Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

Conventions

Problem, fixed model, and precise scope › Conventions · lines 202–240

VII-C@1#eq:C

Let

M = R^{1 + 3}, η = diag (- 1, 1, 1, 1), □ = - \partial_{t}^{2} + Δ_{x}, Σ_{0} = {t = 0}, M_{+} = {t > 0} .

Indices are raised with $η$ , and

(C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}) .

The zero-Cauchy inverse satisfies

□ E_{0} j = j, (E_{0} j) ∣_{Σ_{0}} = 0, \partial_{t} (E_{0} j) ∣_{Σ_{0}} = 0.

We take $f \in C_{c}^{\infty} (M_{+})$ . Thus its support has a positive distance in the time coordinate from $Σ_{0}$ .

dictionary. Relative to the convention $η_{B} = diag (1, - 1, - 1, - 1)$ ,

η = - η_{B}, □ = - □_{B}, E_{0} = - E_{0}^{B} .

All raised-index expressions are recomputed using $η$ . If one also changes $h = - h_{B}$ so that the full metric changes sign, then $C [h] = - C_{B} [h_{B}]$ and $E_{0} C [h] = E_{0}^{B} C_{B} [h_{B}]$ .

Displayed formulas in this passage

VII-C@1#eq:C · line 213
$(C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}) .$
unlabeled · line 205
$M = R^{1 + 3}, η = diag (- 1, 1, 1, 1), □ = - \partial_{t}^{2} + Δ_{x}, Σ_{0} = {t = 0}, M_{+} = {t > 0} .$
unlabeled · line 220
$□ E_{0} j = j, (E_{0} j) ∣_{Σ_{0}} = 0, \partial_{t} (E_{0} j) ∣_{Σ_{0}} = 0.$
unlabeled · line 231
$η = - η_{B}, □ = - □_{B}, E_{0} = - E_{0}^{B} .$