Manuscript XX-D@1 · XX-D.tex · section · c83a526d

Original spatial rods

Complete classical coefficient through second order › Original spatial rods · lines 519–597

XX-D@1#eq:B12, XX-D@1#eq:rod, XX-D@1#eq:Newton, XX-D@1#eq:Y, XX-D@1#eq:C, XX-D@1#eq:P

Let

C = a^{- 2} γ_{1}, D = a^{- 2} γ_{2} .

Expanding the original elliptic density gives

det γ_{s} γ_{s}^{- 1} = a (I + s B_{1} + \frac{s ^{2}}{2} B_{2}),

where

B_{1} B_{2} = \frac{1}{2} (tr C) I - C, = (\frac{1}{2} tr D + \frac{1}{4} (tr C)^{2} - \frac{1}{2} tr (C^{2})) I - (tr C) C + 2 C^{2} - D .

There is no lapse multiplier in this density.

The rods satisfy

\partial_{i} (det γ_{s} γ_{s}^{ij} \partial_{j} R_{s}^{a}) = 0, R_{s}^{a} - x^{a} ⟶ 0 (∣ x ∣ \to \infty) .

For

R_{s}^{a} = x^{a} + s Y_{1}^{a} + \frac{s ^{2}}{2} Y_{2}^{a},

retain the spatial inverse

(G_{sp} b) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{b ( y )}{∣ x - y ∣} d^{3} y, Δ G_{sp} b = b .

It follows that

Y_{1}^{a} Y_{2}^{a} = - G_{sp} \partial_{i} B_{1}^{ia}, = - G_{sp} \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .

Define the local rod-source map

(\calCz)^{ia} = \frac{1}{2 a ^{2}} δ^{ia} δ^{j k} h_{j k} - a^{- 2} h_{ia} - \frac{p}{A} δ^{ia} φ .

Then $B_{1} = \calCz$ , and

Y_{1} = \calRz, (\calRz)^{a} = - G_{sp} \partial_{i} (\calCz)^{ia} .

After integration by parts,

(\calRz)^{a} (t, x) P_{i} (r) = \int_{R^{3}} P_{i} (x - y) (\calCz)^{ia} (t, y) d^{3} y, = - \frac{r _{i}}{4 π ∣ r ∣ ^{3}} .

The kernel $P_{i}$ is locally integrable and has spatial scaling degree two.

Displayed formulas in this passage

XX-D@1#eq:B12 · line 531
$B_{1} B_{2} = \frac{1}{2} (tr C) I - C, = (\frac{1}{2} tr D + \frac{1}{4} (tr C)^{2} - \frac{1}{2} tr (C^{2})) I - (tr C) C + 2 C^{2} - D .$
XX-D@1#eq:rod · line 543
$\partial_{i} (det γ_{s} γ_{s}^{ij} \partial_{j} R_{s}^{a}) = 0, R_{s}^{a} - x^{a} ⟶ 0 (∣ x ∣ \to \infty) .$
XX-D@1#eq:Newton · line 556
$(G_{sp} b) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{b ( y )}{∣ x - y ∣} d^{3} y, Δ G_{sp} b = b .$
XX-D@1#eq:Y · line 565
$Y_{1}^{a} Y_{2}^{a} = - G_{sp} \partial_{i} B_{1}^{ia}, = - G_{sp} \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .$
XX-D@1#eq:C · line 574
$(\calCz)^{ia} = \frac{1}{2 a ^{2}} δ^{ia} δ^{j k} h_{j k} - a^{- 2} h_{ia} - \frac{p}{A} δ^{ia} φ .$
XX-D@1#eq:P · line 586
$(\calRz)^{a} (t, x) P_{i} (r) = \int_{R^{3}} P_{i} (x - y) (\calCz)^{ia} (t, y) d^{3} y, = - \frac{r _{i}}{4 π ∣ r ∣ ^{3}} .$
unlabeled · line 522
$C = a^{- 2} γ_{1}, D = a^{- 2} γ_{2} .$
unlabeled · line 526
$det γ_{s} γ_{s}^{- 1} = a (I + s B_{1} + \frac{s ^{2}}{2} B_{2}),$
unlabeled · line 552
$R_{s}^{a} = x^{a} + s Y_{1}^{a} + \frac{s ^{2}}{2} Y_{2}^{a},$
unlabeled · line 581
$Y_{1} = \calRz, (\calRz)^{a} = - G_{sp} \partial_{i} (\calCz)^{ia} .$