Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Action–stress sign

One action and its normalization dictionary › Action–stress sign · lines 742–817

Em01-F@3#eq:EHvariation, Em01-F@3#eq:mattervariation, Em01-F@3#eq:Einstein-normalization, Em01-F@3#eq:Newton-dictionary

For a compact covariant metric variation,

δ_{g} S_{EH} = \frac{2}{λ ^{2}} \int - det g G^{μν} [g] δ g_{μν} d^{4} x .

The curved matter stress in the invariant matter density is

T_{μν} [g, ϕ] = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} g_{μν} (g^{ρ σ} \partial_{ρ} ϕ \partial_{σ} ϕ - m^{2} ϕ^{2}) .

Its covariant metric variation satisfies

δ_{g} S_{m} = - \frac{1}{2} \int - det g T^{μν} [g, ϕ] δ g_{μν} d^{4} x .

Indeed,

δ - det g = \frac{1}{2} - det g g^{μν} δ g_{μν},

δ g^{α β} = - g^{α μ} g^{β ν} δ g_{μν} .

These identities give (eq:mattervariation). For (eq:EHvariation), vary the Ricci tensor by

δ R_{μν} = \nabla_{ρ} δ Γ^{ρ}_{μν} - \nabla_{ν} δ Γ^{ρ}_{μ ρ} .

Metric compatibility puts the derivative contribution into a divergence; the remaining inverse-metric and determinant variations give the displayed Einstein tensor and sign.

Where $g = 1$ , the invariant part of the model therefore has metric equation

G_{μν} [g] = \frac{λ ^{2}}{4} T_{μν} [g, ϕ] .

This equation concerns the invariant part, not the additional gauge-fixing Euler equation.

The comparison dictionary, when adopting the Newtonian normalization reported in the supplied XI/XII passages, is

λ = κ, λ^{2} = 32 π G_{N}, \frac{λ ^{2}}{4} = 8 π G_{N} (c = 1) .

It is a convention agreement with those passages Packet, not an identification with every use of $κ$ elsewhere.

Displayed formulas in this passage

Em01-F@3#eq:EHvariation · line 745
$δ_{g} S_{EH} = \frac{2}{λ ^{2}} \int - det g G^{μν} [g] δ g_{μν} d^{4} x .$
Em01-F@3#eq:mattervariation · line 763
$δ_{g} S_{m} = - \frac{1}{2} \int - det g T^{μν} [g, ϕ] δ g_{μν} d^{4} x .$
Em01-F@3#eq:Einstein-normalization · line 797
$G_{μν} [g] = \frac{λ ^{2}}{4} T_{μν} [g, ϕ] .$
Em01-F@3#eq:Newton-dictionary · line 806
$λ = κ, λ^{2} = 32 π G_{N}, \frac{λ ^{2}}{4} = 8 π G_{N} (c = 1) .$
unlabeled · line 754
$T_{μν} [g, ϕ] = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} g_{μν} (g^{ρ σ} \partial_{ρ} ϕ \partial_{σ} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 773
$δ - det g = \frac{1}{2} - det g g^{μν} δ g_{μν},$
unlabeled · line 778
$δ g^{α β} = - g^{α μ} g^{β ν} δ g_{μν} .$
unlabeled · line 785
$δ R_{μν} = \nabla_{ρ} δ Γ^{ρ}_{μν} - \nabla_{ν} δ Γ^{ρ}_{μ ρ} .$