Explicit calculations at a level similar to homogeneous models, at least for matrix elements of individual operators, are possible in inhomogeneous models, too. In particular the spherically symmetric model and cylindrically symmetric Einstein-Rosen waves are of this class, where the symmetry or other conditions are strong enough to result in a simple volume operator. In the spherically symmetric model, this simplification comes from the remaining isotropy subgroup isomorphic to U(1) in generic points, while the Einstein-Rosen model is simplified by polarization conditions that play a role analogous to the diagonalization of homogeneous models. With these models one obtains access to applications for black holes and gravitational waves, but also to inhomogeneities in cosmology.
In spherical coordinates , , a spherically symmetric spatial metric takes the form
with . This is related to densitized triad components by [196, 136]
which are conjugate to the other basic variables given by the Ashtekar connection component and the extrinsic curvature component :
Note that we use the Ashtekar connection for the inhomogeneous direction but extrinsic curvature for the homogeneous direction along symmetry orbits . Connection and extrinsic curvature components for the -direction are related by with the spin connection component
with the momentum conjugate to a U(1)-gauge angle . This is a rather complicated function of both triad and connection variables such that the volume would have a rather complicated quantization. It would still be possible to compute the full volume spectrum, but with the disadvantage that volume eigenstates would not be given by triad eigenstates such that computations of many operators would be complicated . This can be avoided by using extrinsic curvature which is conjugate to the triad component . Moreover, this is also in accordance with a general scheme to construct Hamiltonian constraint operators for the full theory as well as symmetric models [194, 42, 58].
The constraint operator in spherical symmetry is given by
Since the Hamiltonian constraint contains the spin connection component given by (38), which contains inverse powers of densitized triad components, one can expect effective classical equations with modifications similar to the Bianchi IX model. However, the situation is now much more complicated since we have a system constrained by many constraints with a non-Abelian algebra. Simply replacing the inverse of with a bounded function as before will change the constraint algebra and thus most likely lead to anomalies. It is currently open if a more refined replacement can be done where not only the spin connection but also the extrinsic curvature terms are modified. This issue has the potential to shed light on many questions related to the anomaly issue. It is one of the cases where models between homogeneous ones, where the anomaly problem trivializes, and the full theory are most helpful.
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