Manuscript XII-C@1 · XII-C.tex · proposition · dead1004

Proposition: Order-one compatibility

C3: the coupled jet for the selected action › The recomputed compatibility condition · lines 1092–1128 · proof included or linked

XII-C@1#prop:compatibility, XII-C@1#eq:new-compatibility, XII-C@1#eq:late-compatibility

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Every element of $J_{g, σ_{ε}}^{D} (ϕ_{0})$ satisfies

(\partial_{μ} g) T_{0}^{μ}_{ν} [ϕ_{0}] = R_{σ_{ε}, ν} [γ_{0}] .

For $t > ε$ , this reduces to

(\partial_{μ} g) T_{0}^{μ}_{ν} [ϕ_{0}] = 0.

Proof. The leading metric equation is $A γ_{0} = 0$ . The next metric equation is

A γ_{1} + J_{3, D} [γ_{0}] - \frac{1}{2} g T_{0} [ϕ_{0}] = 0.

Apply $N_{0}$ . The linearized Bianchi identity gives $N_{0} (A γ_{1}) = 0$ . Lemma lem:divergence also gives

N_{0, ν} (J_{3, D}) = - σ_{ε} Q_{ν} (A γ_{0}) + R_{σ_{ε}, ν} = R_{σ_{ε}, ν} .

Since $P_{m} ϕ_{0} = 0$ , $\partial_{μ} T_{0}^{μ}_{ν} [ϕ_{0}] = 0$ . The resulting equation is (eq:new-compatibility). Above the ramp, (eq:Rampdef-time) vanishes identically.

Displayed formulas in this passage

XII-C@1#eq:new-compatibility · line 1096
$(\partial_{μ} g) T_{0}^{μ}_{ν} [ϕ_{0}] = R_{σ_{ε}, ν} [γ_{0}] .$
XII-C@1#eq:late-compatibility · line 1102
$(\partial_{μ} g) T_{0}^{μ}_{ν} [ϕ_{0}] = 0.$
unlabeled · line 1111
$A γ_{1} + J_{3, D} [γ_{0}] - \frac{1}{2} g T_{0} [ϕ_{0}] = 0.$
unlabeled · line 1118
$N_{0, ν} (J_{3, D}) = - σ_{ε} Q_{ν} (A γ_{0}) + R_{σ_{ε}, ν} = R_{σ_{ε}, ν} .$

Nearby in this section

section: The recomputed compatibility condition