Manuscript XX-F@1 · XX-F.tex · section · de0bd24f

Original rods and the complete second coefficient

Frozen geometric and free-field data › Original rods and the complete second coefficient · lines 310–413

XX-F@1#eq:C, XX-F@1#eq:R, XX-F@1#eq:ell, XX-F@1#eq:q, XX-F@1#eq:qloc, XX-F@1#eq:O

The rods are harmonic on the actual matter-clock slices:

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

There is no lapse weight.

The first inverse clock displacement is $u_{1} = - tφ / A$ . The first induced metric is

(γ_{1})_{ij} = h_{ij} + 2 a^{2} H u_{1} δ_{ij} .

Its original density-source map is

(\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 .

Retain

(G_{sp} b) (x) (\calRz)^{a} P_{i} (r) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d^{3} y, = - G_{sp} \partial_{i} (\calCz)^{ia} = \int P_{i} (x - y) (\calCz)^{ia} (t, y) d^{3} y, = - \frac{r _{i}}{4 π ∣ r ∣ ^{3}} .

The linear lapse coefficient is

ℓ z = - \frac{t}{2 A} h_{00} - \frac{t ^{2}}{A ^{2}} φ ˙ - \frac{t}{A ^{2}} φ .

The full second coefficient is

n_{2} = ℓ ζ + q (z), q (z) = q_{loc} (z) - 2 \calRz \cdot \nabla ℓ z,

where

q_{loc} (z) = - \frac{t}{4 A} h_{00}^{2} + \frac{t}{A a ^{2}} h_{0 i} h_{0 i} + \frac{t ^{2}}{A ^{2}} h_{00} φ ˙ + \frac{2 t ^{3}}{A ^{3}} φ ˙^{2} + \frac{2 t ^{2}}{A ^{2} a ^{2}} h_{0 i} \partial_{i} φ + \frac{t ^{3}}{A ^{3} a ^{2}} \partial_{i} φ \partial_{i} φ + \frac{t ^{2}}{A ^{2}} φ h ˙_{00} + \frac{t}{A ^{2}} φ h_{00} + \frac{2 t ^{3}}{A ^{3}} φ φ ¨ + \frac{6 t ^{2}}{A ^{3}} φ φ ˙ + \frac{t}{A ^{3}} φ^{2} .

For a compact label test $f$ , let

L_{f} (z) = \int f ℓ z d t d^{3} x .

If $b$ is the symmetric polarization of $q$ , set

B_{f} (z, w) = \int f b (z, w) d t d^{3} x .

Then

O_{f} = O_{0, f} + s L_{f} (z) + \frac{s ^{2}}{2} (L_{f} (ζ) + B_{f} (z, z)) (mod s^{3}) .

In particular,

b (z, w) = b_{loc} (z, w) - \calRz \cdot \nabla ℓ w - \calRw \cdot \nabla ℓ z .

The second spatial rod is obtained from

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

It does not occur in $n_{2}$ , because its second inverse-map coefficient multiplies $\partial_{i} N_{0} = 0$ , with $N_{0} = t / A$ . Nothing in this manuscript discards its construction.

Displayed formulas in this passage

XX-F@1#eq:C · line 328
$(\calCz)^{ia} = \frac{1}{2 a ^{2}} δ^{ia} δ^{j k} h_{j k} - a^{- 2} h_{ia} - \frac{p}{A} δ^{ia} φ .$
XX-F@1#eq:R · line 335
$(G_{sp} b) (x) (\calRz)^{a} P_{i} (r) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d^{3} y, = - G_{sp} \partial_{i} (\calCz)^{ia} = \int P_{i} (x - y) (\calCz)^{ia} (t, y) d^{3} y, = - \frac{r _{i}}{4 π ∣ r ∣ ^{3}} .$
XX-F@1#eq:ell · line 350
$ℓ z = - \frac{t}{2 A} h_{00} - \frac{t ^{2}}{A ^{2}} φ ˙ - \frac{t}{A ^{2}} φ .$
XX-F@1#eq:q · line 357
$n_{2} = ℓ ζ + q (z), q (z) = q_{loc} (z) - 2 \calRz \cdot \nabla ℓ z,$
XX-F@1#eq:qloc · line 364
$q_{loc} (z) = - \frac{t}{4 A} h_{00}^{2} + \frac{t}{A a ^{2}} h_{0 i} h_{0 i} + \frac{t ^{2}}{A ^{2}} h_{00} φ ˙ + \frac{2 t ^{3}}{A ^{3}} φ ˙^{2} + \frac{2 t ^{2}}{A ^{2} a ^{2}} h_{0 i} \partial_{i} φ + \frac{t ^{3}}{A ^{3} a ^{2}} \partial_{i} φ \partial_{i} φ + \frac{t ^{2}}{A ^{2}} φ h ˙_{00} + \frac{t}{A ^{2}} φ h_{00} + \frac{2 t ^{3}}{A ^{3}} φ φ ¨ + \frac{6 t ^{2}}{A ^{3}} φ φ ˙ + \frac{t}{A ^{3}} φ^{2} .$
XX-F@1#eq:O · line 392
$O_{f} = O_{0, f} + s L_{f} (z) + \frac{s ^{2}}{2} (L_{f} (ζ) + B_{f} (z, z)) (mod s^{3}) .$
unlabeled · line 313
$\partial_{i} (det γ_{s} γ_{s}^{ij} \partial_{j} R_{s}^{a}) = 0, R_{s}^{a} - x^{a} ⟶ 0 (∣ x ∣ \to \infty) .$
unlabeled · line 324
$(γ_{1})_{ij} = h_{ij} + 2 a^{2} H u_{1} δ_{ij} .$
unlabeled · line 384
$L_{f} (z) = \int f ℓ z d t d^{3} x .$
unlabeled · line 388
$B_{f} (z, w) = \int f b (z, w) d t d^{3} x .$
unlabeled · line 399
$b (z, w) = b_{loc} (z, w) - \calRz \cdot \nabla ℓ w - \calRw \cdot \nabla ℓ z .$
unlabeled · line 406
$Y_{2}^{a} = - G_{sp} \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .$