Step 26. The momentum constraints: keep physics independent of how we label the slice

On every time-slice, invariance under relabeling points on the slice (spatial diffeomorphisms) is enforced by the momentum constraints

\[ \boxed{\;\mathcal{H}_{i}\;=\;-\,2\,\nabla_{j}\,\pi^{j}{}_{i}\;\approx\;0\;} \]

(geometric units). Here \(\pi^{ij}\) is the momentum conjugate to the spatial metric \(\gamma_{ij}\), and \(\nabla\) is the covariant derivative built from \(\gamma_{ij}\).

Plain-English idea: these equations say no spurious momentum is created by mere coordinate shuffling on a slice. Equivalently, \(\mathcal{H}_{i}\) generates infinitesimal relabelings: letting \(\xi^{i}(x)\) be any vector field on the slice,

\[ \delta_{\xi}\gamma_{ij} \;=\; \{\gamma_{ij},\,\int \xi^{k}\mathcal{H}_{k}\,d^{3}x\} \;=\; \mathcal{L}_{\xi}\gamma_{ij}, \]

so demanding \(\mathcal{H}_{i}\!\approx\!0\) means physical data are unchanged by these label changes. In practice, solving \(\mathcal{H}_{i}=0\) kills the longitudinal pieces of the metric momentum (and, in linear theory, of the metric perturbation), helping isolate the transverse–traceless (TT) modes as the only propagating gravitational degrees of freedom.

Useful alternative form (using \(\pi^{ij}\propto\sqrt{\gamma}(K^{ij}-\gamma^{ij}K)\)):

\[ \mathcal{H}_{i}\;=\;-\,2\,\nabla_{j}\pi^{j}{}_{i} \;\propto\; -\,\sqrt{\gamma}\,\nabla_{j}\!\left(K^{j}{}_{i}-\delta^{j}{}_{i}K\right)\approx 0, \]

i.e., the divergence of the traceless part of the extrinsic curvature must vanish.

Mini-Glossary