Manuscript X-A-ST@1 · X-A-ST.tex · proposition · d9ccbde1

Proposition: First-order off-plateau display

Varied-action derivation: Route II · lines 1362–1392 · proof included or linked

X-A-ST@1#prop:offplateau, X-A-ST@1#eq:offplateau

For a $κ$ -independent scalar and

T^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - η^{μν} L_{η} [ϕ],

the first coefficient of (eq:varied) is

[δ_{F} S_{g}^{sc}]_{κ} = \int (1 - g) T^{μν} \partial_{μ} ξ_{ν} d^{4} x = \int (\partial_{μ} g) T^{μν} ξ_{ν} d^{4} x - \int (1 - g) ξ^{ν} (P_{m} ϕ) \partial_{ν} ϕ d^{4} x .

Proof. Insert the first terms of (eq:flatvariation)–(eq:currentvariation). Integrate $\partial_{μ} ((1 - g) ξ^{μ} L_{η})$ . The result is the first line. The identity

\partial_{μ} T^{μν} = (P_{m} ϕ) \partial^{ν} ϕ

and another integration by parts give the second.

Displayed formulas in this passage

X-A-ST@1#eq:offplateau · line 1371
$[δ_{F} S_{g}^{sc}]_{κ} = \int (1 - g) T^{μν} \partial_{μ} ξ_{ν} d^{4} x = \int (\partial_{μ} g) T^{μν} ξ_{ν} d^{4} x - \int (1 - g) ξ^{ν} (P_{m} ϕ) \partial_{ν} ϕ d^{4} x .$
unlabeled · line 1365
$T^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - η^{μν} L_{η} [ϕ],$
unlabeled · line 1388
$\partial_{μ} T^{μν} = (P_{m} ϕ) \partial^{ν} ϕ$

Proposition: First-order off-plateau display

Displayed formulas in this passage

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