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

Conventions

E1: the gravitational action and the metric Euler equation › Conventions · lines 521–545

For a Lorentzian metric $q_{μν}$ of signature $(+, -, -, -)$ ,

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

Under $q \mapsto - q$ the Christoffel symbols, $R^{ρ}_{σ μν}$ , $R_{μν}$ , and $G_{μν}$ are unchanged; $R$ and $- q R$ change sign. Signs therefore cannot be copied across signatures and are fixed below by a positivity check.

Displayed formulas in this passage

unlabeled · line 524
$Γ^{ρ}_{μν} R^{ρ}_{σ μν} R_{σ ν} R = \frac{1}{2} q^{ρ σ} (\partial_{μ} q_{σ ν} + \partial_{ν} q_{σ μ} - \partial_{σ} q_{μν}), = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μ λ} Γ^{λ}_{ν σ} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ σ}, = R^{ρ}_{σ ρ ν} = \partial_{ρ} Γ^{ρ}_{σ ν} - \partial_{ν} Γ^{ρ}_{ρ σ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{σ ν} - Γ^{ρ}_{ν λ} Γ^{λ}_{ρ σ}, = q^{σ ν} R_{σ ν}, G_{μν} = R_{μν} - \frac{1}{2} q_{μν} R, - q = - det q .$

Nearby in this section

definition: Definition: Gravitational action