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

Compact curvature currents

Restricted compact representatives on a two-dimensional TT plane › Compact curvature currents · lines 1747–1782

V@1#eq:electric, V@1#eq:curvaturecurrent, V@1#eq:curvaturepairing

Define the electric linearized curvature

E_{el} (h)_{ij} = \frac{1}{2} (\partial_{0} \partial_{i} h_{0 j} + \partial_{0} \partial_{j} h_{0 i} - \partial_{i} \partial_{j} h_{00} - \partial_{0}^{2} h_{ij}) .

It vanishes on $K ξ$ . For a TT wave,

E_{el} (k)_{ij} = - \frac{1}{2} \partial_{t}^{2} k_{ij} = - \frac{1}{2} Δ k_{ij} .

For a compact smooth symmetric spatial tensor $C^{ij}$ , set

J_{C}^{00} = - \frac{1}{2} \partial_{i} \partial_{j} C^{ij}, J_{C}^{0 j} = J_{C}^{j 0} = \frac{1}{2} \partial_{0} \partial_{i} C^{ij}, J_{C}^{ij} = - \frac{1}{2} \partial_{0}^{2} C^{ij} .

Direct differentiation shows $\partial_{μ} J_{C}^{μν} = 0$ , and two integrations by parts show

F_{J_{C}} (h) = \int C^{ij} E_{el} (h)_{ij} d^{4} x .

In the $0 j$ terms, both symmetric components in the full tensor pairing account for the factor $1/2$ . Thus $J_{C} \in E$ .

Displayed formulas in this passage

V@1#eq:electric · line 1750
$E_{el} (h)_{ij} = \frac{1}{2} (\partial_{0} \partial_{i} h_{0 j} + \partial_{0} \partial_{j} h_{0 i} - \partial_{i} \partial_{j} h_{00} - \partial_{0}^{2} h_{ij}) .$
V@1#eq:curvaturecurrent · line 1766
$J_{C}^{00} = - \frac{1}{2} \partial_{i} \partial_{j} C^{ij}, J_{C}^{0 j} = J_{C}^{j 0} = \frac{1}{2} \partial_{0} \partial_{i} C^{ij}, J_{C}^{ij} = - \frac{1}{2} \partial_{0}^{2} C^{ij} .$
V@1#eq:curvaturepairing · line 1775
$F_{J_{C}} (h) = \int C^{ij} E_{el} (h)_{ij} d^{4} x .$
unlabeled · line 1760
$E_{el} (k)_{ij} = - \frac{1}{2} \partial_{t}^{2} k_{ij} = - \frac{1}{2} Δ k_{ij} .$