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5.12 Spherical symmetry

For spherically-symmetric models, a connection has the form (Appendix B.3)
A = Ax(x )τ3 dr + (A1(x)τ1 + A2(x )τ2) dϑ + (A1(x)τ2 − A2 (x )τ1) sin ϑ dϕ + τ3cos ϑd ϕ, (71 )
whose field-dependent terms automatically have perpendicular internal directions. In this case, it is not diagonalization as in the polarization condition for Einstein–Rosen waves but a non-trivial isotropy subgroup, which leads to this property. The kinematical quantization is then simplified as discussed before, with the only difference being that there is only one type of vertex holonomy
hv(A) = exp (iγ μvK ϕ(v))

as a consequence of a non-trivial isotropy subgroup. The Hamiltonian constraint can again be computed explicitly [109].

Spherically-symmetric models are usually used for applications to non-rotating black holes, but they can also be useful for cosmological purposes. They are particularly interesting as models for the evolution of inhomogeneities as perturbations, which can be applied to gravitational collapse but also to cosmology. In such a context, one often reduces the spherically-symmetric configuration even further by requiring a spatial metric

2 2 2 ds = qxx(x,t)dx + qϕϕ (x, t) dΩ ,

where qxx is related to qϕϕ by ∂x√qxx-= √q-ϕϕ-. One example for such a metric is the spatial part of a flat Friedmann–Robertson–Walker spacetime, where q (x, t) = x2a (t)2 ϕϕ. This allows one to study perturbations around a homogeneous spacetime, which can also be done at the quantum level.

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