The quadratic normalization

completion Einstein jet and its compact detector

equation space being tested

For a smooth free scalar ${\varphi }_0$, let ${\mathcal{J}}_g({\varphi }_0)$ consist of smooth coefficient families

such that the coefficients through order ${\kappa }^2$ of

vanish on the completion slab. The scalar satisfies $P_m{\varphi }_0=0$, and ${\varphi }_1,{\varphi }_2$ have the inherited zero Cauchy data at $\Sigma$. Metric Cauchy data are unrestricted.

This is the invariant Einstein equation, not the gauge-fixed Euler equation.

For clarity, the scalar coefficient equations include

These equations specify the inherited classical Cauchy problem. No inverse is constructed here.

The metric coefficients include

where

Thus the scalar solution correction is not confused with the metric correction to the stress density.

Nonemptiness of ${\mathcal{J}}_g({\varphi }_0)$ requires

In the subclass ${\gamma }_0=0$, this is already necessary for

Proof. This is XI-A :compatibility CheungXIA. For ${\gamma }_0=0$, take the divergence of (eq:sourcejet) and use ${\partial }_{\mu }{G_{\mathrm{lin}}}^{\mu \nu }=0$ and ${\partial }_{\mu }{T_0}^{\mu \nu }[{\varphi }_0]=0$. For arbitrary ${\gamma }_0$, use the order-one coefficient of (eq:noether). On a purported jet its left side vanishes; ${E_{\varphi }}^{[0]}=-P_m{\varphi }_0=0$, while ${{\mathcal{B}}_{\nu }}^{[0]}=({\partial }_{\mu }g){T_0}^{\mu }{}_{\nu }[{\varphi }_0]$. The same condition follows without restricting the leading vacuum metric data.

inherited empty fiber

Define