Manuscript III@2 · III.tex · section · 82dad3e5

Residual vector-wave gauges

The C1 triangular Cauchy quotient › Residual vector-wave gauges · lines 1835–1889

III@2#eq:metric-gauge-data, III@2#eq:scalar-gauge-data

Set

G_{C} = C_{c}^{\infty} (R^{3}; R^{4}) \oplus C_{c}^{\infty} (R^{3}; R^{4})

with its usual test-function topology. For $(α, β)$ , let

□_{η} ξ_{μ} = 0, ξ_{μ} ∣_{t_{0}} = α_{μ}, \partial_{t} ξ_{μ} ∣_{t_{0}} = β_{μ} .

These fields are spacelike compact but generally not past trivial.

The induced metric data are

δ A_{00} δ B_{00} = 2 β_{0}, = 2Δ α_{0}, δ A_{0 i} δ B_{0 i} = β_{i} + \partial_{i} α_{0}, = Δ α_{i} + \partial_{i} β_{0}, δ A_{ij} δ B_{ij} = \partial_{i} α_{j} + \partial_{j} α_{i}, = \partial_{i} β_{j} + \partial_{j} β_{i} .

The scalar data transform by

δ u δ v = α^{μ} (\partial_{μ} Φ)_{0}, = β^{μ} (\partial_{μ} Φ)_{0} + α^{μ} (\partial_{t} \partial_{μ} Φ)_{0} .

Here the subscript $0$ means evaluation at $t_{0}$ .

Let

g : G_{C} ⟶ X_{C}

be this combined continuous linear map. Its image is constrained because $(K ξ, X_{ξ} Φ)$ solves (eq:C1harmonic).

The invariant datum is the entire class

[A, B, u, v] \in X_{C} / g (G_{C}) .

The scalar data do not descend separately.

Displayed formulas in this passage

III@2#eq:metric-gauge-data · line 1852
$δ A_{00} δ B_{00} = 2 β_{0}, = 2Δ α_{0}, δ A_{0 i} δ B_{0 i} = β_{i} + \partial_{i} α_{0}, = Δ α_{i} + \partial_{i} β_{0}, δ A_{ij} δ B_{ij} = \partial_{i} α_{j} + \partial_{j} α_{i}, = \partial_{i} β_{j} + \partial_{j} β_{i} .$
III@2#eq:scalar-gauge-data · line 1867
$δ u δ v = α^{μ} (\partial_{μ} Φ)_{0}, = β^{μ} (\partial_{μ} Φ)_{0} + α^{μ} (\partial_{t} \partial_{μ} Φ)_{0} .$
unlabeled · line 1838
$G_{C} = C_{c}^{\infty} (R^{3}; R^{4}) \oplus C_{c}^{\infty} (R^{3}; R^{4})$
unlabeled · line 1844
$□_{η} ξ_{μ} = 0, ξ_{μ} ∣_{t_{0}} = α_{μ}, \partial_{t} ξ_{μ} ∣_{t_{0}} = β_{μ} .$
unlabeled · line 1877
$g : G_{C} ⟶ X_{C}$
unlabeled · line 1885
$[A, B, u, v] \in X_{C} / g (G_{C}) .$