In the ADM picture the lapse \(N=e^{\Phi}\) and the shift \(N^{i}\) do not carry their own dynamics. They multiply the constraints and act as Lagrange multipliers. Varying them enforces the Hamiltonian and momentum constraints (\(\mathcal H_\perp\!\approx\!0,\ \mathcal H_i\!\approx\!0\)) on every slice. Solving these, plus fixing the residual coordinate freedom on the slice, removes all nonphysical pieces (the scalar and longitudinal parts). What remains are exactly two transverse–traceless (TT) tensor modes. The usual gravitational waves.
In linearized language around flat space, write the spatial metric perturbation as \(h_{ij}\). The constraints + gauge let you impose
\[ \partial^i h^{\mathrm{TT}}_{ij}=0,\qquad \delta^{ij}h^{\mathrm{TT}}_{ij}=0, \]
leaving only the two polarizations of \(h^{\mathrm{TT}}_{ij}\) to propagate. The time potential \(\Phi\) still matters. It enforces the rules (e.g., the flux law ties \(\partial_t\Phi\) to energy flow), but it is not a new propagating scalar graviton. So “time-first” reorganizes GR’s bookkeeping; it doesn’t change GR’s physical content.
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(N=e^{\Phi}\) | lapse (multiplier) | advances proper time between slices | dimensionless | “Clock gear that enforces the rules” |
| \(N^{i}\) | shift (multiplier) | slides coordinates along the slice | velocity-like | “Conveyor belt for labels” |
| \(\mathcal H_\perp\approx 0\) | Hamiltonian constraint | energy/normal constraint on a slice | constraint (no units) | “Balance the slice” |
| \(\mathcal H_i\approx 0\) | momentum constraints | enforce spatial diffeo invariance | constraint (no units) | “Ignore pure relabeling” |
| \(h_{ij}^{\mathrm{TT}}\) | TT metric perturbation | physical GW modes after constraints/gauge | length\(^2\) (perturbative) | “Pure ripples with no gauge wiggle” |
| TT conditions | transversality & tracelessness | \(\partial^i h^{\mathrm{TT}}_{ij}=0,\ \delta^{ij}h^{\mathrm{TT}}_{ij}=0\) | — | “No compression, no divergence—just shear” |
| DOF count | physical graviton modes | per point: 2 polarizations | 2 | “Two violin strings, not three” |