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

Definition: Gravitational action

E1: the gravitational action and the metric Euler equation › Conventions · lines 546–565

X-E@1#def:action

S_{gr} [q] = - \frac{1}{2 κ ^{2}} \int_{M} - q R [q] d^{4} x, S = S^{sc}_{g} [q, ϕ] + S_{gr} [q],

S^{sc}_{g} [q, ϕ] = \int_{M} [(1 - g) L_{η} [ϕ] + g L_{q} [q, ϕ]] d^{4} x, L_{q} = \frac{1}{2} - q (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}),

with $L_{η} = L_{q} ∣_{q = η}$ . Both actions are Lagrangian densities; their Euler–Lagrange derivatives are defined through compactly supported variations, and no decay of $h, k$ is assumed. $κ$ is formal. The dictionary $κ^{2} = 8 π G_{N}$ is used in Diagnostic diag:sign only.

Displayed formulas in this passage

unlabeled · line 548
$S_{gr} [q] = - \frac{1}{2 κ ^{2}} \int_{M} - q R [q] d^{4} x, S = S^{sc}_{g} [q, ϕ] + S_{gr} [q],$
unlabeled · line 553
$S^{sc}_{g} [q, ϕ] = \int_{M} [(1 - g) L_{η} [ϕ] + g L_{q} [q, ϕ]] d^{4} x, L_{q} = \frac{1}{2} - q (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}),$

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section: Conventions