Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

The metric transformation law

Plateau covariance and the second-order intertwiner › The metric transformation law · lines 1106–1127

VII-A@1#eq:hprime, VII-A@1#eq:kprime

Let $F_{κ}$ be as in (eq:formalF). The transformed metric jet is

g^{'} = F_{κ}^{*} g = η + κ h^{'} + κ^{2} k^{'} + O (κ^{3}),

with

h^{'} k^{'} = h + K ξ, = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η .

The second equation is indispensable. Keeping $k$ fixed while changing $h$ by $K ξ$ does not describe the second-order pullback of a metric.

Displayed formulas in this passage

VII-A@1#eq:hprime · line 1116
$h^{'} k^{'} = h + K ξ, = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η .$
VII-A@1#eq:kprime · line 1116
$h^{'} k^{'} = h + K ξ, = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η .$
unlabeled · line 1110
$g^{'} = F_{κ}^{*} g = η + κ h^{'} + κ^{2} k^{'} + O (κ^{3}),$