Manuscript X-E-ST@1 · X-E-ST.tex · section · 44614221

Coordinates, scalar operators, and tests

Frozen data and the inherited scalar completion › Coordinates, scalar operators, and tests · lines 364–430

X-E-ST@1#eq:coordinates, X-E-ST@1#eq:scalarL, X-E-ST@1#eq:completed

Let

□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2} .

Use the positive retarded and advanced inverses $E_{m}^{R}, E_{m}^{A}$ of $P_{m}$ , and $E_{0}^{R}, E_{0}^{A}$ of $□$ . In particular,

P_{m} E_{m}^{R} = P_{m} E_{m}^{A} = 1, E_{m}^{R}^{T} = E_{m}^{A} .

Their Cauchy, support, and continuity properties are used only on the completion BGP.

For a smooth symmetric tensor $l$ , write

w^{ρ} [l] = - E_{0}^{R} \partial_{μ} (σ a_{1}^{μ ρ} [l]) .

The second coordinate coefficient solves

□ v^{ρ} [h, k] = - \partial_{μ} {σ (A^{μ ρ} [h, k] + a_{1}^{μν} [h] \partial_{ν} w^{ρ} [h])},

with zero position and velocity at $Σ$ . Retarded expressions for nondecaying backgrounds are defined locally by the finite-time Cauchy problem. Every retained dependence lies in $D_{f}$ .

The compactly switched Euler coefficients are

L_{1} [l] u L_{2} [h, k] u = \partial_{μ} (g a_{1}^{μν} [l] \partial_{ν} u) + g m^{2} b_{1} [l] u, = \partial_{μ} (g A^{μν} [h, k] \partial_{ν} u) + g m^{2} B [h, k] u .

Each divergence retains its $\partial g$ terms.

Define

c_{1} [h] c_{2} [h, k] d_{1} [h] d_{2} [h, k] = \partial_{μ} (f w^{μ} [h]), = - \partial_{μ} (f (w^{ν} [h] \partial_{ν} w^{μ} [h] - v^{μ} [h, k])) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w^{μ} [h] w^{ν} [h]), = c_{1} [h] - L_{1} [h] E_{m}^{A} f, = c_{2} [h, k] - L_{1} [h] E_{m}^{A} c_{1} [h] - L_{2} [h, k] E_{m}^{A} f + L_{1} [h] E_{m}^{A} L_{1} [h] E_{m}^{A} f .

These are the completed tests of X-C-ST XCST,XC. They are restated to fix the insertion target, not reconstructed as additional response maps.

Displayed formulas in this passage

X-E-ST@1#eq:coordinates · line 385
$□ v^{ρ} [h, k] = - \partial_{μ} {σ (A^{μ ρ} [h, k] + a_{1}^{μν} [h] \partial_{ν} w^{ρ} [h])},$
X-E-ST@1#eq:scalarL · line 399
$L_{1} [l] u L_{2} [h, k] u = \partial_{μ} (g a_{1}^{μν} [l] \partial_{ν} u) + g m^{2} b_{1} [l] u, = \partial_{μ} (g A^{μν} [h, k] \partial_{ν} u) + g m^{2} B [h, k] u .$
X-E-ST@1#eq:completed · line 412
$c_{1} [h] c_{2} [h, k] d_{1} [h] d_{2} [h, k] = \partial_{μ} (f w^{μ} [h]), = - \partial_{μ} (f (w^{ν} [h] \partial_{ν} w^{μ} [h] - v^{μ} [h, k])) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w^{μ} [h] w^{ν} [h]), = c_{1} [h] - L_{1} [h] E_{m}^{A} f, = c_{2} [h, k] - L_{1} [h] E_{m}^{A} c_{1} [h] - L_{2} [h, k] E_{m}^{A} f + L_{1} [h] E_{m}^{A} L_{1} [h] E_{m}^{A} f .$
unlabeled · line 367
$□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2} .$
unlabeled · line 373
$P_{m} E_{m}^{R} = P_{m} E_{m}^{A} = 1, E_{m}^{R}^{T} = E_{m}^{A} .$
unlabeled · line 380
$w^{ρ} [l] = - E_{0}^{R} \partial_{μ} (σ a_{1}^{μ ρ} [l]) .$

Coordinates, scalar operators, and tests

Displayed formulas in this passage

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