Manuscript XI-A@1 · XI-A.tex · section · b8211ecc

Frozen geometric data

X-A premises, conventions, and non-inputs › Frozen geometric data · lines 307–352

Use the X-A conventions:

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ = {t = 0},

η_{μν} = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.

The mass is fixed for each instance of the model. All coefficient fields are real and smooth on a neighbourhood of the slab.

Fix $g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1])$ with nonempty plateau $P_{g} = int {g = 1}$ . Fix $0 < ε < T$ and

σ_{ε} (t) = χ (t / ε), χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.

Choose an open relatively compact $U$ such that

U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .

The group $G_{U}$ is generated by the corresponding compactly supported flows. Every admitted transformation is the identity near $Σ$ and preserves $g$ and $σ_{ε}$ .

For coefficient calculations, genuine flows of $κ ξ + κ^{2} ζ$ , $ξ, ζ \in g_{U}$ , provide all Taylor coefficients needed below. They lift X-A's second-order formal transformations without enlarging its support class.

Displayed formulas in this passage

unlabeled · line 310
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ = {t = 0},$
unlabeled · line 315
$η_{μν} = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.$
unlabeled · line 328
$σ_{ε} (t) = χ (t / ε), χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.$
unlabeled · line 339
$U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .$