Manuscript XI-A-ST@1 · XI-A-ST.tex · section · c3f0929d

The quadratic normalization

The signed action and its Euler derivatives › The quadratic normalization · lines 793–881

XI-A-ST@1#rest:euler, XI-A-ST@1#eq:eexact, XI-A-ST@1#eq:eexp, XI-A-ST@1#eq:Ephiexact, XI-A-ST@1#eq:scalar-switch, XI-A-ST@1#eq:sourcedivergence, XI-A-ST@1#sec:noether

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The invariant metric Euler density is

e^{μν} := \frac{δ S _{inv}}{δ γ _{μν}} = \frac{2}{κ} ρ_{q} G^{μν} (q) - \frac{κ}{2} g ρ_{q} T_{q}^{μν}, = 2 H_{1}^{μν} + κ (2 H_{2}^{μν} - \frac{1}{2} g T_{0}^{μν}) + κ^{2} (2 H_{3}^{μν} - \frac{1}{2} g J_{1}^{μν}) + O (κ^{3}) .

It contains no derivatives of $g$ .

By contrast, the scalar Euler density is

E_{ϕ} = - \partial_{μ} [((1 - g) η^{μν} + g ρ_{q} q^{μν}) \partial_{ν} ϕ] - m^{2} ((1 - g) + g ρ_{q}) ϕ .

Equivalently,

E_{ϕ} = - (P_{m} + κ V_{1, g} + κ^{2} V_{2, g}) ϕ + O (κ^{3}),

where

V_{j, g} ϕ = g [\partial_{μ} (a_{j}^{μν} \partial_{ν} ϕ) + m^{2} b_{j} ϕ] + (\partial_{μ} g) a_{j}^{μν} \partial_{ν} ϕ, j = 1, 2.

Finally,

\partial_{μ} (g T_{0}^{μν}) = g (P_{m} ϕ) \partial^{ν} ϕ + (\partial_{μ} g) T_{0}^{μν} .

Proof. The formulas are XI-A :euler and :scalar-switch CheungXIA. Combining (eq:stressvariation) with $δ q = κ δ γ$ gives (eq:eexact); no metric integration by parts in the matter action is involved. Scalar integration by parts instead gives (eq:Ephiexact), and hence (eq:scalar-switch). The identity $\partial_{μ} T_{0}^{μν} = (P_{m} ϕ) \partial^{ν} ϕ$ gives (eq:sourcedivergence). All operations are completion-chart operations and transport by Lemma lem:transfer.

The action is fixed before its derivative is projected. The classical cubic and quartic Einstein coefficients supply $2 H_{2}$ and $2 H_{3}$ . Because (eq:eexact) contains $1/ κ$ , a covariance check through order $κ^{2}$ must retain the required order-three metric curvature data. A premature order-two truncation of the pulled-back metric is not sufficient.

identities and tensor-density covariance

correct adjoint operator

Displayed formulas in this passage

XI-A-ST@1#eq:eexact · line 798
$e^{μν} := \frac{δ S _{inv}}{δ γ _{μν}} = \frac{2}{κ} ρ_{q} G^{μν} (q) - \frac{κ}{2} g ρ_{q} T_{q}^{μν}, = 2 H_{1}^{μν} + κ (2 H_{2}^{μν} - \frac{1}{2} g T_{0}^{μν}) + κ^{2} (2 H_{3}^{μν} - \frac{1}{2} g J_{1}^{μν}) + O (κ^{3}) .$
XI-A-ST@1#eq:eexp · line 798
$e^{μν} := \frac{δ S _{inv}}{δ γ _{μν}} = \frac{2}{κ} ρ_{q} G^{μν} (q) - \frac{κ}{2} g ρ_{q} T_{q}^{μν}, = 2 H_{1}^{μν} + κ (2 H_{2}^{μν} - \frac{1}{2} g T_{0}^{μν}) + κ^{2} (2 H_{3}^{μν} - \frac{1}{2} g J_{1}^{μν}) + O (κ^{3}) .$
XI-A-ST@1#eq:Ephiexact · line 815
$E_{ϕ} = - \partial_{μ} [((1 - g) η^{μν} + g ρ_{q} q^{μν}) \partial_{ν} ϕ] - m^{2} ((1 - g) + g ρ_{q}) ϕ .$
XI-A-ST@1#eq:scalar-switch · line 830
$V_{j, g} ϕ = g [\partial_{μ} (a_{j}^{μν} \partial_{ν} ϕ) + m^{2} b_{j} ϕ] + (\partial_{μ} g) a_{j}^{μν} \partial_{ν} ϕ, j = 1, 2.$
XI-A-ST@1#eq:sourcedivergence · line 839
$\partial_{μ} (g T_{0}^{μν}) = g (P_{m} ϕ) \partial^{ν} ϕ + (\partial_{μ} g) T_{0}^{μν} .$
unlabeled · line 824
$E_{ϕ} = - (P_{m} + κ V_{1, g} + κ^{2} V_{2, g}) ϕ + O (κ^{3}),$

The quadratic normalization

Displayed formulas in this passage

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