Manuscript XIX-E@1 · XIX-E.tex · section · 702f146e

A fixed density ramp can fail the harmonic response test

Prescribed profiles, convolution, and clocks › A fixed density ramp can fail the harmonic response test · lines 401–441

XIX-E@1#eq:ramp-equation

Use the flat background

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

on $M_{T} = [0, T] \times R^{3}$ . For a smooth symmetric perturbation $h$ , set

A_{1}^{α μ} [h] = \frac{1}{2} (η^{ρ σ} h_{ρ σ}) η^{α μ} - h^{α μ} .

The first density-ramped harmonic correction is the smooth zero-data solution of

□_{η} W_{σ}^{μ} [h] = - \partial_{α} (σ A_{1}^{α μ} [h]) .

Ordinary coordinate-time position and velocity vanish at $t = 0$ .

The zero-data solution operator for a smooth spatially compact forcing $F$ can be written componentwise as

G_{-} F (t, x) = - \frac{1}{4 π} \int_{0}^{t} (t - s) \int_{S^{2}} F (s, x + (t - s) ω) d ω d s .

Differentiation of the spherical means verifies $□_{η} G_{-} F = F$ and the data.

For a compact smooth vector field $ξ$ vanishing in an initial collar, let

(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .

Displayed formulas in this passage

XIX-E@1#eq:ramp-equation · line 418
$□_{η} W_{σ}^{μ} [h] = - \partial_{α} (σ A_{1}^{α μ} [h]) .$
unlabeled · line 404
$η = diag (- 1, 1, 1, 1), □_{η} = - \partial_{t}^{2} + Δ$
unlabeled · line 411
$A_{1}^{α μ} [h] = \frac{1}{2} (η^{ρ σ} h_{ρ σ}) η^{α μ} - h^{α μ} .$
unlabeled · line 427
$G_{-} F (t, x) = - \frac{1}{4 π} \int_{0}^{t} (t - s) \int_{S^{2}} F (s, x + (t - s) ω) d ω d s .$
unlabeled · line 437
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .$