Harmonic initial data relative to a fixed smooth background

Let $s\ge 5$ be an integer and $L_0>0$. Let the underlying periodic manifold be $\mathrm{\Sigma }={\mathbb{T}}_{2\pi L_0}^3$ with length coordinates $X^i$, and use the diffeomorphism $D_{L_0}:{\mathbb{T}}_{2\pi }^3\to \mathrm{\Sigma }$, $[\theta ]\mapsto [L_0\theta ]$, to parametrize its component functions as in Periodic high-Sobolev state spaces. This map is well defined with smooth inverse $[X]\mapsto [X\mathrm{/}{L}_0]$. Let $(h,K,e,b)$ be periodic data of Constrained electrovacuum initial data with $h\in H^{s+1}$, $K,e,b\in H^s$ and a uniform positive lower eigenvalue bound for the continuous representative of $h$. Here Sobolev membership means membership of the length-frame component arrays $h_{ij},K_{ij}$ and the specified orthonormal arrays $e_A,b_A$, composed with $D_{L_0}$, in the spaces on the dimensionless torus; it does not mean the components of the pulled-back tensors. All spatial derivatives below are ${\mathrm{\partial }}_{X^i}=L_0^{-1}{\mathrm{\partial }}_{{\theta }^i}$. Fix a smooth positive reference metric $\gamma$ on $\mathrm{\Sigma }$ and use the product background $\bar{g}=-(d{x}^0{)}^2+{\gamma }_{ij}(X)d{X}^{i}d{X}^j$ near $\{x^0=0\}\times \mathrm{\Sigma }$. Define

$Q^i:=h^{jk}(\mathrm{\Gamma }(h{)}^i{}_{jk}-\mathrm{\Gamma }(\gamma {)}^i{}_{jk}),$

where the difference of the two spatial Levi-Civita connections is a tensor. Prescribe the spacetime metric Cauchy $1$-jet at $x^0=0$ by

$g_{ij}=h_{ij},g_{00}=-1,g_{0i}=0,{\mathrm{\partial }}_0{g}_{ij}=2K_{ij},{\mathrm{\partial }}_0{g}_{00}=-2{\mathrm{tr}}_{h}K,{\mathrm{\partial }}_0{g}_{0i}=h_{ij}{Q}^j\mathrm{.}$

For this jet, let $g_0^{ab}=\mathrm{diag}(-1,h^{ij})$ be the inverse metric block on the slice and define its Christoffel value by

${\mathrm{\Gamma }}^a{}_{bc}[j_0^{1}g]:=\frac{1}{2}{g}_0^{ad}({\mathrm{\partial }}_b{g}_{dc}+{\mathrm{\partial }}_c{g}_{db}-{\mathrm{\partial }}_d{g}_{bc}){\mid }_{x^0=0}\mathrm{.}$

Define the initial-slice harmonic-gauge vector directly from the prescribed $1$-jet by

$H_0^a:=g_0^{bc}({\mathrm{\Gamma }}^a{}_{bc}[j_0^{1}g]-{\bar{\mathrm{\Gamma }}}^a{}_{bc}){\mid }_{x^0=0}\mathrm{.}$

This is the initial value of the harmonic-gauge vector of The harmonic gauge vector, the reduced Einstein equation and the Einstein residual for every $C^1$ metric on a neighborhood realizing the prescribed jet; no neighborhood extension is needed to define $H_0$. Each metric value in the jet is $H^{s+1}$ and each prescribed first derivative is $H^s$, so the displayed algebraic expression has an $H^s$ representative. It vanishes on the initial slice. The background $\gamma$ is required to be smooth; choosing $\gamma =h$ is not asserted to be valid for finite-regularity $h$.