Manuscript IX-A@1 · IX-A.tex · section · 55aa6f6d

Geometry and Cauchy response

Fixed spacetime, fields, and kernel carrier › Geometry and Cauchy response · lines 393–449

IX-A@1#eq:C, IX-A@1#eq:Ezero, IX-A@1#eq:Df

Let

M = R^{1 + 3}, η = \diag (- 1, 1, 1, 1), Σ = {t = 0}, M_{+} = {t > 0},

and

□_{η} = - \partial_{t}^{2} + Δ_{x} .

All spaces are complexified when needed. Classical real fields are recovered by complex conjugation.

For a smooth symmetric covariant tensor $h$ , set

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

The zero-Cauchy operator is

(E_{0} s) (t) = - \int_{0}^{t} \frac{sin (( t - u ) - Δ )}{- Δ} s (u) d u, t \geq 0.

At zero spatial frequency the sine quotient has its continuous value $t - u$ . Thus

□_{η} E_{0} s = s, E_{0} s ∣_{Σ} = 0, \partial_{t} E_{0} s ∣_{Σ} = 0.

For smooth sources without decay, this formula is interpreted locally by finite propagation and compatible compact cutoffs. All observations in this paper concern compact truncated causal regions. Smooth Cauchy theory and finite propagation justify this use BGP.

For $f \in C_{c}^{\infty} (M_{+})$ , put

D_{f} = J^{-} (supp f) \cap {t \geq 0} .

This is compact. The functional $B_{f}$ depends on $h$ only in $D_{f}$ and on $ϕ$ only in $supp f$ .

Displayed formulas in this passage

IX-A@1#eq:C · line 409
$(C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}) .$
IX-A@1#eq:Ezero · line 418
$(E_{0} s) (t) = - \int_{0}^{t} \frac{sin (( t - u ) - Δ )}{- Δ} s (u) d u, t \geq 0.$
IX-A@1#eq:Df · line 441
$D_{f} = J^{-} (supp f) \cap {t \geq 0} .$
unlabeled · line 396
$M = R^{1 + 3}, η = \diag (- 1, 1, 1, 1), Σ = {t = 0}, M_{+} = {t > 0},$
unlabeled · line 402
$□_{η} = - \partial_{t}^{2} + Δ_{x} .$
unlabeled · line 428
$□_{η} E_{0} s = s, E_{0} s ∣_{Σ} = 0, \partial_{t} E_{0} s ∣_{Σ} = 0.$