Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

All retained Euler switch terms

The switched action and its coefficient equations › All retained Euler switch terms · lines 756–819

X-A-conv@1#eq:metricEuler, X-A-conv@1#eq:Dg, X-A-conv@1#eq:Vjg, X-A-conv@1#eq:Wg

Let $E_{h}$ be the ordinary first-derivative Euler operator in the tensor variable. Since every $ℓ_{j}$ depends on $h$ and $\partial h$ , but not on higher derivatives of $h$ ,

\frac{δ S _{g} ^{[2]}}{δ h _{α β}} = E_{h}^{α β} ℓ_{0} + κ [g E_{h}^{α β} ℓ_{1} - (\partial_{μ} g) \frac{\partial ℓ _{1}}{\partial ( \partial _{μ} h _{α β} )}] + κ^{2} [g E_{h}^{α β} ℓ_{2} - (\partial_{μ} g) \frac{\partial ℓ _{2}}{\partial ( \partial _{μ} h _{α β} )}] .

Equation (eq:metricEuler), together with (eq:gravzero)–(eq:gravtwo), is an explicit finite definition of the switched metric Euler equation. No coupled existence assertion is inferred from it.

The scalar Euler equation is $- D_{g} [q] ϕ = 0$ , where

D_{g} [q] ϕ = \partial_{μ} ([η^{μν} + κ g a_{1} (h)^{μν} + κ^{2} g a_{2} (h)^{μν}] \partial_{ν} ϕ) .

Its output is density valued.

Define

V_{j, g} [h] χ = \partial_{μ} (g a_{j} (h)^{μν} \partial_{ν} χ) = g \partial_{μ} (a_{j} (h)^{μν} \partial_{ν} χ) + (\partial_{μ} g) a_{j} (h)^{μν} \partial_{ν} χ .

These are all switch-derivative terms in the displayed scalar coefficient equations. There is no mass term because $m = 0$ .

For a metric jet, $V_{2, g} [h]$ is replaced by

W_{g} [h, k] χ = V_{1, g} [k] χ + V_{2, g} [h] χ .

Displayed formulas in this passage

X-A-conv@1#eq:metricEuler · line 761
$\frac{δ S _{g} ^{[2]}}{δ h _{α β}} = E_{h}^{α β} ℓ_{0} + κ [g E_{h}^{α β} ℓ_{1} - (\partial_{μ} g) \frac{\partial ℓ _{1}}{\partial ( \partial _{μ} h _{α β} )}] + κ^{2} [g E_{h}^{α β} ℓ_{2} - (\partial_{μ} g) \frac{\partial ℓ _{2}}{\partial ( \partial _{μ} h _{α β} )}] .$
X-A-conv@1#eq:Dg · line 784
$D_{g} [q] ϕ = \partial_{μ} ([η^{μν} + κ g a_{1} (h)^{μν} + κ^{2} g a_{2} (h)^{μν}] \partial_{ν} ϕ) .$
X-A-conv@1#eq:Vjg · line 799
$V_{j, g} [h] χ = \partial_{μ} (g a_{j} (h)^{μν} \partial_{ν} χ) = g \partial_{μ} (a_{j} (h)^{μν} \partial_{ν} χ) + (\partial_{μ} g) a_{j} (h)^{μν} \partial_{ν} χ .$
X-A-conv@1#eq:Wg · line 812
$W_{g} [h, k] χ = V_{1, g} [k] χ + V_{2, g} [h] χ .$