\[ \boxed{\;\mathcal{H}\;=\;N\,\mathcal{H}_{\perp}\;+\;N^{i}\,\mathcal{H}_{i},\qquad N=e^{\Phi}\;} \]
After slicing spacetime into “now”-surfaces (the 3-metric \(\gamma_{ij}\)) plus a way to step to the next slice, gravity’s Hamiltonian density is just a weighted sum of constraints. The weights are the lapse \(N=e^{\Phi}\) (how much proper time you advance) and the shift \(N^{i}\) (how much you slide points sideways on the slice). Because \(N\) and \(N^{i}\) multiply the constraints \(\mathcal{H}_{\perp}\) (energy/“normal” constraint) and \(\mathcal{H}_{i}\) (momentum/“diffeo” constraints), varying them enforces
\[ \mathcal{H}_{\perp}\approx 0,\qquad \mathcal{H}_{i}\approx 0, \]
so \(N\) and \(N^{i}\) carry no new dynamics. They are Lagrange multipliers that steer the foliation. In time-first language: the time potential \(\Phi\) enters only through \(N=e^{\Phi}\) to enforce the right physics; the propagating content lives in the spatial geometry’s true degrees of freedom (the TT modes you’ll meet in the next step).
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(\mathcal{H}\) | Hamiltonian density | generator of time evolution on the slice | energy density (geom. units) | “Mixer that combines the constraints” |
| \(N=e^{\Phi}\) | lapse | how far you advance proper time per coordinate step | dimensionless | “Clock gear ratio” |
| \(N^{i}\) | shift | sideways slide of coordinates along the slice | velocity-like | “Conveyor belt of the grid” |
| \(\mathcal{H}_{\perp}\) | Hamiltonian (normal) constraint | enforces Einstein’s energy equation on each slice | \(\approx 0\) (constraint) | “Keep the slice physically balanced” |
| \(\mathcal{H}_{i}\) | momentum (diffeo) constraint | enforces spatial diffeomorphism invariance | \(\approx 0\) (constraint) | “Keep patterns from depending on labels” |
| \(\gamma_{ij}\) | spatial metric | geometry of a “now”-surface | length\(^2\) components | “Ruler grid on the slice” |
| \(\pi^{ij}\) | canonical momentum | conjugate to \(\gamma_{ij}\) (related to extrinsic curvature) | density-like | “How fast the ruler grid is bending” |
| \(\Phi\) | time potential | sets the lapse only (here) | dimensionless | “Altitude of time” choosing the step size” |
| \(\approx\) | weak equality | holds on the constraint surface (Dirac) | notation | “Equal once the rules are enforced” |