Manuscript V-B@1 · V-B-v1.tex · theorem · 3fadc1ad

Theorem: Distributional current and initial-surface term

Raw current, surface term, and complete Ward defect · lines 435–477 · proof included or linked

V-B@1#thm:raw, V-B@1#eq:Jraw, V-B@1#eq:surface

For every smooth $h$ ,

B_{f} (h, ϕ) = ⟨ J_{raw}, h ⟩,

where

J_{raw}^{μν} = \partial^{(μ} V \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (V \partial^{α} ϕ) .

This is a compact order-zero tensor distribution. It has the form

J_{raw} = Θ (t) J_{bulk} + δ (t) j_{Σ},

with

j_{Σ}^{00} = \frac{1}{2} u_{0} ϕ ˙_{0}, j_{Σ}^{0 i} = - \frac{1}{2} u_{0} \partial_{i} ϕ_{0}, j_{Σ}^{ij} = \frac{1}{2} δ^{ij} u_{0} ϕ ˙_{0} .

Proof. Integrate the metric derivative in $⟨ V, L_{h} ϕ ⟩$ by parts. The terms containing $V h^{μν} \partial_{μ} \partial_{ν} ϕ$ cancel, leaving (eq:Jraw). A cutoff equal to one near $K_{f}$ extends the calculation from compact to arbitrary smooth $h$ .

Since

\partial^{μ} (Θ u) = Θ \partial^{μ} u - δ (t) δ^{μ}_{0} u_{0},

the singular contribution is exactly (eq:surface). The remaining density is bounded and piecewise smooth on a compact set.

Displayed formulas in this passage

V-B@1#eq:Jraw · line 442
$J_{raw}^{μν} = \partial^{(μ} V \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (V \partial^{α} ϕ) .$
V-B@1#eq:surface · line 454
$j_{Σ}^{00} = \frac{1}{2} u_{0} ϕ ˙_{0}, j_{Σ}^{0 i} = - \frac{1}{2} u_{0} \partial_{i} ϕ_{0}, j_{Σ}^{ij} = \frac{1}{2} δ^{ij} u_{0} ϕ ˙_{0} .$
unlabeled · line 438
$B_{f} (h, ϕ) = ⟨ J_{raw}, h ⟩,$
unlabeled · line 450
$J_{raw} = Θ (t) J_{bulk} + δ (t) j_{Σ},$
unlabeled · line 470
$\partial^{μ} (Θ u) = Θ \partial^{μ} u - δ (t) δ^{μ}_{0} u_{0},$

Theorem: Distributional current and initial-surface term

Displayed formulas in this passage

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