Manuscript X-E@1 · X-E.tex · definition · 293fae6a

Definition: Metric Euler operator

E1: the gravitational action and the metric Euler equation › The metric Euler equation · lines 726–751

X-E@1#def:metric-euler, X-E@1#eq:metric-euler, X-E@1#eq:metric-euler-expanded

E^{μν} [q, ϕ] := \frac{δ S}{δ h _{μν}} = \frac{1}{2 κ} - q G^{μν} [q] - \frac{κ}{2} g - q T^{μν} [q, ϕ], \frac{δ S}{δ k _{μν}} = κ E^{μν} .

The metric equation is $E^{μν} = 0$ ; the $k$ -equation is the same equation. Expanded,

E^{μν} = \frac{1}{2} G_{lin}^{μν} [h] + κ (\frac{1}{2} G_{lin}^{μν} [k] + \frac{1}{2} G_{2}^{μν} [h, h] - \frac{1}{2} g T^{μν} [ϕ]) + κ^{2} (\frac{1}{2} [- q G^{μν}]_{(3)} [h, k] - \frac{1}{2} g T^{μν}_{(1)} [h, ϕ]) + O (κ^{3}),

where $[- q G^{μν}]_{(3)}$ is the cubic coefficient of the closed form, defined by Lemma lem:palatini and the conventions above.

Displayed formulas in this passage

X-E@1#eq:metric-euler · line 728
$E^{μν} [q, ϕ] := \frac{δ S}{δ h _{μν}} = \frac{1}{2 κ} - q G^{μν} [q] - \frac{κ}{2} g - q T^{μν} [q, ϕ], \frac{δ S}{δ k _{μν}} = κ E^{μν} .$
X-E@1#eq:metric-euler-expanded · line 738
$E^{μν} = \frac{1}{2} G_{lin}^{μν} [h] + κ (\frac{1}{2} G_{lin}^{μν} [k] + \frac{1}{2} G_{2}^{μν} [h, h] - \frac{1}{2} g T^{μν} [ϕ]) + κ^{2} (\frac{1}{2} [- q G^{μν}]_{(3)} [h, k] - \frac{1}{2} g T^{μν}_{(1)} [h, ϕ]) + O (κ^{3}),$