The “energy balance” that must hold on every time-slice is
\[ \boxed{\; \mathcal{H}_{\perp} = \frac{16\pi}{\sqrt{\gamma}}\!\left(\pi_{ij}\pi^{ij}-\tfrac12\,\pi^{2}\right) - \frac{\sqrt{\gamma}}{16\pi}\,{}^{(3)}\!R \;\approx\;0 \;} \]
(geometric units \(G=c=1\)). It says: the kinetic curvature of the slice (the quadratic momenta \(\pi^{ij}\)) must exactly balance the intrinsic curvature of the 3-geometry (the scalar curvature \({}^{(3)}\!R\)). Because the lapse \(N=e^{\Phi}\) multiplies this constraint in the Hamiltonian (Step 24), varying \(N\) enforces \(\mathcal{H}_{\perp}\!\approx\!0\); \(\Phi\) itself does not add a propagating scalar degree of freedom. Solving \(\mathcal{H}_{\perp}=0\) together with the momentum constraints (next step) removes all non-TT parts, leaving only the two transverse–traceless graviton modes to propagate.
(If you like, recall \(\pi^{ij}\!\propto\!\sqrt{\gamma}(K^{ij}-\gamma^{ij}K)\): the momentum encodes the slice’s extrinsic curvature—how the slice bends in spacetime. Then \(\mathcal{H}_{\perp}=0\) is a Gauss-law–like relation between “bending rate” and “stored curvature.”)
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(\mathcal{H}_{\perp}\) | Hamiltonian (normal) constraint | Energy balance on a slice; must vanish | \(\approx 0\) | “Gauss law for gravity on the slice” |
| \(\gamma_{ij}\) | spatial metric | 3-metric on the time slice | length\(^2\) components | “Ruler grid of the slice” |
| \(\gamma \equiv \det\gamma_{ij}\) | metric determinant | volume element \(\sqrt{\gamma}\,d^3x\) | (length)\(^3\) | “Cell size of the grid” |
| \({}^{(3)}\!R\) | scalar curvature of slice | intrinsic curvature built from \(\gamma_{ij}\) | 1/length\(^2\) | “How the grid is curved within itself” |
| \(\pi^{ij}\) | canonical momentum | conjugate to \(\gamma_{ij}\); \(\sim\!\sqrt{\gamma}(K^{ij}-\gamma^{ij}K)\) | density-like | “How fast the grid is bending” |
| \(\pi \equiv \gamma_{ij}\pi^{ij}\) | trace of momentum | trace part of \(\pi^{ij}\) | density-like | “Overall squish/stretch rate” |
| \(K_{ij},\,K\) | extrinsic curvature & trace | embedding curvature of the slice in spacetime | 1/length | “Bend of the slice into tomorrow” |
| \(N=e^{\Phi}\) | lapse (time potential) | multiplies \(\mathcal{H}_{\perp}\) in \(\mathcal{H}\) | dimensionless | “Knob that enforces the rule, not a new field” |