Manuscript X-G@1 · X-G.tex · proposition · 01566e14

Proposition: Full switched-matter breaking

The switched scalar action: an evaluated breaking · lines 972–1021 · proof included or linked

X-G@1#prop:breaking, X-G@1#eq:sSfirst, X-G@1#eq:sS, X-G@1#eq:B

For compact ghost coefficient functions, modulo integrated total derivatives,

s S_{ref} = s S_{g}^{sc} = κ \int [(1 - g) T_{0}^{μν} \partial_{μ} c_{ν} + (\partial_{μ} g) c^{μ} (L_{η} - L_{q})] d^{4} x .

Equivalently,

s S_{g}^{sc} = κ \int c^{ν} B_{ν} d^{4} x,

where

B_{ν} = (\partial_{μ} g) T_{0}^{μ}_{ν} - (1 - g) (P_{m} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) (L_{η} - L_{q}) .

Proof. Naturality gives $s L_{q} = κ \partial_{μ} (c^{μ} L_{q})$ . The flat metric is held fixed, so its scalar density instead has variation

s L_{η} = κ [T_{0}^{μν} \partial_{μ} c_{ν} + \partial_{μ} (c^{μ} L_{η})] .

Weight these identities by $g$ and $1 - g$ and integrate the divergence terms. This proves (eq:sSfirst). One more integration by parts and the displayed stress divergence give (eq:B).

The Einstein–Hilbert variation is a compact divergence. The variation of $s Ψ_{q ˉ}$ is zero by Lemma lem:nonlinear.

Displayed formulas in this passage

X-G@1#eq:sSfirst · line 976
$s S_{ref} = s S_{g}^{sc} = κ \int [(1 - g) T_{0}^{μν} \partial_{μ} c_{ν} + (\partial_{μ} g) c^{μ} (L_{η} - L_{q})] d^{4} x .$
X-G@1#eq:sS · line 987
$s S_{g}^{sc} = κ \int c^{ν} B_{ν} d^{4} x,$
X-G@1#eq:B · line 993
$B_{ν} = (\partial_{μ} g) T_{0}^{μ}_{ν} - (1 - g) (P_{m} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) (L_{η} - L_{q}) .$
unlabeled · line 1007
$s L_{η} = κ [T_{0}^{μν} \partial_{μ} c_{ν} + \partial_{μ} (c^{μ} L_{η})] .$

Proposition: Full switched-matter breaking

Displayed formulas in this passage

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