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

Inherited coordinate data and bare tests

Frozen analytical data and predecessor handoff › Inherited coordinate data and bare tests · lines 538–602

X-C-ST@1#eq:coordinates, X-C-ST@1#eq:coordinate-laws, X-C-ST@1#eq:bare

Fix $f \in C_{c}^{\infty} (O; R)$ . The coordinate handoff is

w^{μ} [h] v^{μ} [h, k] = - E_{0, R} \partial_{α} (σ a_{1}^{α μ} [h]), = - E_{0, R} \partial_{α} [σ A_{2}^{α μ} [h, k] + σ a_{1}^{α β} [h] \partial_{β} w^{μ} [h]] .

The massless inverse here is the supplied zero-data coordinate prescription at $t = 0$ . Smoothness without spatial decay, fixed-width continuity, and retained-past dependence are inherited chart data.

On $G_{p}$ , the supplied laws are

w [h^{'}] v [h^{'}, k^{'}] = w [h] + ξ, = v [h, k] + X_{ξ} w [h] + ζ + \frac{1}{2} X_{ξ} ξ, X_{ξ} = ξ^{μ} \partial_{μ} .

No proof of X-A-ST's coordinate covariance is claimed again here.

Let

B^{μ} = w^{ν} \partial_{ν} w^{μ} - v^{μ} .

The inherited compact bare tests are

c_{0} c_{2} [h, k] = f, c_{1} [h] = \partial_{μ} (f w^{μ} [h]), = - \partial_{μ} (f B^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w^{μ} [h] w^{ν} [h]) .

They are smooth, supported in $K$ , and continuous in the smooth backgrounds at fixed $ε$ . Their scalar-only and joint admissions were established in X-B-ST XBST. The label test $f$ remains fixed.

Define

c [q] = f + κ c_{1} [h] + κ^{2} c_{2} [h, k] .

For every smooth scalar $ψ$ ,

⟨ c [q], ψ ⟩ = \int f ψ \circ (x + κ w + κ^{2} v)^{- 1} d^{4} x (mod κ^{3}) .

This fixes the insertion representatives, including their signs.

Displayed formulas in this passage

X-C-ST@1#eq:coordinates · line 542
$w^{μ} [h] v^{μ} [h, k] = - E_{0, R} \partial_{α} (σ a_{1}^{α μ} [h]), = - E_{0, R} \partial_{α} [σ A_{2}^{α μ} [h, k] + σ a_{1}^{α β} [h] \partial_{β} w^{μ} [h]] .$
X-C-ST@1#eq:coordinate-laws · line 560
$w [h^{'}] v [h^{'}, k^{'}] = w [h] + ξ, = v [h, k] + X_{ξ} w [h] + ζ + \frac{1}{2} X_{ξ} ξ, X_{ξ} = ξ^{μ} \partial_{μ} .$
X-C-ST@1#eq:bare · line 575
$c_{0} c_{2} [h, k] = f, c_{1} [h] = \partial_{μ} (f w^{μ} [h]), = - \partial_{μ} (f B^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w^{μ} [h] w^{ν} [h]) .$
unlabeled · line 571
$B^{μ} = w^{ν} \partial_{ν} w^{μ} - v^{μ} .$
unlabeled · line 590
$c [q] = f + κ c_{1} [h] + κ^{2} c_{2} [h, k] .$
unlabeled · line 594
$⟨ c [q], ψ ⟩ = \int f ψ \circ (x + κ w + κ^{2} v)^{- 1} d^{4} x (mod κ^{3}) .$