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A 3+1 decomposition of spacetime describes spacetime as spatial hypersurfaces labelled by a time coordinate $t$, with a positive-definite spatial induced metric ${\gamma }_{ij}$. In units $c=1$ and metric signature $(-,+,+,+)$,The positive lapse function $\alpha$ measures proper time between nearby slices along the future unit normal; the shift vector ${\beta }^i$ describes the spatial displacement of the coordinate grid relative to that normal. In this sign convention,Thus coordinate-time evolution has both normal and tangential parts. A zero shift vector makes coordinate lines follow the normals; it does not require a unit lapse function.

Flat Minkowski spacetime illustrates the distinction. In Cartesian coordinates $\alpha =1$, $\beta =0$ and ${\gamma }_{ij}={\delta }_{ij}$. Changing spatial labels to ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{v}t$ gives the same geometry with $\beta =\mathbf{v}$ and the same spatial induced metric. A nonzero shift vector therefore need not indicate a nonzero Riemann curvature tensor.

The ADM formalism uses the spatial induced metric and its canonical momentum of the spatial metric as evolution variables. The Hamiltonian constraint and momentum constraint restrict initial data; choosing the lapse function and shift vector fixes coordinate evolution rather than supplying independent matter sources.

The Arnowitt-Deser-Misner formalism expresses general relativity as constrained Hamiltonian evolution of a spatial metric and its conjugate momentum after a 3+1 decomposition of spacetime. Lapse and shift enforce the Hamiltonian and momentum constraints. For an asymptotically flat spacetime its boundary Hamiltonian gives the Arnowitt-Deser-Misner energy.

Varying the shift in the 3+1 decomposition of spacetime gives a nondynamical momentum constraint. For a $P(X,\varphi )$ field it equates the spatial divergence of $K_i{}^j-{\delta }_i^{j}K$ to the scalar momentum density.

With vectors and independent tensors neglected, finite scalar cosmological perturbations of the local spatial scale can be encoded by $h_{ij}=a_{\mathrm{bg}}^2{e}^{2\zeta }{\delta }_{ij}$. Scalar lapse and shift potentials complete its 3+1 decomposition of spacetime. The exact shift-square term is $h^{ij}{N}_i{N}_j$, independently of the parametrization of the scalar shift.

Numerical relativity computes solutions of the Einstein field equations when an analytic solution or a small-perturbation approximation is insufficient. A 3+1 decomposition of spacetime separates initial spatial geometry and matter from their subsequent evolution. Initial data must satisfy the Hamiltonian constraint and momentum constraint, while the lapse function and shift vector specify the coordinate evolution.

The calculation also needs boundary conditions and a numerical evolution formulation. Resolution studies, constraint monitoring and comparison with known solutions test whether the computed result approximates the intended spacetime. Binary black-hole mergers provide an observational application: numerical waveforms connect strongly dynamical geometry with interferometer signals.^[ref] Extracting a physical wave requires controlling the frame and propagation model as well as solving grid equations.

Use the equivalent density form of harmonic slicing from part (ii). With $\sqrt{-g}=\alpha \sqrt{\gamma }$, positive lapse function $\alpha$, and the corrected inverse metric tensor, its time component becomesThus $D_t(\sqrt{\gamma }/\alpha )=(\sqrt{\gamma }/\alpha ){\partial }_i{\beta }^i$, orThe spatial volume evolution identity gives $D_t\log \sqrt{\gamma }={\partial }_i{\beta }^i-\alpha K$. Subtraction yields $D_t\log \alpha =-\alpha K$, hence the harmonic lapse evolution isThe shift vector need not vanish; its transport term is essential in a general 3+1 decomposition of spacetime.

Take the orientation induced by the future-directed unit normal and positive lapse function $\alpha$. In coordinates adapted to the 3+1 decomposition of spacetime, any component ${ϵ}_{\ell ijk}$ with all four indices spatial vanishes: only three distinct spatial indices are available. The shift vector therefore contributes nothing, andHere $[ijk]$ is the Levi-Civita symbol, and the last equality uses the metric determinant in a 3+1 decomposition, $|g|={\alpha }^2\gamma$. The right side is precisely the spatial metric volume tensor, with component $\sqrt{\gamma }$ at $ijk=123$, the appropriate permutation sign, and zero for repeated indices. HenceMore invariantly, the spatial volume form is the restriction to the hypersurface of the contraction of the spacetime volume form with $n$.

Write $h_{ij}$ for the spatial metric tensor and lower the shift vector with $h_{ij}$. For the negative-shift convention, the four-dimensional metric and its inverse have componentsThe normal has covariant components $n_{\mu }=(-N,0,0,0)$, so $n_{\mu }{n}^{\mu }=-1$ and it annihilates every tangent vector to the spatial slice. Since $n_i=0$, its covariant derivative gives $n_{i;j}=N{\Gamma }^0{}_{ij}$ and therefore $K_{ij}=-N{\Gamma }^0{}_{ij}$.

Compute the required Christoffel symbol using the symmetric connection formula. Its time-index term and spatial-index term combine to giveThe spatial expression in parentheses is ${\dot{h}}_{ij}+D_i{N}_j+D_j{N}_i$, where $D$ is the covariant derivative of $h$. Thus the extrinsic curvature with a negative shift isThis derivation fixes both signs from the actual 3+1 decomposition of spacetime, rather than transferring a formula using the opposite shift vector convention. The connection formula used here has $g_{\lambda \kappa ,\nu }$ as its second differentiated term; the printed repeated-index version is a typographical error.

Comparison with the 3+1 decomposition of spacetime gives the diagonal spatial metricThe mixed metric coefficient is ${\beta }_r=g_{tr}=M/r$. Raising its index with ${\gamma }^{rr}=r^2/(r+M{)}^2$ yields the shift vectorNowand $g_{tt}=-{\alpha }^2+{\beta }_i{\beta }^i=-(r-M)/(r+M)$. The positive lapse function is consequently

The inverse 3+1 decomposition of spacetime metric givesThe first term is the squared derivative along the hypersurface normal and the second is the spatial-gradient contribution.

With a future unit timelike normal in signature $(-,+,+,+)$, this convention makes the extrinsic curvature trace equal to the normal congruence's expansion, $K={\nabla }_{\mu }{n}^{\mu }$. An expanding flat Friedmann-Lemaître-Robertson-Walker metric has $K_{ij}=a^{\prime }{\delta }_{ij}$ and $K=3a^{\prime }/a^2$. This is the opposite sign to the negative-expansion convention used in many 3+1 decomposition of spacetime formulas. Projection identities and momentum constraints must use the same convention throughout.

In a 3+1 decomposition of spacetime with $K_{ij}=-({\partial }_t{\gamma }_{ij}-{\mathcal{L}}_{\beta }{\gamma }_{ij})/(2\alpha )$, contract the metric evolution with ${\gamma }^{ij}$ and use the Jacobi determinant derivative formula. Here $\gamma =\det {\gamma }_{ij}$, $K={\gamma }^{ij}{K}_{ij}$, $\alpha$ is the lapse function, and $\beta$ is the shift vector. The identity separates spatial transport and coordinate compression from the geometric volume change caused by extrinsic curvature.