Manuscript XII-B@1 · XII-B.tex · section · 229bc869

Inherited sheet and the absent changed object

Inherited sheet and the absent changed object · lines 396–471

The inherited geometry is

M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2} .

Fix $0 < ε < T$ and

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

The ramp is the one supplied in XI-B and XI-E:

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.

It is smooth and flat at both endpoints. Moreover $σ_{ε}^{'} (t) > 0$ for $0 < t < ε$ . Indeed, on $(0, 1)$ ,

\frac{d}{d s} lo g \frac{χ ( s )}{1 - χ ( s )} = \frac{1}{s ^{2}} + \frac{1}{( 1 - s ) ^{2}} > 0.

The switch and plateau satisfy

g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset,

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

This class is not reduced after the audit.

Trace reversal and the differential normalization are

P r = r - \frac{1}{2} η tr_{η} r, R = \frac{1}{2} K, s_{0} γ = R c, G_{gr}^{R} = - P E_{0}^{R} .

Parentheses include $\frac{1}{2}$ . The equation inverse and action inverse remain distinct as in XI-B XIB. The factor $- 4$ is not imported.

The original cubic source is

J_{3} [γ] = 2 H_{2} [γ],

with the curvature convention and Einstein–Hilbert action

S_{EH} [q] = - \frac{2}{κ ^{2}} \int - q R [q] d^{4} x, q = η + κγ .

These data specify the original local coefficient, not its new weighted variational completion.

The scalar completion $A_{1}$ and the coordinate ramp are those of XI-E XIE. The new completed insertion would need both its scalar-response leg and its coordinate leg. Neither is discarded merely because a local metric source has a smooth weight. No completed new kernel is defined here, because B0 has stopped.

Displayed formulas in this passage

unlabeled · line 400
$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2} .$
unlabeled · line 407
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3} .$
unlabeled · line 412
$σ_{ε} (t) = χ (t / ε), χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.$
unlabeled · line 424
$\frac{d}{d s} lo g \frac{χ ( s )}{1 - χ ( s )} = \frac{1}{s ^{2}} + \frac{1}{( 1 - s ) ^{2}} > 0.$
unlabeled · line 430
$g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset,$
unlabeled · line 434
$U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 441
$P r = r - \frac{1}{2} η tr_{η} r, R = \frac{1}{2} K, s_{0} γ = R c, G_{gr}^{R} = - P E_{0}^{R} .$
unlabeled · line 452
$J_{3} [γ] = 2 H_{2} [γ],$
unlabeled · line 456
$S_{EH} [q] = - \frac{2}{κ ^{2}} \int - q R [q] d^{4} x, q = η + κγ .$