To perform the first step we need a Hilbert space of functionals of spatial metrics. Unfortunately,
the space of metrics, or alternatively extrinsic curvature tensors, is mathematically poorly understood and
not much is known about suitable inner products. At this point, a new set of variables introduced by
Ashtekar [7, 8, 30
] becomes essential. This is a triad formulation, but uses the triad in a densitized form
(i.e., it is multiplied with an additional factor of a Jacobian under coordinate transformations). The
densitized triad
is then related to the triad by
but has the same
properties concerning gauge rotations and its orientation (note the absolute value which is often
omitted). The densitized triad is conjugate to extrinsic curvature coefficients
:
Spatial geometry is then obtained directly from the densitized triad, which is related to the spatial metric by
There is more freedom in a triad since it can be rotated without changing the metric. The theory is independent of such rotations provided the Gauss constraint
is satisfied. Independence from any spatial coordinate system or background is implemented by the diffeomorphism constraint (modulo Gauss constraint) with the curvatureSpace-time geometry, however, is more complicated to deduce since it requires a good knowledge of the dynamics. In a canonical setting, dynamics is implemented by the Hamiltonian constraint
where extrinsic curvature components have to be understood as functions of the Ashtekar connection and the densitized triad through the spin connection.![]() |
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