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

Retarded cone measures

The independently defined marked insertion › Retarded cone measures · lines 1774–1842

Em01-F@3#eq:RB, Em01-F@3#eq:cone-identities, Em01-F@3#eq:rresponse

Define

R (ψ) = \frac{1}{4 π} \int_{R^{3}} \frac{ψ ( ∣ x ∣ , x )}{∣ x ∣} d^{3} x, B (t, x) = \frac{1}{8 π} 1_{{t \geq ∣ x ∣}} .

The measure defining $R$ is locally finite at the tip. Away from it,

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

Appendix app:cone proves

□_{η} R = δ_{D}^{(4)}, R * R = B, \partial_{s^{α}} B (s) = \frac{s _{α}}{2} R (s) (s^{0} > 0, s \neq = 0) .

For compact smooth $h$ , put

X_{h}^{μ} = - R * \partial_{α} a_{1}^{α μ} [h] .

Its kernel is

X_{h}^{μ} (b) = \int L^{μ, I} (b, a) h_{I} (a) d^{4} a, L^{μ, I} (s, a) = A^{α μ, I} \partial_{a^{α}} R (s - a) .

It solves

□_{η} X_{h}^{μ} = - \partial_{α} a_{1}^{α μ} [h]

with retarded support.

Define

r [h, G] = P R * Q (h, G) .

Then

(- □_{η} P) r [h, G] = - Q (h, G) .

This is the response to the negative nonlinear Euler source. Its coupling-normalized field correction carries a factor $λ$ . This forcing normalization is separate from the explicit prefactor assigned to the insertion.

The retarded Green operators are used on compact sources and, for the nested response, on the resulting past-compact sources. All relevant intermediate integrations lie in causal diamonds over compact endpoint sets.

Displayed formulas in this passage

Em01-F@3#eq:RB · line 1777
$R (ψ) = \frac{1}{4 π} \int_{R^{3}} \frac{ψ ( ∣ x ∣ , x )}{∣ x ∣} d^{3} x, B (t, x) = \frac{1}{8 π} 1_{{t \geq ∣ x ∣}} .$
Em01-F@3#eq:cone-identities · line 1797
$□_{η} R = δ_{D}^{(4)}, R * R = B, \partial_{s^{α}} B (s) = \frac{s _{α}}{2} R (s) (s^{0} > 0, s \neq = 0) .$
Em01-F@3#eq:rresponse · line 1825
$r [h, G] = P R * Q (h, G) .$
unlabeled · line 1791
$R (t, x) = \frac{Θ ( t )}{2 π} δ_{D} (t^{2} - ∣ x ∣^{2}) .$
unlabeled · line 1806
$X_{h}^{μ} = - R * \partial_{α} a_{1}^{α μ} [h] .$
unlabeled · line 1810
$X_{h}^{μ} (b) = \int L^{μ, I} (b, a) h_{I} (a) d^{4} a, L^{μ, I} (s, a) = A^{α μ, I} \partial_{a^{α}} R (s - a) .$
unlabeled · line 1819
$□_{η} X_{h}^{μ} = - \partial_{α} a_{1}^{α μ} [h]$
unlabeled · line 1830
$(- □_{η} P) r [h, G] = - Q (h, G) .$