Manuscript Em06-A@1 · Em06-A-v1.tex · section · ee9923c6

Green operators and the harmonic coordinate coefficient

The classical object before the limit › Green operators and the harmonic coordinate coefficient · lines 237–292

Em06-A@1#eq:retarded, Em06-A@1#eq:advanced, Em06-A@1#eq:w, Em06-A@1#eq:phi1

Let $G_{m}^{R}$ and $G_{m}^{A}$ be the retarded and advanced Green operators of $P_{m}$ . For compact smooth sources,

P_{m} G_{m}^{R} v = P_{m} G_{m}^{A} v = v, (G_{m}^{R})^{*} = G_{m}^{A} .

With $D_{m} = - Δ + m^{2}$ , the chosen signs are

(G_{m}^{R} v) (t) (G_{m}^{A} v) (t) = \int_{- \infty}^{t} \frac{sin (( t - s ) D _{m} )}{D _{m}} v (s) d s, = \int_{t}^{\infty} \frac{sin (( s - t ) D _{m} )}{D _{m}} v (s) d s .

These formulas give the Green identities and finite propagation. They also fix the signs independently of differently normalized Green operators in comparison manuscripts. Standard general Cauchy and Green-operator results may be found in BGP.

Define $w [h]$ , for $t \geq 0$ , by the componentwise massless Cauchy problem

□ w [h]^{μ} = C (h)^{μ}, w [h]^{μ} ∣_{Σ_{0}} = 0, \partial_{t} w [h]^{μ} ∣_{Σ_{0}} = 0.

This problem has a unique smooth solution on every finite time slab. For arbitrary smooth $h$ , it is understood locally through finite propagation; no spatial Fourier transform of an arbitrary nondecaying $h$ is required.

The formal metric is $g_{κ} = η + κh$ . Indeed,

\partial_{κ} □_{g_{κ}} v ∣_{κ = 0} = - h^{μν} \partial_{μ} \partial_{ν} v - C (h)^{ν} \partial_{ν} v .

Consequently, $X_{κ}^{μ} = x^{μ} + κ w [h]^{μ}$ obeys the harmonic coordinate equation through first order. Its inverse has first coefficient $- w [h]$ .

The switched first scalar coefficient is

ϕ_{1, χ} [h; ϕ_{0}] = - G_{m}^{R} Q_{χ h} ϕ_{0} .

Its source is compact and lies strictly to the future of $Σ_{0}$ , so it has zero Cauchy data there.

Displayed formulas in this passage

Em06-A@1#eq:retarded · line 247
$(G_{m}^{R} v) (t) (G_{m}^{A} v) (t) = \int_{- \infty}^{t} \frac{sin (( t - s ) D _{m} )}{D _{m}} v (s) d s, = \int_{t}^{\infty} \frac{sin (( s - t ) D _{m} )}{D _{m}} v (s) d s .$
Em06-A@1#eq:advanced · line 247
$(G_{m}^{R} v) (t) (G_{m}^{A} v) (t) = \int_{- \infty}^{t} \frac{sin (( t - s ) D _{m} )}{D _{m}} v (s) d s, = \int_{t}^{\infty} \frac{sin (( s - t ) D _{m} )}{D _{m}} v (s) d s .$
Em06-A@1#eq:w · line 264
$□ w [h]^{μ} = C (h)^{μ}, w [h]^{μ} ∣_{Σ_{0}} = 0, \partial_{t} w [h]^{μ} ∣_{Σ_{0}} = 0.$
Em06-A@1#eq:phi1 · line 285
$ϕ_{1, χ} [h; ϕ_{0}] = - G_{m}^{R} Q_{χ h} ϕ_{0} .$
unlabeled · line 241
$P_{m} G_{m}^{R} v = P_{m} G_{m}^{A} v = v, (G_{m}^{R})^{*} = G_{m}^{A} .$
unlabeled · line 276
$\partial_{κ} □_{g_{κ}} v ∣_{κ = 0} = - h^{μν} \partial_{μ} \partial_{ν} v - C (h)^{ν} \partial_{ν} v .$

Green operators and the harmonic coordinate coefficient

Displayed formulas in this passage

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