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

The patch, source incidence, and retarded kernels

Component conventions and the retained internal expression › The patch, source incidence, and retarded kernels · lines 512–586

Em01-C@3#eq:source-polynomial, Em01-C@3#eq:Q, Em01-C@3#eq:retarded-kernels, Em01-C@3#eq:T

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Work on

W = A \times B \times Z \times Y, w = (a, b, z, y),

where $A, B, Y$ have separated closures, $Z \subset A$ , and

b^{0} > y^{0} > a^{0} > ε, z^{0} > ε, b^{0} > z^{0} .

These inequalities hold throughout the patch. The smooth ramp $σ_{ε}$ equals one on $t > ε$ . The observation functions $f, g$ are fixed and smooth.

Let $J_{D} = EL_{γ} (σ_{ε} ℓ_{3})$ , with the finite specification in Appendix app:source. Under (eq:source-conversion), its derivative has the local form

(J^{'} [G] h)^{L} (v) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial^{α} G_{J} (v) \partial^{β} h_{I} (v) .

Here and below Greek multiindices in $\partial^{α}$ are distinct from Greek spacetime indices in indexed tensor contractions.

Define the source incidence

Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ_{D}^{(4)} (v - z) \partial_{v}^{β} δ_{D}^{(4)} (v - a) .

The superscript $(4)$ here denotes the dimension of the Dirac mass. Primes, such as $δ_{D}^{'} (q)$ , denote one-dimensional derivatives.

The kernels are

R (t, x) = \frac{Θ ( t )}{2 π} δ_{D} (t^{2} - ∣ x ∣^{2}), B (t, x) = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .

They satisfy $B = R * R$ . Indeed, the spatial transform of $R$ is $Θ (t) sin (ρt) / ρ$ . Its causal time convolution is

Θ (t) \frac{sin ( ρt ) - ρt cos ( ρt )}{2 ρ ^{3}},

with value $Θ (t) t^{3} /6$ at $ρ = 0$ . Radial integration gives the same transform for the displayed $B$ .

Put

L^{μ, I} (s, a) = σ_{ε} (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a),

T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .

Fix also a smooth advanced observation root $F = E_{m}^{A} f$ , with $m \geq 0$ . The analysis below only uses its smoothness and its fixed value.

Displayed formulas in this passage

Em01-C@3#eq:source-polynomial · line 530
$(J^{'} [G] h)^{L} (v) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial^{α} G_{J} (v) \partial^{β} h_{I} (v) .$
Em01-C@3#eq:Q · line 542
$Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ_{D}^{(4)} (v - z) \partial_{v}^{β} δ_{D}^{(4)} (v - a) .$
Em01-C@3#eq:retarded-kernels · line 555
$R (t, x) = \frac{Θ ( t )}{2 π} δ_{D} (t^{2} - ∣ x ∣^{2}), B (t, x) = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .$
Em01-C@3#eq:T · line 577
$T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .$
unlabeled · line 515
$W = A \times B \times Z \times Y, w = (a, b, z, y),$
unlabeled · line 520
$b^{0} > y^{0} > a^{0} > ε, z^{0} > ε, b^{0} > z^{0} .$
unlabeled · line 565
$Θ (t) \frac{sin ( ρt ) - ρt cos ( ρt )}{2 ρ ^{3}},$
unlabeled · line 573
$L^{μ, I} (s, a) = σ_{ε} (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a),$