Manuscript IX-C-ST@1 · IX-C-ST.tex · section · 892f50c4

Shared analytical inputs and notation

Shared analytical inputs and notation · lines 384–416

IX-C-ST@1#eq:generators

Configurations $Ψ = (h, ϕ)$ and configuration directions are smooth and off shell on the supported presentation. Metric configurations are symmetric covariant tensors. All compact coefficient kernels are understood with their actual tensor symmetries and retained boundary terms.

Set

(C h)^{μ} B_{f} [h, ϕ] = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), = - \int_{V} f (x) w^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, L W_{g} [h] = E_{0} C, = \int_{V} g (y) h_{00} (y)^{2} d^{4} y . w = L h,

The notation $E_{0}$ refers only to the supplied supported comparison response.

Write

x = (t, x), y = (s, y), x^{'} = (t^{'}, x^{'}), X = V^{3} .

Distributional pairings are complex linear in their test. Complex tests give complex-valued functionals on real configurations; complexified directions are equivalent by polarization.

Displayed formulas in this passage

IX-C-ST@1#eq:generators · line 394
$(C h)^{μ} B_{f} [h, ϕ] = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), = - \int_{V} f (x) w^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, L W_{g} [h] = E_{0} C, = \int_{V} g (y) h_{00} (y)^{2} d^{4} y . w = L h,$
unlabeled · line 408
$x = (t, x), y = (s, y), x^{'} = (t^{'}, x^{'}), X = V^{3} .$