Manuscript XIV-J@1 · XIV-J.tex · input · 1c0e72ca

Input statement: Source incidence and gravitational response maps

J0: copied premises and their domains › XIV-H's exact maps · lines 398–470

XIV-J@1#input:Hmaps, XIV-J@1#eq:Q, XIV-J@1#eq:T, XIV-J@1#eq:K8, XIV-J@1#eq:K9, XIV-J@1#eq:convolution

The untested source incidence is

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

It is defined and supported on $v = a = z$ . Its smooth coefficients are those of the actual source.

Using an explicit output index, the response kernel is

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

The coordinate and scalar response legs are

K_{8}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], B = R * R,

and

K_{9}^{I, J} (a, b, z) = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z), F = E_{m}^{A} f .

The derivative and mass terms are both retained.

On the separated chronological patch, $K_{8}, K_{9}$ are finite sums of derivatives of total order at most three of smooth coefficients times

δ ((b - a)^{2}) δ^{(4)} (a - z) .

Their regular upper conormal order is $3/2$ on

Λ_{resp} = N^{*} {(b - a)^{2} = 0, a = z} ∖ 0.

The copied convolution is

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

It is an identity of existing internal kernels XIVH,XIVI.

Displayed formulas in this passage

XIV-J@1#eq:Q · line 401
$Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a) .$
XIV-J@1#eq:T · line 413
$T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .$
XIV-J@1#eq:K8 · line 420
$K_{8}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], B = R * R,$
XIV-J@1#eq:K9 · line 433
$K_{9}^{I, J} (a, b, z) = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z), F = E_{m}^{A} f .$
XIV-J@1#eq:convolution · line 460
$R (t, x) = \frac{Θ ( t )}{2 π} δ (t^{2} - ∣ x ∣^{2}), R * R = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .$
unlabeled · line 451
$δ ((b - a)^{2}) δ^{(4)} (a - z) .$
unlabeled · line 455
$Λ_{resp} = N^{*} {(b - a)^{2} = 0, a = z} ∖ 0.$