Manuscript Em02-D@4 · Em02-D-v4.tex · section · fcf2fac7

Second and third temporal derivatives

Integrated-vertex jets and the known six rows › Second and third temporal derivatives · lines 1235–1284

Em02-D@4#eq:P2, Em02-D@4#eq:Coperator, Em02-D@4#eq:Voperator, Em02-D@4#eq:P3

At $S = s = q ℓ_{n}$ , write

g_{ϕ} = k_{n} \cdot \nabla_{x} ϕ, D_{w} ϕ = \partial_{S^{0}} ϕ with a, r, x fixed .

The notation $g_{ϕ}$ here is unrelated to the observation cutoff $g$ .

Define

Q ϕ (a, s, r) = \int_{0}^{1} g_{ϕ} (a, s, r, θ s) d θ, P_{2} = - \frac{κ}{2} G [g_{ϕ}] = - \frac{κ}{16 π} Q ϕ .

Also set

C [w] V [ϕ] P_{3} = 4 A [w] + 3 B [w] = A [w] + \frac{3}{2} T [w] + \frac{3}{4} G [k_{n} \cdot \nabla_{x} w] + \frac{3}{2 q} G [w], = \frac{1}{8 π} \int_{0}^{1} θ [(n \cdot \nabla_{x}) \partial_{x^{0}} ϕ + Δ_{x} ϕ] (θ s) d θ, = \frac{κ}{2} {G [ϕ] - V [ϕ] - C [g_{ϕ}] - 3 G [D_{w} g_{ϕ}]} .

The operations in these formulas act on the $x$ -weight, with $a, s, r$ as displayed spectators.

Displayed formulas in this passage

Em02-D@4#eq:P2 · line 1248
$Q ϕ (a, s, r) = \int_{0}^{1} g_{ϕ} (a, s, r, θ s) d θ, P_{2} = - \frac{κ}{2} G [g_{ϕ}] = - \frac{κ}{16 π} Q ϕ .$
Em02-D@4#eq:Coperator · line 1258
$C [w] V [ϕ] P_{3} = 4 A [w] + 3 B [w] = A [w] + \frac{3}{2} T [w] + \frac{3}{4} G [k_{n} \cdot \nabla_{x} w] + \frac{3}{2 q} G [w], = \frac{1}{8 π} \int_{0}^{1} θ [(n \cdot \nabla_{x}) \partial_{x^{0}} ϕ + Δ_{x} ϕ] (θ s) d θ, = \frac{κ}{2} {G [ϕ] - V [ϕ] - C [g_{ϕ}] - 3 G [D_{w} g_{ϕ}]} .$
Em02-D@4#eq:Voperator · line 1258
$C [w] V [ϕ] P_{3} = 4 A [w] + 3 B [w] = A [w] + \frac{3}{2} T [w] + \frac{3}{4} G [k_{n} \cdot \nabla_{x} w] + \frac{3}{2 q} G [w], = \frac{1}{8 π} \int_{0}^{1} θ [(n \cdot \nabla_{x}) \partial_{x^{0}} ϕ + Δ_{x} ϕ] (θ s) d θ, = \frac{κ}{2} {G [ϕ] - V [ϕ] - C [g_{ϕ}] - 3 G [D_{w} g_{ϕ}]} .$
Em02-D@4#eq:P3 · line 1258
$C [w] V [ϕ] P_{3} = 4 A [w] + 3 B [w] = A [w] + \frac{3}{2} T [w] + \frac{3}{4} G [k_{n} \cdot \nabla_{x} w] + \frac{3}{2 q} G [w], = \frac{1}{8 π} \int_{0}^{1} θ [(n \cdot \nabla_{x}) \partial_{x^{0}} ϕ + Δ_{x} ϕ] (θ s) d θ, = \frac{κ}{2} {G [ϕ] - V [ϕ] - C [g_{ϕ}] - 3 G [D_{w} g_{ϕ}]} .$
unlabeled · line 1238
$g_{ϕ} = k_{n} \cdot \nabla_{x} ϕ, D_{w} ϕ = \partial_{S^{0}} ϕ with a, r, x fixed .$