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

Specialization, not reproduction

Substitution and the weighted comparison panel › Specialization, not reproduction · lines 2005–2107

Em01-F@3#eq:weightedL, Em01-F@3#eq:weightedK5p, Em01-F@3#eq:weightedK5m, Em01-F@3#eq:weightedK6, Em01-F@3#eq:weightedsource, Em01-F@3#eq:weightedsourcevariation, Em01-F@3#eq:weightedintermediate

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The seven-row display that follows belongs to the independently defined marked insertion with

σ_{c} = σ_{grav} = 1.

Its resemblance to supplied weighted rows is not an identification of the insertions.

For comparison only, the supplied coordinate leg has the form

L_{σ_{c}}^{μ, I} (b, a) = σ_{c} (a) A^{α μ, I} \partial_{a^{α}} R (b - a) .

Corresponding displayed outer factors are

K_{5, +}^{[σ_{c}]} K_{5, -}^{[σ_{c}]} K_{6}^{[σ_{c}]} = \partial_{b^{μ}} [f (b) σ_{c} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L_{σ_{c}}^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{c} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L_{σ_{c}}^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{c} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)] .

These equations record factors in the supplied passages Packet. Equality of the comparison density map with (eq:Didentified) is not inferred.

The reported density-weighted cubic source has the form

J_{3, σ_{grav}}^{μν} D_{σ_{grav}}^{μν} = σ_{grav} J_{3}^{μν} + D_{σ_{grav}}^{μν}, = 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{grav} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{grav} .

For fixed weights its mixed variation is

D J_{3, σ_{grav}} [G] h = σ_{grav} D J_{3} [G] h + D D_{σ_{grav}} [G] h .

The tensors $C_{2}, M_{2}$ belong to the reported unintegrated density representative. Their general components are not reconstructed here.

The extra source incidence would generate additional contributions schematically denoted

K_{8, layer}, K_{9, der, layer}, K_{9, mass, layer} .

These names identify comparison obligations, not newly derived rows in the theorem of this paper. At constant $σ_{grav} = 1$ , the terms containing its derivatives vanish identically. A source weighting $σ_{grav} J_{3}$ and a density weighting are otherwise different prescriptions.

coordinate weight. Even if a weighted source array were supplied, a nonconstant coordinate weight changes the intermediate composition. The relevant scalar kernel is of the form

H_{σ_{c}, α} (b, v) = \int R (b - s) \partial_{s^{α}} (σ_{c} (s) R (s - v)) d^{4} s .

The tensor product in $(b - s, s - v)$ , its differentiated smooth weight, and causal-diamond properness define this kernel. At $σ_{c} = 1$ ,

H_{1, α} (b, v) = \partial_{b^{α}} B (b - v) .

For general $σ_{c}$ , (eq:weightedintermediate) contains both $\partial σ_{c}$ terms and a weighted retarded integration. Replacing $q$ by $q_{σ_{grav}}$ in the translation-invariant $B$ -formula does not account for it.

Displayed formulas in this passage

Em01-F@3#eq:weightedL · line 2016
$L_{σ_{c}}^{μ, I} (b, a) = σ_{c} (a) A^{α μ, I} \partial_{a^{α}} R (b - a) .$
Em01-F@3#eq:weightedK5p · line 2023
$K_{5, +}^{[σ_{c}]} K_{5, -}^{[σ_{c}]} K_{6}^{[σ_{c}]} = \partial_{b^{μ}} [f (b) σ_{c} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L_{σ_{c}}^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{c} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L_{σ_{c}}^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{c} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)] .$
Em01-F@3#eq:weightedK5m · line 2023
$K_{5, +}^{[σ_{c}]} K_{5, -}^{[σ_{c}]} K_{6}^{[σ_{c}]} = \partial_{b^{μ}} [f (b) σ_{c} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L_{σ_{c}}^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{c} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L_{σ_{c}}^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{c} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)] .$
Em01-F@3#eq:weightedK6 · line 2023
$K_{5, +}^{[σ_{c}]} K_{5, -}^{[σ_{c}]} K_{6}^{[σ_{c}]} = \partial_{b^{μ}} [f (b) σ_{c} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L_{σ_{c}}^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{c} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L_{σ_{c}}^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{c} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)] .$
Em01-F@3#eq:weightedsource · line 2050
$J_{3, σ_{grav}}^{μν} D_{σ_{grav}}^{μν} = σ_{grav} J_{3}^{μν} + D_{σ_{grav}}^{μν}, = 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{grav} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{grav} .$
Em01-F@3#eq:weightedsourcevariation · line 2062
$D J_{3, σ_{grav}} [G] h = σ_{grav} D J_{3} [G] h + D D_{σ_{grav}} [G] h .$
Em01-F@3#eq:weightedintermediate · line 2089
$H_{σ_{c}, α} (b, v) = \int R (b - s) \partial_{s^{α}} (σ_{c} (s) R (s - v)) d^{4} s .$
unlabeled · line 2009
$σ_{c} = σ_{grav} = 1.$
unlabeled · line 2074
$K_{8, layer}, K_{9, der, layer}, K_{9, mass, layer} .$
unlabeled · line 2099
$H_{1, α} (b, v) = \partial_{b^{α}} B (b - v) .$

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section: Specialization, not reproduction