Manuscript XX-A@1 · XX-A.tex · section · 3a586110

The actual coupled linearized equations

Quantum field input: the operators that are specified and those that remain missing › The actual coupled linearized equations · lines 960–1010

XX-A@1#eq:dGamma, XX-A@1#eq:dRic, XX-A@1#eq:dR, XX-A@1#eq:dG, XX-A@1#eq:dT, XX-A@1#eq:dScalar

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Let

E_{μν} = M^{2} G_{μν} - T_{μν}, E_{ϕ} = □_{g} ϕ - V^{'} (ϕ),

and define $P z = (δ E_{μν}, δ E_{ϕ})$ at the displayed background. This defines the coupled linearized operator without discarding mixed metric–scalar terms.

For completeness, write

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

Here $\nabla$ is the background Levi-Civita derivative. The matter part is

δ T_{μν} δ E_{ϕ} = \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} .

The formulas follow by differentiating the metric inverse, connection, curvature, and scalar stress tensor.

Displayed formulas in this passage

XX-A@1#eq:dGamma · line 972
$(δ Γ)_{μν}^{ρ} δ R_{μν} δ R δ G_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} (δ Γ)_{μν}^{ρ} - \nabla_{ν} (δ Γ)_{μ ρ}^{ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} δ R_{μν}, = δ R_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} δ R .$
XX-A@1#eq:dRic · line 972
$(δ Γ)_{μν}^{ρ} δ R_{μν} δ R δ G_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} (δ Γ)_{μν}^{ρ} - \nabla_{ν} (δ Γ)_{μ ρ}^{ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} δ R_{μν}, = δ R_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} δ R .$
XX-A@1#eq:dR · line 972
$(δ Γ)_{μν}^{ρ} δ R_{μν} δ R δ G_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} (δ Γ)_{μν}^{ρ} - \nabla_{ν} (δ Γ)_{μ ρ}^{ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} δ R_{μν}, = δ R_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} δ R .$
XX-A@1#eq:dG · line 972
$(δ Γ)_{μν}^{ρ} δ R_{μν} δ R δ G_{μν} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} (δ Γ)_{μν}^{ρ} - \nabla_{ν} (δ Γ)_{μ ρ}^{ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} δ R_{μν}, = δ R_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} δ R .$
XX-A@1#eq:dT · line 991
$δ T_{μν} δ E_{ϕ} = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .$
XX-A@1#eq:dScalar · line 991
$δ T_{μν} δ E_{ϕ} = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .$
unlabeled · line 963
$E_{μν} = M^{2} G_{μν} - T_{μν}, E_{ϕ} = □_{g} ϕ - V^{'} (ϕ),$

Nearby in this section

lemma: Lemma: Gauge kernel of the unfixed operator