Manuscript VII-I@1 · VII-I.tex · section · 907e96ab

Conventions

One classical observable and its actual Cauchy data › Conventions · lines 293–369

VII-I@1#eq:C, VII-I@1#eq:coordinates, VII-I@1#eq:coordinate-coefficients

Throughout our constructions,

M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, Σ = {t = 0} .

Put $M_{+}^{\circ} = {t > 0}$ . The retarded and advanced inverses of $□$ are $E^{R}_{0}, E^{A}_{0}$ , and $E_{0} = E^{R}_{0} - E^{A}_{0}$ .

The zero-data solution operator is

□ E_{0} j = j (t > 0), E_{0} j ∣_{Σ} = \partial_{t} E_{0} j ∣_{Σ} = 0.

On future smooth sources it is

E_{0} j = E^{R}_{0} (θ j)_{{t \geq 0}} .

For compact observation sets, no spatial decay is required: finite propagation reduces each calculation to a compact causal region BGP.

Let

q = η + κh + κ^{2} k (mod κ^{3}) .

The coefficient $k$ is metric-jet bookkeeping, not a second independent quantum metric field.

Set

τ (h) = η^{μν} h_{μν},

a_{1}^{μν} [h] a_{2}^{μν} [h] C^{μ} h = \frac{1}{2} τ (h) η^{μν} - h^{μν}, = h^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} τ (h) h^{μν} + (\frac{1}{8} τ (h)^{2} - \frac{1}{4} h_{ρ σ} h^{ρ σ}) η^{μν}, = \partial_{ν} h^{ν μ} - \frac{1}{2} \partial^{μ} τ (h) = - \partial_{ν} a_{1}^{ν μ} [h] .

The coordinates solve

\partial_{α} (- q q^{α β} \partial_{β} X_{q}^{μ}) = 0, X_{q}^{μ} ∣_{Σ} = x^{μ}, \partial_{t} X_{q}^{μ} ∣_{Σ} = δ^{μ}_{0} .

Their formal expansion is

X_{q}^{μ} w^{μ} [h] v^{μ} [h, k] = x^{μ} + κ w^{μ} [h] + κ^{2} v^{μ} [h, k], = E_{0} C^{μ} h, = E_{0} C^{μ} k - E_{0} [\partial_{α} a_{2}^{α μ} [h] + \partial_{α} (a_{1}^{α β} [h] \partial_{β} w^{μ} [h])] .

Displayed formulas in this passage

VII-I@1#eq:C · line 330
$a_{1}^{μν} [h] a_{2}^{μν} [h] C^{μ} h = \frac{1}{2} τ (h) η^{μν} - h^{μν}, = h^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} τ (h) h^{μν} + (\frac{1}{8} τ (h)^{2} - \frac{1}{4} h_{ρ σ} h^{ρ σ}) η^{μν}, = \partial_{ν} h^{ν μ} - \frac{1}{2} \partial^{μ} τ (h) = - \partial_{ν} a_{1}^{ν μ} [h] .$
VII-I@1#eq:coordinates · line 346
$\partial_{α} (- q q^{α β} \partial_{β} X_{q}^{μ}) = 0, X_{q}^{μ} ∣_{Σ} = x^{μ}, \partial_{t} X_{q}^{μ} ∣_{Σ} = δ^{μ}_{0} .$
VII-I@1#eq:coordinate-coefficients · line 355
$X_{q}^{μ} w^{μ} [h] v^{μ} [h, k] = x^{μ} + κ w^{μ} [h] + κ^{2} v^{μ} [h, k], = E_{0} C^{μ} h, = E_{0} C^{μ} k - E_{0} [\partial_{α} a_{2}^{α μ} [h] + \partial_{α} (a_{1}^{α β} [h] \partial_{β} w^{μ} [h])] .$
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$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, Σ = {t = 0} .$
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$□ E_{0} j = j (t > 0), E_{0} j ∣_{Σ} = \partial_{t} E_{0} j ∣_{Σ} = 0.$
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$E_{0} j = E^{R}_{0} (θ j)_{{t \geq 0}} .$
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$q = η + κh + κ^{2} k (mod κ^{3}) .$
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$τ (h) = η^{μν} h_{μν},$