Manuscript VII-A@1 · VII-A.tex · example · 11da3fe6

Example: An explicit sourced jet

Restriction to a named Einstein–scalar coupling jet › A nonzero-source example · lines 1519–1569

VII-A@1#ex:sourced, VII-A@1#eq:sourcedk

Take $m = 0$ , a nonzero constant covector $a_{μ}$ , and

ϕ_{0} = a_{μ} x^{μ}, h = 0.

Then $P_{0} ϕ_{0} = 0$ , and

T_{μν} = a_{μ} a_{ν} - \frac{1}{2} η_{μν} a_{ρ} a^{ρ}

is constant. Let $T = η^{μν} T_{μν}$ and set

σ (x) = (- \frac{1}{4} T_{α β} + \frac{1}{12} η_{α β} T) x^{α} x^{β}, k_{μν} = 2 σ η_{μν} .

For a conformal linearized perturbation in four dimensions,

G_{μν}^{(1)} [2 σ η] = - 2 \partial_{μ} \partial_{ν} σ + 2 η_{μν} □_{η} σ .

Here

\partial_{μ} \partial_{ν} σ = - \frac{1}{2} T_{μν} + \frac{1}{6} η_{μν} T, □_{η} σ = \frac{1}{6} T,

so $G^{(1)} [k] = T$ as a tensor equation. Furthermore,

ϕ_{1, g} = 0, ϕ_{2, g} = - E_{0} Q_{gk}^{(0)} ϕ_{0} .

This is a nonzero-source member of $J_{2}^{g}$ .

The polynomial metric coefficient is used formally on finite causal domains. No asymptotic-flatness assertion or global smallness assertion is attached to this example.

Displayed formulas in this passage

VII-A@1#eq:sourcedk · line 1532
$σ (x) = (- \frac{1}{4} T_{α β} + \frac{1}{12} η_{α β} T) x^{α} x^{β}, k_{μν} = 2 σ η_{μν} .$
unlabeled · line 1522
$ϕ_{0} = a_{μ} x^{μ}, h = 0.$
unlabeled · line 1526
$T_{μν} = a_{μ} a_{ν} - \frac{1}{2} η_{μν} a_{ρ} a^{ρ}$
unlabeled · line 1544
$G_{μν}^{(1)} [2 σ η] = - 2 \partial_{μ} \partial_{ν} σ + 2 η_{μν} □_{η} σ .$
unlabeled · line 1551
$\partial_{μ} \partial_{ν} σ = - \frac{1}{2} T_{μν} + \frac{1}{6} η_{μν} T, □_{η} σ = \frac{1}{6} T,$
unlabeled · line 1560
$ϕ_{1, g} = 0, ϕ_{2, g} = - E_{0} Q_{gk}^{(0)} ϕ_{0} .$