= 3+1 decomposition of spacetime
= 3+1 decomposition
{synonym}
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> $(-,+,+,+)$,
$$
ds^2=-\alpha^2dt^2+\gamma_{ij}(dx^i+\beta^idt)(dx^j+\beta^jdt).
$$
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,
$$
n=\frac1\alpha(\partial_t-\beta^i\partial_i),\qquad \partial_t=\alpha n+\beta^i\partial_i.
$$
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 $\mathbf x'=\mathbf x-\mathbf vt$ 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.
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