Chapter 26: ADM Formalism
The ADM (Arnowitt-Deser-Misner) formalism recasts general relativity as a Hamiltonian system by foliating spacetime into spatial slices. Essential for numerical relativity and canonical quantum gravity.
3+1 Decomposition
\( ds^2 = -N^2 dt^2 + h_{ij}(dx^i + N^i dt)(dx^j + N^j dt) \)
N = lapse, Ni = shift, hij = spatial metric
Lapse Function N
Rate of proper time vs coordinate time between slices
Shift Vector Ni
How spatial coordinates shift between slices
Hamiltonian and Momentum Constraints
Hamiltonian Constraint
\( \mathcal{H} = {}^{(3)}R + K^2 - K_{ij}K^{ij} = 16\pi G \rho \)
Momentum Constraint
\( \mathcal{M}_i = D_j(K^j_i - \delta^j_i K) = 8\pi G j_i \)
These constraints must be satisfied on each spatial slice. They're the "initial data" constraints for numerical evolution.