Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

Full linearization

The coupled Euler operator and the normalized action Hessian › Full linearization · lines 489–553

XX-B@1#eq:Gamma1, XX-B@1#eq:R1, XX-B@1#eq:Rscalar1, XX-B@1#eq:G1, XX-B@1#eq:T1, XX-B@1#eq:scalar1, XX-B@1#eq:H, XX-B@1#eq:S2

Back to manuscript

The first curvature variations are

Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1} .

The matter variations are

(T_{1})_{μν} (E_{ϕ})_{1} = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .

All potentials in these expressions are evaluated at $ϕ ˉ$ .

The coupled action Hessian is the operator

H z = (- \frac{1}{2} (M^{2} G_{1} - T_{1})_{μν}, (E_{ϕ})_{1}) .

The metric factor $- 1/2$ is essential. It follows from

δ S = \int (- \frac{1}{2} E^{μν} δ g_{μν} + E_{ϕ} δ ϕ) d V_{g} .

At an on-shell background, the variations of the volume and of the indices multiplying $E$ contribute no extra term to the second variation. Therefore

S_{2} (z) = \frac{1}{2} (z, H z)_{0}

is the actual quadratic action.

Displayed formulas in this passage

XX-B@1#eq:Gamma1 · line 492
$Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1} .$
XX-B@1#eq:R1 · line 492
$Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1} .$
XX-B@1#eq:Rscalar1 · line 492
$Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1} .$
XX-B@1#eq:G1 · line 492
$Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1} .$
XX-B@1#eq:T1 · line 513
$(T_{1})_{μν} (E_{ϕ})_{1} = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .$
XX-B@1#eq:scalar1 · line 513
$(T_{1})_{μν} (E_{ϕ})_{1} = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .$
XX-B@1#eq:H · line 533
$H z = (- \frac{1}{2} (M^{2} G_{1} - T_{1})_{μν}, (E_{ϕ})_{1}) .$
XX-B@1#eq:S2 · line 548
$S_{2} (z) = \frac{1}{2} (z, H z)_{0}$
unlabeled · line 540
$δ S = \int (- \frac{1}{2} E^{μν} δ g_{μν} + E_{ϕ} δ ϕ) d V_{g} .$

Full linearization

Displayed formulas in this passage

Nearby in this section