A, B) = - \int_{{t} \times R^{3}} J_{k}^{0} (A, B) d^{3} x .

The form is independent of $t$ , antisymmetric, $D$ -bilinear, and weakly nondegenerate.

Proof. Symmetry of $a_{k}^{μν}$ gives

\partial_{μ} J_{k}^{μ} (A, B) = A P_{k} B - B P_{k} A = 0.

Support is bounded on finite slabs, so integration has no spatial boundary term.

At a fixed time set

q_{A} = A ∣_{t_{0}}, p_{A} = - a_{k}^{0 ν} \partial_{ν} A ∣_{t_{0}} .

The leading coefficient of $a_{k}^{00}$ is $- 1$ , hence it is invertible in $D$ . Arbitrary compact formal $(q_{A}, p_{A})$ determine the coefficient Cauchy data. Solve first for $A_{0}$ , then for the forced $A_{1}$ .

The pairing is

σ_{k} (A, B) = \int (q_{A} p_{B} - q_{B} p_{A}) d^{3} x .

If it vanishes for every $B$ , take $q_{B} = 0$ and arbitrary real compact $p_{B}$ . Both coefficients of $q_{A}$ vanish distributionally. Then take $p_{B} = 0$ and arbitrary real compact $q_{B}$ , obtaining $p_{A} = 0$ . Cauchy uniqueness gives $A = 0$ .

Displayed formulas in this passage

unlabeled · line 1427
$J_{k}^{μ} (A, B) = a_{k}^{μν} (A \partial_{ν} B - B \partial_{ν} A),$
unlabeled · line 1431
$σ_{k, t} (A, B) = - \int_{{t} \times R^{3}} J_{k}^{0} (A, B) d^{3} x .$
unlabeled · line 1441
$\partial_{μ} J_{k}^{μ} (A, B) = A P_{k} B - B P_{k} A = 0.$
unlabeled · line 1449
$q_{A} = A ∣_{t_{0}}, p_{A} = - a_{k}^{0 ν} \partial_{ν} A ∣_{t_{0}} .$
unlabeled · line 1460
$σ_{k} (A, B) = \int (q_{A} p_{B} - q_{B} p_{A}) d^{3} x .$

Corollary: Prescribed-metric scalar symplectic form

Displayed formulas in this passage

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