Manuscript VI-B@1 · VI-B-v1.tex · definition · 33798723

Definition: Named local solution-jet space

The second-order Einstein–scalar coefficient space › Normalization and the scalar second variation · lines 588–610

VI-B@1#def:jet, VI-B@1#eq:jet1, VI-B@1#eq:jet2, VI-B@1#eq:jet3, VI-B@1#eq:jet4, VI-B@1#eq:jet5

Let $J_{R, b}$ be the real smooth tuples

(h, j, ϕ_{0}, ϕ_{1}, ϕ_{2})

on the supported presentation satisfying

G^{(1)} (h) G^{(1)} (j) + G^{(2)} [h, h] P ϕ_{0} P ϕ_{1} P ϕ_{2} = 0, = \frac{1}{4} T (η, ϕ_{0}), = 0, = - L_{h} ϕ_{0}, = - L_{h} ϕ_{1} - (L_{j} + M_{h}) ϕ_{0} .

Each tuple includes its coefficient initial data. No assertion is made that arbitrary unconstrained initial metric data extend to this space.

Displayed formulas in this passage

VI-B@1#eq:jet1 · line 595
$G^{(1)} (h) G^{(1)} (j) + G^{(2)} [h, h] P ϕ_{0} P ϕ_{1} P ϕ_{2} = 0, = \frac{1}{4} T (η, ϕ_{0}), = 0, = - L_{h} ϕ_{0}, = - L_{h} ϕ_{1} - (L_{j} + M_{h}) ϕ_{0} .$
VI-B@1#eq:jet2 · line 595
$G^{(1)} (h) G^{(1)} (j) + G^{(2)} [h, h] P ϕ_{0} P ϕ_{1} P ϕ_{2} = 0, = \frac{1}{4} T (η, ϕ_{0}), = 0, = - L_{h} ϕ_{0}, = - L_{h} ϕ_{1} - (L_{j} + M_{h}) ϕ_{0} .$
VI-B@1#eq:jet3 · line 595
$G^{(1)} (h) G^{(1)} (j) + G^{(2)} [h, h] P ϕ_{0} P ϕ_{1} P ϕ_{2} = 0, = \frac{1}{4} T (η, ϕ_{0}), = 0, = - L_{h} ϕ_{0}, = - L_{h} ϕ_{1} - (L_{j} + M_{h}) ϕ_{0} .$
VI-B@1#eq:jet4 · line 595
$G^{(1)} (h) G^{(1)} (j) + G^{(2)} [h, h] P ϕ_{0} P ϕ_{1} P ϕ_{2} = 0, = \frac{1}{4} T (η, ϕ_{0}), = 0, = - L_{h} ϕ_{0}, = - L_{h} ϕ_{1} - (L_{j} + M_{h}) ϕ_{0} .$
VI-B@1#eq:jet5 · line 595
$G^{(1)} (h) G^{(1)} (j) + G^{(2)} [h, h] P ϕ_{0} P ϕ_{1} P ϕ_{2} = 0, = \frac{1}{4} T (η, ϕ_{0}), = 0, = - L_{h} ϕ_{0}, = - L_{h} ϕ_{1} - (L_{j} + M_{h}) ϕ_{0} .$
unlabeled · line 591
$(h, j, ϕ_{0}, ϕ_{1}, ϕ_{2})$