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

Lemma: Retarded cone identities

Cone normalization and causal convolution · lines 3096–3145 · proof included or linked

Em01-F@3#lem:cone

The measures (eq:RB) satisfy (eq:cone-identities). Their causal convolution is support-proper over compact external sets.

Proof. With spatial Fourier phase $e^{- i k \cdot x}$ ,

R (t, k) = Θ (t) \frac{sin ( ϖ t )}{ϖ}, ϖ = ∣ k ∣.

The limiting value is $Θ (t) t$ . Its value at $0 +$ is zero and its derivative there is one, so

(\partial_{t}^{2} + ϖ^{2}) R = δ_{D} (t) .

Fourier injectivity gives $□_{η} R = δ_{D}^{(4)}$ .

If $s, x - s$ both lie in the closed future cone, $s$ lies in the compact causal diamond from $0$ to $x$ . These diamonds are uniformly bounded over compact $x$ -sets.

The temporal convolution is

\int_{0}^{t} \frac{sin ( ϖ ( t - τ ))}{ϖ} \frac{sin ( ϖ τ )}{ϖ} d τ = \frac{sin ( ϖ t ) - ϖ t cos ( ϖ t )}{2 ϖ ^{3}},

with limit $t^{3} /6$ . The spatial transform of $B$ gives the same expression, multiplied by $Θ (t)$ . Hence $R * R = B$ .

For $s^{0} > 0, s \neq = 0$ , the regular cone equation gives

\partial_{s^{α}} B (s) = \frac{2 s _{α}}{8 π} δ_{D} (s^{2}) = \frac{s _{α}}{2} R (s) .

This separated derivative formula does not replace the cone-tip measure definition of $R$ .

Displayed formulas in this passage

unlabeled · line 3105
$R (t, k) = Θ (t) \frac{sin ( ϖ t )}{ϖ}, ϖ = ∣ k ∣.$
unlabeled · line 3114
$(\partial_{t}^{2} + ϖ^{2}) R = δ_{D} (t) .$
unlabeled · line 3124
$\int_{0}^{t} \frac{sin ( ϖ ( t - τ ))}{ϖ} \frac{sin ( ϖ τ )}{ϖ} d τ = \frac{sin ( ϖ t ) - ϖ t cos ( ϖ t )}{2 ϖ ^{3}},$
unlabeled · line 3136
$\partial_{s^{α}} B (s) = \frac{2 s _{α}}{8 π} δ_{D} (s^{2}) = \frac{s _{α}}{2} R (s) .$