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4.14 Inhomogeneous models

The full theory is complicated at several different levels of both conceptual and technical nature. For instance, one has to deal with infinitely many degrees of freedom, most operators have complicated actions, and interpreting solutions to all constraints in a geometrical manner can be difficult. Most of these complications are avoided in homogeneous models, in particular when effective classical equations are employed. These equations use approximations of expectation values of quantum geometrical operators which need to be known rather explicitly. The question then arises whether one can still work at this level while relaxing the symmetry conditions and bringing in more complications of the full theory.

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 x, h, f a spherically symmetric spatial metric takes the form

2 2 2 ds = qxx(x, t) dx + qff(x,t)d_O_

with 2 2 2 2 d_O_ = dh + sin h df. This is related to densitized triad components by [196136]

|Ex |= qff, (Ef)2 = qxxqff,

which are conjugate to the other basic variables given by the Ashtekar connection component Ax and the extrinsic curvature component Kf:

x f {Ax(x),E (y)}= 8pGgd(x, y), {gKf(x), E (y)}= 16pGgd(x, y).

Note that we use the Ashtekar connection for the inhomogeneous direction x but extrinsic curvature for the homogeneous direction along symmetry orbits [75Jump To The Next Citation Point]. Connection and extrinsic curvature components for the f-direction are related by 2 2 2 2 A f = Gf + g K f with the spin connection component

-Ex'- Gf = - 2Ef . (38)
Unlike in the full theory or homogeneous models, Af is not conjugate to a triad component but to [52Jump To The Next Citation Point]
f V~ ------------------- P = 4(Ef)2 - A -f2(Pb)2

with the momentum Pb conjugate to a U(1)-gauge angle b. This is a rather complicated function of both triad and connection variables such that the volume integral V~ -x-- f V = 4p |E |E dx 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 [74Jump To The Next Citation Point]. This can be avoided by using extrinsic curvature which is conjugate to the triad component [75Jump To The Next Citation Point]. Moreover, this is also in accordance with a general scheme to construct Hamiltonian constraint operators for the full theory as well as symmetric models [194Jump To The Next Citation Point42Jump To The Next Citation Point58Jump To The Next Citation Point].

The constraint operator in spherical symmetry is given by

integral ( ) H[N ] = - (2G) -1 dxN (x)| Ex |-1/2 (K2fEf + 2KfKxEx) + (1 - G2f)Ef + 2G'fEx (39) B
accompanied by the diffeomorphism constraint
integral x -1 x f ' x' D[N ] = (2G) N (x)(2E K f - KxE ). (40) B
We have expressed this in terms of Kx for simplicity, keeping in mind that as the basic variable for quantization we will later use the connection component Ax.

Since the Hamiltonian constraint contains the spin connection component Gf given by (38View Equation), 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 Ef 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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