Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

Classical backgrounds and the frozen algebra

Frozen analytical data and predecessor handoff › Classical backgrounds and the frozen algebra · lines 329–377

X-C-ST@1#eq:plateau, X-C-ST@1#eq:metriclaw

Let $h, k$ be arbitrary smooth real symmetric tensors on the completion, with no spatial decay or metric field equation imposed, and set

q = η + κh + κ^{2} k .

Their topology is the smooth compact-open topology.

Retain the same fixed $ε > 0$ , smooth ramp $σ (t) = χ (t / ε)$ , and compact switch

g \in C_{c}^{\infty} ((0, T_{*}) \times R^{3}; [0, 1])

as in the predecessors, where

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

The plateau and algebra are unchanged:

U ⋐ L \cap int {g = 1} \cap {ε < t < T_{*}}, g_{p} = C_{c}^{\infty} (U; T M) .

For $ξ, ζ \in g_{p}$ , the formal pullback generated by $κ ξ + κ^{2} ζ$ belongs to $G_{p}$ . The full background transformation is

h^{'} k^{'} = h + K ξ, (K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η .

Outside the plateau, the same formula will be used only to calculate defects for arbitrary compact interior generators. Those generators are not added to $g_{p}$ .

Displayed formulas in this passage

X-C-ST@1#eq:plateau · line 355
$U ⋐ L \cap int {g = 1} \cap {ε < t < T_{*}}, g_{p} = C_{c}^{\infty} (U; T M) .$
X-C-ST@1#eq:metriclaw · line 367
$h^{'} k^{'} = h + K ξ, (K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η .$
unlabeled · line 334
$q = η + κh + κ^{2} k .$
unlabeled · line 341
$g \in C_{c}^{\infty} ((0, T_{*}) \times R^{3}; [0, 1])$
unlabeled · line 345
$χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.$