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

Fully expanded native coefficient

Complete classical coefficient through second order › Fully expanded native coefficient · lines 658–701

XX-D@1#eq:ell, XX-D@1#eq:ncomplete, XX-D@1#eq:qloc

Define the local linear map

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

Then the complete result is

n_{1} = ℓ z, n_{2} = ℓ ζ + q_{loc} (z) - 2 (\calRz)^{i} \partial_{i} (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} .

The spatial metric enters through the rod term, not through an artificially restricted scalar metric subclass.

Displayed formulas in this passage

XX-D@1#eq:ell · line 661
$ℓ z = - \frac{t}{2 A} h_{00} - \frac{t ^{2}}{A ^{2}} φ ˙ - \frac{t}{A ^{2}} φ .$
XX-D@1#eq:ncomplete · line 668
$n_{1} = ℓ z, n_{2} = ℓ ζ + q_{loc} (z) - 2 (\calRz)^{i} \partial_{i} (ℓ z),$
XX-D@1#eq:qloc · line 675
$q_{loc} (z) = - \frac{t}{4 A} h_{00}^{2} + \frac{t}{A a ^{2}} h_{0 i} h_{0 i} + \frac{t ^{2}}{A ^{2}} h_{00} φ ˙ + \frac{2 t ^{3}}{A ^{3}} φ ˙^{2} + \frac{2 t ^{2}}{A ^{2} a ^{2}} h_{0 i} \partial_{i} φ + \frac{t ^{3}}{A ^{3} a ^{2}} \partial_{i} φ \partial_{i} φ + \frac{t ^{2}}{A ^{2}} φ h ˙_{00} + \frac{t}{A ^{2}} φ h_{00} + \frac{2 t ^{3}}{A ^{3}} φ φ ¨ + \frac{6 t ^{2}}{A ^{3}} φ φ ˙ + \frac{t}{A ^{3}} φ^{2} .$