Manuscript VII-A-ST@1 · VII-A-ST.tex · proposition · c39a67a3

Proposition: A nonempty sourced massless subclass

The named plateau Einstein–scalar coupling jet · lines 1187–1225 · proof included or linked

VII-A-ST@1#prop:nonempty, VII-A-ST@1#eq:sourced-example

For $m = 0$ , let $a_{μ} \neq = 0$ be constant and put

ϕ_{0} = a_{μ} x^{μ}, h = 0, T_{μν} = a_{μ} a_{ν} - \frac{1}{2} η_{μν} a_{ρ} a^{ρ}, T = η^{μν} T_{μν} .

Define

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} .

Together with the assigned switched scalar coefficients, this is a nonzero-source member of $J_{g}^{2}$ .

Proof. The leading scalar is free and its stress is the displayed constant nonzero tensor. Direct linearized curvature gives

G^{(1)} [2 σ η]_{μν} = - 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 .

Thus $G^{(1)} [k] = T$ . The scalar coefficients exist by Proposition prop:scalar; $u_{1, g} = 0$ and $u_{2, g} = - E_{0} Q_{g k}^{(0)} ϕ_{0}$ .

Displayed formulas in this passage

VII-A-ST@1#eq:sourced-example · line 1197
$σ (x) = (- \frac{1}{4} T_{α β} + \frac{1}{12} η_{α β} T) x^{α} x^{β}, k_{μν} = 2 σ η_{μν} .$
unlabeled · line 1190
$ϕ_{0} = a_{μ} x^{μ}, h = 0, T_{μν} = a_{μ} a_{ν} - \frac{1}{2} η_{μν} a_{ρ} a^{ρ}, T = η^{μν} T_{μν} .$
unlabeled · line 1211
$G^{(1)} [2 σ η]_{μν} = - 2 \partial_{μ} \partial_{ν} σ + 2 η_{μν} □_{η} σ .$
unlabeled · line 1217
$\partial_{μ} \partial_{ν} σ = - \frac{1}{2} T_{μν} + \frac{1}{6} η_{μν} T, □_{η} σ = \frac{1}{6} T .$