A discrete Hamiltonian and momentum constraint diagnostic

Use the dimensionless angular coordinates $x^i$ and the periodic spatial three-torus from the earlier fixed-gauge data definition. Choose integers $N_i\ge 3$, coordinate spacings ${\mathrm{\Delta }}_i:=2\pi \mathrm{/}{N}_i$, and the finite periodic grid

At each grid node $p$, supply a real symmetric positive-definite matrix $h_{ij}(p)$ and its matrix inverse $h^{ij}(p)$ (so $h^{ik}(p)h_{kj}(p)={\delta }^i{}_j$), a real symmetric array $K_{ij}(p)$, a scalar $\rho (p)$, a covector array $j_i(p)$, and real constants $\mathrm{\Lambda }$ and $\kappa >0$. These are the sampled spatial metric, second fundamental form, energy density and momentum density. For a future unit normal $n$ and the coordinate basis ${\mathrm{\partial }}_i$, use the convention $K(X,Y)=g({\mathrm{\nabla }}_{X}n,Y)$, $\rho =T(n,n)$ and $j_i=-T(n,{\mathrm{\partial }}_i)$, where $T$ is a smooth symmetric covariant two-tensor. The array definition itself needs only the listed samples and does not assume the continuum field equation. The vacuum case is $\rho =j_i=\mathrm{\Lambda }=0$. In the SI angular-coordinate convention used here, $[x^i]=1$, $[h_{ij}]={\mathrm{m}}^2$, $[K_{ij}]=\mathrm{m}$, $[\rho ]=\mathrm{J}{\mathrm{m}}^{-3}$, $[j_i]=\mathrm{J}{\mathrm{m}}^{-2}$, $[\mathrm{\Lambda }]={\mathrm{m}}^{-2}$, and $[\kappa ]={\mathrm{m}}^{-1}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^2$. Thus the scalar residual below has units ${\mathrm{m}}^{-2}$ and its covector components in this angular basis have units ${\mathrm{m}}^{-1}$.

Let ${\mathbb{R}}^{G_{\mathbf{\Delta }}}$ denote the real arrays indexed by the grid. Choose and specify linear maps ${\delta }_i:{\mathbb{R}}^{G_{\mathbf{\Delta }}}\to {\mathbb{R}}^{G_{\mathbf{\Delta }}}$ with ${\delta }_{i}1=0$; apply them componentwise to tensor arrays. Periodic index wrapping is used here. Define the discrete connection components, curvature, Ricci array and scalar curvature by nodewise operations, with every index $i,j,k,\mathrm{\ell },a,b$ ranging over $1,2,3$ and repeated upper/lower indices summed:

Set

The signed discrete Hamiltonian and momentum residuals are

These signs match the Hamiltonian and momentum constraint equations when the pointwise equation $G+\mathrm{\Lambda }g=\kappa T$ is assumed. The displayed discrete formulas are array operations: they do not define a coordinate-covariant discrete connection or assume a discrete Leibniz rule. A reconstruction map has the exact type

where ${\mathcal{D}}_{\mathbf{\Delta }}$ is the set of all grid arrays $(h_{ij},K_{ij},\rho ,j_i)$ with the pointwise symmetry and positive-definite conditions above, and $\mathcal{C}$ is the set of pointwise assignments on ${\mathbb{T}}_{2\pi }^3$ of a positive-definite symmetric covariant two-tensor $h$, a symmetric covariant two-tensor $K$, a scalar $\rho$ and a covector $j$, with the same units. No continuity or differentiability of a member of $\mathcal{C}$ is implied; if continuum derivatives are used, the needed regularity must be stated separately. Specify ${\mathcal{I}}_{\mathbf{\Delta }}$ as part of each diagnostic, since it records which continuum field the grid is intended to represent. For every reported diagnostic, also record the grid norm and its weights, the mesh hierarchy, the units, the derivative maps, the periodic boundary treatment, and ${\mathcal{I}}_{\mathbf{\Delta }}$. A small residual by itself does not assert closeness to an exact constraint-satisfying development.

Fixed-gauge finite-slab data and solution topology

Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form

The geometric convention in the constraint theorem fixes the signs above. Gourgoulhon's 3+1 lecture notes use the opposite sign for the second fundamental form in their convention; their displayed equations are therefore context and cross-checks only, not a source for silently replacing the local sign convention.