Manuscript Em01-B@3 · Em01-B-v3.tex · section · 69189992

Spacetime, normalization, and geometry

Frozen expression and complete contraction › Spacetime, normalization, and geometry · lines 380–455

Em01-B@3#eq:inverse, Em01-B@3#eq:pair, Em01-B@3#eq:N

Let

M = R^{1 + 3}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x} .

Indices are raised and lowered with $η$ . Trace reversal is

(P r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} tr_{η} r, P^{2} = id .

Retain

G_{gr}^{R} = - P E_{0}^{R}, r_{D} = P E_{0}^{R} J_{D}, r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]) .

The frozen pair is

V_{g} = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x, O = Φ_{D}^{(2)} + A_{1} [f, g] (r_{g} [ϕ]; ϕ),

where

Φ_{D}^{(2)} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D} .

The inverse and insertion coefficients remain $- 1$ , and the scalar-sourced slot occurs once.

The width $ε > 0$ , observations, switches, masses, component pairings, and partial-diagonal class $C_{part}$ remain fixed. The smooth switch $σ_{ε}$ equals one after the frozen onset region.

Work on a sufficiently small patch

W = A \times B \times Z \times Y, w = (a, b, z, y),

with separated closures of $A, B, Y$ , $Z \subset A$ , and

b^{0} > y^{0} > a^{0} > ε .

The outer separations $b - a, b - z$ are nonzero and future directed. The residual variable $z$ is retained.

Put

q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2},

and

N = {(a, b, z, y) : z = a, y = a + α ℓ_{n}, b = a + β ℓ_{n}, 0 < α < β, ℓ_{n} = (1, n), n \in S^{2}} .

This is a smooth submanifold of dimension eight in the sixteen-dimensional $W$ .

The inherited canonical domain $U_{can}$ is not identified with $W ∖ N$ or any larger characteristic complement.

Displayed formulas in this passage

Em01-B@3#eq:inverse · line 397
$G_{gr}^{R} = - P E_{0}^{R}, r_{D} = P E_{0}^{R} J_{D}, r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]) .$
Em01-B@3#eq:pair · line 404
$V_{g} = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x, O = Φ_{D}^{(2)} + A_{1} [f, g] (r_{g} [ϕ]; ϕ),$
Em01-B@3#eq:N · line 440
$N = {(a, b, z, y) : z = a, y = a + α ℓ_{n}, b = a + β ℓ_{n}, 0 < α < β, ℓ_{n} = (1, n), n \in S^{2}} .$
unlabeled · line 383
$M = R^{1 + 3}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x} .$
unlabeled · line 390
$(P r)_{μν} = r_{μν} - \frac{1}{2} η_{μν} tr_{η} r, P^{2} = id .$
unlabeled · line 412
$Φ_{D}^{(2)} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D} .$
unlabeled · line 424
$W = A \times B \times Z \times Y, w = (a, b, z, y),$
unlabeled · line 429
$b^{0} > y^{0} > a^{0} > ε .$
unlabeled · line 436
$q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2},$