Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

Layer coefficients and the weighted source

Finite source specification and implementation inventory › Layer coefficients and the weighted source · lines 2022–2101

Em01-C@3#eq:C-coefficient-rule, Em01-C@3#eq:M2, Em01-C@3#eq:actualsource

Define

U^{μν} (s) = u_{0}^{μν} + s u_{1}^{μν} + s^{2} u_{2}^{μν}, D (s) = 1 + s b_{1} + s^{2} b_{2},

and

C_{j}^{μν α β} = [s^{j}] D (s) (U^{μν} (s) U^{α β} (s) - U^{μ (α} (s) U^{β) ν} (s)), j = 0, 1, 2.

Put

M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .

The full weighted tensor source is

J_{D}^{μν} = σ_{ε} J_{3}^{μν} + 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .

For completeness, the coefficient argument is as follows. For $q = η + s γ$ , compact covariant metric variation gives

\frac{δ}{δ γ _{μν}} (- \frac{2}{s ^{2}} \int σ - q R [q] d^{4} x) = \frac{2}{s} - q [σ G^{μν} + q^{μν} □_{q} σ - \nabla^{μ} \nabla^{ν} σ] .

Indeed, if $h = δ q$ ,

δ R = - R^{μν} h_{μν} + \nabla_{μ} \nabla_{ν} h^{μν} - □_{q} (q^{μν} h_{μν}),

and two compact integrations by parts yield the displayed variation.

The order- $s$ coefficient on the left is $EL_{γ} (σ ℓ_{3})$ . The Einstein term on the right gives $σ J_{3}$ , by expanding $- q$ and the two inverse metrics raising $G_{μν}$ . The remaining term is

\frac{2}{s} C^{μν α β} (q) (\partial_{α} \partial_{β} σ - Γ^{ρ}_{α β} (q) \partial_{ρ} σ),

whose order- $s$ coefficient is the layer part of (eq:actualsource). This calculation does not select a full-density weighted gravitational action.

Displayed formulas in this passage

Em01-C@3#eq:C-coefficient-rule · line 2031
$C_{j}^{μν α β} = [s^{j}] D (s) (U^{μν} (s) U^{α β} (s) - U^{μ (α} (s) U^{β) ν} (s)), j = 0, 1, 2.$
Em01-C@3#eq:M2 · line 2043
$M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .$
Em01-C@3#eq:actualsource · line 2051
$J_{D}^{μν} = σ_{ε} J_{3}^{μν} + 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .$
unlabeled · line 2025
$U^{μν} (s) = u_{0}^{μν} + s u_{1}^{μν} + s^{2} u_{2}^{μν}, D (s) = 1 + s b_{1} + s^{2} b_{2},$
unlabeled · line 2065
$\frac{δ}{δ γ _{μν}} (- \frac{2}{s ^{2}} \int σ - q R [q] d^{4} x) = \frac{2}{s} - q [σ G^{μν} + q^{μν} □_{q} σ - \nabla^{μ} \nabla^{ν} σ] .$
unlabeled · line 2078
$δ R = - R^{μν} h_{μν} + \nabla_{μ} \nabla_{ν} h^{μν} - □_{q} (q^{μν} h_{μν}),$
unlabeled · line 2091
$\frac{2}{s} C^{μν α β} (q) (\partial_{α} \partial_{β} σ - Γ^{ρ}_{α β} (q) \partial_{ρ} σ),$