Manuscript IX-E-3@1 · IX-E-3.tex · section · 8c41417a

A fully specified flat realization and the solid-ball error

A fully specified flat realization and the solid-ball error · lines 630–664

IX-E-3@1#eq:C-flat, IX-E-3@1#eq:E-flat

This section is a diagnostic realization, not a replacement of the inherited operator.

Let

η = diag (1, - 1, - 1, - 1), P = \partial_{t}^{2} - Δ,

and define

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

Let $E_{0}$ be the componentwise solution operator for $P$ with zero displacement and velocity at $t = 0$ . Its retarded representation is

(E_{0} j) (t, x) = \int_{0}^{t} \int_{R^{3}} \frac{δ ( t - v - ∣ x - w ∣ )}{4 π ∣ x - w ∣} j (v, w) d^{3} w d v .

This is the standard flat three-dimensional wave realization. Its propagation is concentrated on null spheres, rather than throughout the interiors of causal balls Ivrii.

Displayed formulas in this passage

IX-E-3@1#eq:C-flat · line 642
$(C h)^{μ} = \partial_{ν} h^{ν μ} - \frac{1}{2} \partial^{μ} (η_{α β} h^{α β}) .$
IX-E-3@1#eq:E-flat · line 652
$(E_{0} j) (t, x) = \int_{0}^{t} \int_{R^{3}} \frac{δ ( t - v - ∣ x - w ∣ )}{4 π ∣ x - w ∣} j (v, w) d^{3} w d v .$
unlabeled · line 636
$η = diag (1, - 1, - 1, - 1), P = \partial_{t}^{2} - Δ,$

Nearby in this section

proposition: Proposition: Verification of (TC) for the flat realization