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4.18 Inhomogeneous models

The full theory is complicated at several different levels of both its 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 phenomenological equations of different types or precise effective 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 of 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, in which 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. Nevertheless, in spite of significant calculational simplifications, several fundamental issues remain to be resolved, such as a satisfactory quantum treatment of the constraint algebras arising in gravity.

In spherical coordinates x, ϑ, ϕ a spherically-symmetric spatial metric takes the form

ds2 = qxx(x, t) dx2 + qϕϕ(x,t)dΩ2

with 2 2 2 2 dΩ = dϑ + sin ϑ dϕ. This is related to densitized triad components by [294199]

x ϕ 2 |E | = qϕϕ , (E ) = qxxqϕϕ,

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

x ϕ {Ax (x),E (y)} = 8πG γδ (x, y), {γK ϕ(x),E (y)} = 16πG γ δ(x,y).

Note that we use the Ashtekar connection for the inhomogeneous direction x but extrinsic curvature for the homogeneous direction along symmetry orbits [109Jump To The Next Citation Point]. Connection and extrinsic curvature components for the ϕ-direction are related by A2 = Γ 2 + γ2K2 ϕ ϕ ϕ with the spin connection component

Ex′ Γ ϕ = −---ϕ-. (41 ) 2E
Unlike in the full theory or homogeneous models, A ϕ is not conjugate to a triad component but to [60Jump To The Next Citation Point]
∘ ------------------- P ϕ = 4(E ϕ)2 − A −ϕ2(Pβ)2

with the momentum β P conjugate to a U(1)-gauge angle β. This is a rather complicated function of both triad and connection variables such that the volume ∫ ∘ ----- V = 4π |Ex |E ϕdx would have a 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 [107Jump To The Next Citation Point]. This can be avoided by using extrinsic curvature, which is conjugate to the triad component [109Jump To The Next Citation Point]. Moreover, it is also in accordance with a general scheme to construct Hamiltonian constraint operators for the full theory as well as symmetric models [292Jump To The Next Citation Point50Jump To The Next Citation Point68Jump To The Next Citation Point].

The constraint operator in spherical symmetry is given by

∫ −1 x −1∕2( 2 ϕ x 2 ϕ ′ x) H [N ] = − (2G ) dxN (x)|E | (K ϕE + 2K ϕKxE ) + (1 − Γϕ )E + 2Γ ϕE (42 ) B
accompanied by the diffeomorphism constraint
∫ D [N x] = (2G )−1 N x(x)(2E ϕK ′ − KxEx ′) . (43 ) B ϕ
We have expressed this in terms of Kx for simplicity, keeping in mind that we will later use the connection component Ax as the basic variable for quantization.

Since the Hamiltonian constraint contains the spin connection component Γ ϕ given by (41View Equation), which contains inverse powers of densitized triad components, one can expect effective classical equations with corrections similar to the Bianchi IX model. Moreover, an inverse of Ex occurs in (42View Equation). However, the situation is now much more complicated, since we have a system bound by many constraints with non-Abelian algebra. Simply replacing the inverse of x E or ϕ E with a bounded function as before will change the constraint algebra and thus most likely lead to anomalies. In addition, higher powers of connection components or extrinsic curvature arise from holonomies and further correction terms from quantum backreaction. This issue has the potential to shed light on many questions related to the anomaly issue, as done for inverse corrections in [102Jump To The Next Citation Point]. It is one of the cases where models that lie between homogeneous ones, where the anomaly problem trivializes, and the full theory are most helpful.

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