Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

Original rods and their separate spatial inverse

The complete background and classical prescription › Original rods and their separate spatial inverse · lines 348–422

XX-B@1#eq:induced, XX-B@1#eq:rods, XX-B@1#eq:gamma1, XX-B@1#eq:gamma2, XX-B@1#eq:B1, XX-B@1#eq:B2, XX-B@1#eq:Newton, XX-B@1#eq:Y1, XX-B@1#eq:Y2

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On each clock slice, use the induced metric

γ_{s, ij} = (g_{s})_{ij} (θ_{s}, x) + (g_{s})_{0 i} (θ_{s}, x) \partial_{j} θ_{s} + (g_{s})_{0 j} (θ_{s}, x) \partial_{i} θ_{s} + (g_{s})_{00} (θ_{s}, x) \partial_{i} θ_{s} \partial_{j} θ_{s} .

The original rods are defined by

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

There is no lapse weight in this equation.

Write

γ_{s} = a^{2} δ + s γ_{1} + \frac{s ^{2}}{2} γ_{2} .

The complete embedding contributions are

(γ_{1})_{ij} (γ_{2})_{ij} = h_{ij} + 2 a^{2} H u_{1} δ_{ij}, = k_{ij} + 2 u_{1} h ˙_{ij} + u_{2} \partial_{t} (a^{2} δ_{ij}) + u_{1}^{2} \partial_{t}^{2} (a^{2} δ_{ij}) + 2 h_{0 i} \partial_{j} u_{1} + 2 h_{0 j} \partial_{i} u_{1} - 2 \partial_{i} u_{1} \partial_{j} u_{1} .

Define

C = a^{- 2} γ_{1}, D = a^{- 2} γ_{2}, a^{- 1} det γ_{s} γ_{s}^{- 1} = I + s B_{1} + \frac{s ^{2}}{2} B_{2} .

Then

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 - D - (tr C) C + 2 C^{2} .

The spatial inverse is

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

For $R_{s} = x + s Y_{1} + s^{2} Y_{2} /2$ ,

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}) .

Displayed formulas in this passage

XX-B@1#eq:induced · line 351
$γ_{s, ij} = (g_{s})_{ij} (θ_{s}, x) + (g_{s})_{0 i} (θ_{s}, x) \partial_{j} θ_{s} + (g_{s})_{0 j} (θ_{s}, x) \partial_{i} θ_{s} + (g_{s})_{00} (θ_{s}, x) \partial_{i} θ_{s} \partial_{j} θ_{s} .$
XX-B@1#eq:rods · line 361
$\partial_{i} (det γ_{s} γ_{s}^{ij} \partial_{j} R_{s}^{a}) = 0, R_{s}^{a} - x^{a} ⟶ 0 (\abs x \to \infty) .$
XX-B@1#eq:gamma1 · line 377
$(γ_{1})_{ij} (γ_{2})_{ij} = h_{ij} + 2 a^{2} H u_{1} δ_{ij}, = k_{ij} + 2 u_{1} h ˙_{ij} + u_{2} \partial_{t} (a^{2} δ_{ij}) + u_{1}^{2} \partial_{t}^{2} (a^{2} δ_{ij}) + 2 h_{0 i} \partial_{j} u_{1} + 2 h_{0 j} \partial_{i} u_{1} - 2 \partial_{i} u_{1} \partial_{j} u_{1} .$
XX-B@1#eq:gamma2 · line 377
$(γ_{1})_{ij} (γ_{2})_{ij} = h_{ij} + 2 a^{2} H u_{1} δ_{ij}, = k_{ij} + 2 u_{1} h ˙_{ij} + u_{2} \partial_{t} (a^{2} δ_{ij}) + u_{1}^{2} \partial_{t}^{2} (a^{2} δ_{ij}) + 2 h_{0 i} \partial_{j} u_{1} + 2 h_{0 j} \partial_{i} u_{1} - 2 \partial_{i} u_{1} \partial_{j} u_{1} .$
XX-B@1#eq:B1 · line 397
$B_{1} B_{2} = \frac{1}{2} (tr C) I - C, = (\frac{1}{2} tr D + \frac{1}{4} (tr C)^{2} - \frac{1}{2} tr (C^{2})) I - D - (tr C) C + 2 C^{2} .$
XX-B@1#eq:B2 · line 397
$B_{1} B_{2} = \frac{1}{2} (tr C) I - C, = (\frac{1}{2} tr D + \frac{1}{4} (tr C)^{2} - \frac{1}{2} tr (C^{2})) I - D - (tr C) C + 2 C^{2} .$
XX-B@1#eq:Newton · line 406
$(G_{sp} b) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{b ( y )}{\abs x - y} d^{3} y, Δ G_{sp} b = b .$
XX-B@1#eq:Y1 · line 415
$Y_{1}^{a} Y_{2}^{a} = - G_{sp} \partial_{i} B_{1}^{ia}, = - G_{sp} \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .$
XX-B@1#eq:Y2 · line 415
$Y_{1}^{a} Y_{2}^{a} = - G_{sp} \partial_{i} B_{1}^{ia}, = - G_{sp} \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .$
unlabeled · line 373
$γ_{s} = a^{2} δ + s γ_{1} + \frac{s ^{2}}{2} γ_{2} .$
unlabeled · line 391
$C = a^{- 2} γ_{1}, D = a^{- 2} γ_{2}, a^{- 1} det γ_{s} γ_{s}^{- 1} = I + s B_{1} + \frac{s ^{2}}{2} B_{2} .$

Original rods and their separate spatial inverse

Displayed formulas in this passage

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