Manuscript III@2 · III.tex · section · 82dad3e5

Canonical temporal normalization

What instantaneous Poisson compensation establishes › Canonical temporal normalization · lines 2342–2367

Write

γ_{ij} = δ_{ij} + κ h_{ij}, β_{i} = κ h_{0 i} + O (κ^{2}), N_{lapse} = 1 + O (κ),

and choose

K_{ij}^{ADM} = \frac{1}{2 N _{lapse}} (γ ˙_{ij} - D_{i} β_{j} - D_{j} β_{i}) .

With

v_{ij} = h ˙_{ij} - \partial_{i} h_{0 j} - \partial_{j} h_{0 i},

declare the quadratic kinetic density

L_{kin}^{(2)} = \frac{1}{2} (v_{ij} v_{ij} - (v_{ii})^{2}) .

This follows from the ADM kinetic term with Einstein–Hilbert coefficient $2/ κ^{2}$ .

Displayed formulas in this passage

unlabeled · line 2345
$γ_{ij} = δ_{ij} + κ h_{ij}, β_{i} = κ h_{0 i} + O (κ^{2}), N_{lapse} = 1 + O (κ),$
unlabeled · line 2351
$K_{ij}^{ADM} = \frac{1}{2 N _{lapse}} (γ ˙_{ij} - D_{i} β_{j} - D_{j} β_{i}) .$
unlabeled · line 2357
$v_{ij} = h ˙_{ij} - \partial_{i} h_{0 j} - \partial_{j} h_{0 i},$
unlabeled · line 2361
$L_{kin}^{(2)} = \frac{1}{2} (v_{ij} v_{ij} - (v_{ii})^{2}) .$