Manuscript V@1 · V.tex · theorem · e7ddf37d

Theorem: Raw current and surface measure

Raw current and complete gauge defect › Distributional reconstruction · lines 671–724 · proof included or linked

V@1#thm:rawcurrent, V@1#eq:Jraw, V@1#eq:surface

For every smooth $h$ ,

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

The current is a compact distribution of order at most zero. It decomposes as

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

where $J_{bulk}$ is (eq:Jraw) with $V = u$ , and

j_{Σ}^{μν} = - δ_{0}^{(μ} u_{0} \partial^{ν)} ϕ ∣_{0} + \frac{1}{2} η^{μν} u_{0} ϕ ˙ ∣_{0} .

Here $u_{0} = u (0)$ . In components,

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. Using (eq:L), integrate the derivative of $h$ in $⟨ V, L_{h} ϕ ⟩$ by parts:

⟨ V, L_{h} ϕ ⟩ = - ⟨ V h^{μν}, \partial_{μ} \partial_{ν} ϕ ⟩ + ⟨ h^{μν}, \partial_{μ} (V \partial_{ν} ϕ) ⟩ - \frac{1}{2} ⟨ tr_{η} h, \partial_{α} (V \partial^{α} ϕ) ⟩ .

The Hessian terms cancel. This proves the current formula first for compact $h$ , and then for arbitrary smooth $h$ by inserting a cutoff equal to one near $K_{f}$ .

The derivative of $Θ u$ is

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

Substitution gives (eq:surface). Only a bounded bulk density and a slice measure occur. Their support is compact by finite propagation.

Displayed formulas in this passage

V@1#eq:Jraw · line 674
$B_{raw} (h) = ⟨ J_{raw}, h ⟩, J_{raw}^{μν} = \partial^{(μ} V \partial^{ν)} ϕ - \frac{1}{2} η^{μν} \partial_{α} (V \partial^{α} ϕ) .$
V@1#eq:surface · line 688
$j_{Σ}^{μν} = - δ_{0}^{(μ} u_{0} \partial^{ν)} ϕ ∣_{0} + \frac{1}{2} η^{μν} u_{0} ϕ ˙ ∣_{0} .$
unlabeled · line 705
$⟨ V, L_{h} ϕ ⟩ = - ⟨ V h^{μν}, \partial_{μ} \partial_{ν} ϕ ⟩ + ⟨ h^{μν}, \partial_{μ} (V \partial_{ν} ϕ) ⟩ - \frac{1}{2} ⟨ tr_{η} h, \partial_{α} (V \partial^{α} ϕ) ⟩ .$
unlabeled · line 684
$J_{raw} = Θ (t) J_{bulk} + δ (t) j_{Σ},$
unlabeled · line 695
$j_{Σ}^{00} = \frac{1}{2} u_{0} ϕ ˙ ∣_{0}, j_{Σ}^{0 i} = - \frac{1}{2} u_{0} \partial_{i} ϕ ∣_{0}, j_{Σ}^{ij} = \frac{1}{2} δ^{ij} u_{0} ϕ ˙ ∣_{0} .$
unlabeled · line 717
$\partial^{μ} (Θ u) = Θ \partial^{μ} u - δ (t) δ_{0}^{μ} u_{0} .$

Theorem: Raw current and surface measure

Displayed formulas in this passage

Nearby in this section