Manuscript X-E-ST@1 · X-E-ST.tex · proposition · 44614221

Proposition: Euler derivatives

E1: action, Palatini variation, and switched Noether identity › Matter variation and the complete Euler equation · lines 572–613 · proof included or linked

X-E-ST@1#prop:euler, X-E-ST@1#eq:E-q, X-E-ST@1#eq:E-phi

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The tensor-density and scalar-density Euler derivatives are

E^{μν} := \frac{δ S}{δ q _{μν}} E_{h}^{μν} := \frac{δ S}{δ h _{μν}} E_{ϕ} = \frac{J _{q}}{2 κ ^{2}} (G^{μν} [q] - κ^{2} g T^{μν} [q, ϕ]), = κ E^{μν}, = - \partial_{μ} ([(1 - g) η^{μν} + g J_{q} q^{μν}] \partial_{ν} ϕ) - m^{2} (1 - g + g J_{q}) ϕ . \frac{δ S}{δ k _{μν}} = κ E_{h}^{μν},

There are no derivatives of $g$ in $E$ ; all of them remain in $E_{ϕ}$ and in the divergence identity below. In particular,

E_{ϕ} = - (P_{m} + κ L_{1} [h] + κ^{2} L_{2} [h, k]) ϕ + O (κ^{3}) .

Proof. Varying the displayed matter density algebraically gives

δ_{q} L_{q} = - \frac{1}{2} J_{q} T^{μν} [q, ϕ] δ q_{μν} .

There are no metric derivatives in $L_{q}$ . Combine this with Lemma lem:palatini. One scalar integration by parts gives (eq:E-phi), and (eq:densities) gives its expansion.

Displayed formulas in this passage

X-E-ST@1#eq:E-q · line 575
$E^{μν} := \frac{δ S}{δ q _{μν}} E_{h}^{μν} := \frac{δ S}{δ h _{μν}} E_{ϕ} = \frac{J _{q}}{2 κ ^{2}} (G^{μν} [q] - κ^{2} g T^{μν} [q, ϕ]), = κ E^{μν}, = - \partial_{μ} ([(1 - g) η^{μν} + g J_{q} q^{μν}] \partial_{ν} ϕ) - m^{2} (1 - g + g J_{q}) ϕ . \frac{δ S}{δ k _{μν}} = κ E_{h}^{μν},$
X-E-ST@1#eq:E-phi · line 575
$E^{μν} := \frac{δ S}{δ q _{μν}} E_{h}^{μν} := \frac{δ S}{δ h _{μν}} E_{ϕ} = \frac{J _{q}}{2 κ ^{2}} (G^{μν} [q] - κ^{2} g T^{μν} [q, ϕ]), = κ E^{μν}, = - \partial_{μ} ([(1 - g) η^{μν} + g J_{q} q^{μν}] \partial_{ν} ϕ) - m^{2} (1 - g + g J_{q}) ϕ . \frac{δ S}{δ k _{μν}} = κ E_{h}^{μν},$
unlabeled · line 596
$E_{ϕ} = - (P_{m} + κ L_{1} [h] + κ^{2} L_{2} [h, k]) ϕ + O (κ^{3}) .$
unlabeled · line 604
$δ_{q} L_{q} = - \frac{1}{2} J_{q} T^{μν} [q, ϕ] δ q_{μν} .$

Proposition: Euler derivatives

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