Manuscript X-G-ST@1 · X-G-ST.tex · section · 3c97567f

The reference action being varied

Stage 2A: the switched scalar has a classical gauge breaking › The reference action being varied · lines 1077–1110

X-G-ST@1#eq:scalarAction

For this calculation only, define the natural reference action

S_{ref} = S_{EH} + S_{sc, g} + s_{kin} Ψ_{q ˉ},

where

S_{sc, g} L_{q} [q, ϕ] = \int ((1 - g) L_{η} [ϕ] + g L_{q} [q, ϕ]) d^{4} x, = \frac{1}{2} J_{q} (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .

It is used to evaluate a natural gauge variation. Its Euler equation is not identified with the post-variation source prescription (eq:mixed).

The flat stress satisfies

T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - η^{μν} L_{η} [ϕ], \partial_{μ} T_{0}^{μ}_{ν} = (P_{m} ϕ) \partial_{ν} ϕ .

Displayed formulas in this passage

X-G-ST@1#eq:scalarAction · line 1087
$S_{sc, g} L_{q} [q, ϕ] = \int ((1 - g) L_{η} [ϕ] + g L_{q} [q, ϕ]) d^{4} x, = \frac{1}{2} J_{q} (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 1081
$S_{ref} = S_{EH} + S_{sc, g} + s_{kin} Ψ_{q ˉ},$
unlabeled · line 1102
$T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - η^{μν} L_{η} [ϕ], \partial_{μ} T_{0}^{μ}_{ν} = (P_{m} ϕ) \partial_{ν} ϕ .$