ds2=N2dt2+hij(dxi+Nidt)(dxj+Njdt),ds^2 = -N^2 dt^2 + h_{ij} (dx^i + N^i dt)(dx^j + N^j dt),
where:
NN: lapse function,
NiN^i: shift vector,
hijh_{ij}: 3-metric of spatial slices.
In the presence of torsion, the canonical variables are extended:
The extrinsic curvature KijK_{ij} is generalized to include torsion contributions,
Canonical momenta receive corrections from spin densities.
The total Hamiltonian becomes:
Htotal=HGR+Htorsion+Hmatter,\mathcal{H}_{\text{total}} = \mathcal{H}_{\text{GR}} + \mathcal{H}_{\text{torsion}} + \mathcal{H}_{\text{matter}},
where:
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