Manuscript XXII-E@1 · XXII-E.tex · definition · 51c99d2f

Definition: Linearized Einstein comparator

A well-defined linearized comparison › Finite-dimensional metric jets · lines 689–718

XXII-E@1#def:einstein, XXII-E@1#eq:ricci-jet, XXII-E@1#eq:scalar-jet, XXII-E@1#eq:einstein-jet

Use $η$ to raise and contract indices, and define

R^{(1)}_{μν} (j) R^{(1)} (j) E_{η} (j)_{μν} = \frac{1}{2} η^{ρ σ} (h_{σ ν ∣ ρ μ} + h_{σ μ ∣ ρ ν} - h_{μν ∣ ρ σ} - h_{ρ σ ∣ μν}), = η^{μν} R^{(1)}_{μν} (j), = R^{(1)}_{μν} (j) - \frac{1}{2} η_{μν} R^{(1)} (j) .

Thus

E_{η} : J^{2} ⟶ Sym^{2} (V_{D}^{*})

is a fixed linear map. For a smooth field $h$ , write $E_{η} [h] (x) = E_{η} (j_{x}^{2} h)$ .

Displayed formulas in this passage

XXII-E@1#eq:ricci-jet · line 692
$R^{(1)}_{μν} (j) R^{(1)} (j) E_{η} (j)_{μν} = \frac{1}{2} η^{ρ σ} (h_{σ ν ∣ ρ μ} + h_{σ μ ∣ ρ ν} - h_{μν ∣ ρ σ} - h_{ρ σ ∣ μν}), = η^{μν} R^{(1)}_{μν} (j), = R^{(1)}_{μν} (j) - \frac{1}{2} η_{μν} R^{(1)} (j) .$
XXII-E@1#eq:scalar-jet · line 692
$R^{(1)}_{μν} (j) R^{(1)} (j) E_{η} (j)_{μν} = \frac{1}{2} η^{ρ σ} (h_{σ ν ∣ ρ μ} + h_{σ μ ∣ ρ ν} - h_{μν ∣ ρ σ} - h_{ρ σ ∣ μν}), = η^{μν} R^{(1)}_{μν} (j), = R^{(1)}_{μν} (j) - \frac{1}{2} η_{μν} R^{(1)} (j) .$
XXII-E@1#eq:einstein-jet · line 692
$R^{(1)}_{μν} (j) R^{(1)} (j) E_{η} (j)_{μν} = \frac{1}{2} η^{ρ σ} (h_{σ ν ∣ ρ μ} + h_{σ μ ∣ ρ ν} - h_{μν ∣ ρ σ} - h_{ρ σ ∣ μν}), = η^{μν} R^{(1)}_{μν} (j), = R^{(1)}_{μν} (j) - \frac{1}{2} η_{μν} R^{(1)} (j) .$
unlabeled · line 712
$E_{η} : J^{2} ⟶ Sym^{2} (V_{D}^{*})$