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

Exact geometric conventions

E1: action, Palatini variation, and switched Noether identity › Exact geometric conventions · lines 479–517

X-E-ST@1#eq:action

For a genuine Lorentzian metric $q$ , use

Γ^{ρ}_{μν} R^{ρ}_{σ μν} R_{σ ν} = \frac{1}{2} q^{ρ λ} (\partial_{μ} q_{λ ν} + \partial_{ν} q_{λ μ} - \partial_{λ} q_{μν}), = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μ λ} Γ^{λ}_{ν σ} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ σ}, = R^{ρ}_{σ ρ ν}, R = q^{σ ν} R_{σ ν}, G_{μν} = R_{μν} - \frac{1}{2} q_{μν} R .

The frozen Lagrangian is

S_{gr} [q] S_{sc, g} [q, ϕ] L_{q} [q, ϕ] = - \frac{1}{2 κ ^{2}} \int J_{q} R [q] d^{4} x, = \int ((1 - g) L_{η} [ϕ] + g L_{q} [q, ϕ]) d^{4} x, = \frac{1}{2} J_{q} (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}), S = S_{gr} + S_{sc, g} .

The notation denotes local Lagrangian densities: Euler derivatives are defined using compact variations. No globally finite action integral or background decay is assumed. This is the X-E action convention XE; its sign is checked in Section sec:free.

Displayed formulas in this passage

unlabeled · line 482
$Γ^{ρ}_{μν} R^{ρ}_{σ μν} R_{σ ν} = \frac{1}{2} q^{ρ λ} (\partial_{μ} q_{λ ν} + \partial_{ν} q_{λ μ} - \partial_{λ} q_{μν}), = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μ λ} Γ^{λ}_{ν σ} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ σ}, = R^{ρ}_{σ ρ ν}, R = q^{σ ν} R_{σ ν}, G_{μν} = R_{μν} - \frac{1}{2} q_{μν} R .$
X-E-ST@1#eq:action · line 498
$S_{gr} [q] S_{sc, g} [q, ϕ] L_{q} [q, ϕ] = - \frac{1}{2 κ ^{2}} \int J_{q} R [q] d^{4} x, = \int ((1 - g) L_{η} [ϕ] + g L_{q} [q, ϕ]) d^{4} x, = \frac{1}{2} J_{q} (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}), S = S_{gr} + S_{sc, g} .$

Nearby in this section

lemma: Lemma: Palatini first variation