Manuscript XIX-A@1 · XIX-A.tex · section · e30d8062

A self-contained second-order composition identity

The scalar-clock lapse and the original spatial rods › A self-contained second-order composition identity · lines 568–620

XIX-A@1#eq:Scoeff, XIX-A@1#eq:Ncoeff, XIX-A@1#eq:ucoeff

All perturbations use the factorial convention

q_{e} = q_{0} + e h + \frac{e ^{2}}{2} k, T_{e} = T_{0} + e θ_{1} + \frac{e ^{2}}{2} θ_{2} .

Write

q_{e}^{- 1} = R_{0} + e R_{1} + \frac{e ^{2}}{2} R_{2}, S [q_{e}, T_{e}] = v^{2} + e S_{1} + \frac{e ^{2}}{2} S_{2} .

Matrix inversion gives

R_{0} = q_{0}^{- 1}, R_{1} = - q_{0}^{- 1} h q_{0}^{- 1}, R_{2} = 2 q_{0}^{- 1} h q_{0}^{- 1} h q_{0}^{- 1} - q_{0}^{- 1} k q_{0}^{- 1} .

Consequently,

S_{1} S_{2} = - R_{1}^{μν} \partial_{μ} T_{0} \partial_{ν} T_{0} - 2 R_{0}^{μν} \partial_{μ} T_{0} \partial_{ν} θ_{1}, = - R_{2}^{μν} \partial_{μ} T_{0} \partial_{ν} T_{0} - 4 R_{1}^{μν} \partial_{μ} T_{0} \partial_{ν} θ_{1} - 2 R_{0}^{μν} \partial_{μ} θ_{1} \partial_{ν} θ_{1} - 2 R_{0}^{μν} \partial_{μ} T_{0} \partial_{ν} θ_{2} .

No vector or tensor metric components have been discarded.

The raw lapse coefficients are

N_{1} = - \frac{S _{1}}{2 v ^{3}}, N_{2} = \frac{3 S _{1} ^{2}}{4 v ^{5}} - \frac{S _{2}}{2 v ^{3}} .

Write

Y_{e}^{i} = x^{i} + e Y_{1}^{i} + \frac{e ^{2}}{2} Y_{2}^{i} .

Define

u_{1} = - \frac{θ _{1}}{v}, u_{2} = - \frac{θ _{2} + 2 u _{1} θ ˙ _{1} + v ˙ u _{1} ^{2}}{v} .

Displayed formulas in this passage

XIX-A@1#eq:Scoeff · line 587
$S_{1} S_{2} = - R_{1}^{μν} \partial_{μ} T_{0} \partial_{ν} T_{0} - 2 R_{0}^{μν} \partial_{μ} T_{0} \partial_{ν} θ_{1}, = - R_{2}^{μν} \partial_{μ} T_{0} \partial_{ν} T_{0} - 4 R_{1}^{μν} \partial_{μ} T_{0} \partial_{ν} θ_{1} - 2 R_{0}^{μν} \partial_{μ} θ_{1} \partial_{ν} θ_{1} - 2 R_{0}^{μν} \partial_{μ} T_{0} \partial_{ν} θ_{2} .$
XIX-A@1#eq:Ncoeff · line 604
$N_{1} = - \frac{S _{1}}{2 v ^{3}}, N_{2} = \frac{3 S _{1} ^{2}}{4 v ^{5}} - \frac{S _{2}}{2 v ^{3}} .$
XIX-A@1#eq:ucoeff · line 615
$u_{1} = - \frac{θ _{1}}{v}, u_{2} = - \frac{θ _{2} + 2 u _{1} θ ˙ _{1} + v ˙ u _{1} ^{2}}{v} .$
unlabeled · line 571
$q_{e} = q_{0} + e h + \frac{e ^{2}}{2} k, T_{e} = T_{0} + e θ_{1} + \frac{e ^{2}}{2} θ_{2} .$
unlabeled · line 576
$q_{e}^{- 1} = R_{0} + e R_{1} + \frac{e ^{2}}{2} R_{2}, S [q_{e}, T_{e}] = v^{2} + e S_{1} + \frac{e ^{2}}{2} S_{2} .$
unlabeled · line 581
$R_{0} = q_{0}^{- 1}, R_{1} = - q_{0}^{- 1} h q_{0}^{- 1}, R_{2} = 2 q_{0}^{- 1} h q_{0}^{- 1} h q_{0}^{- 1} - q_{0}^{- 1} k q_{0}^{- 1} .$
unlabeled · line 611
$Y_{e}^{i} = x^{i} + e Y_{1}^{i} + \frac{e ^{2}}{2} Y_{2}^{i} .$