### 5.12 Spherical symmetry

For spherically-symmetric models, a connection has the form (Appendix B.3)
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
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

where is related to by . One example for such a metric is the spatial part of a
flat Friedmann–Robertson–Walker spacetime, where . This allows one to
study perturbations around a homogeneous spacetime, which can also be done at the quantum
level.