4.3 Spherical symmetry

Spherically-symmetric systems in GR are another type of midisuperspace models (in a sense “the other type”) that has received a lot of attention. They are enjoying a second youth these days as very useful test beds for LQG. A 3+1 spacetime (M, g) is called spherically symmetric if its isometry group contains a subgroup isomorphic to SO(3) and the orbits of this subgroup are 2-spheres such that the metric g induces Riemannian metrics on them that are proportional to the unit round metric on S2. Notice that in the standard definition of spherically symmetry the spacetime manifold is taken to be diffeomorphic to R × Σ, where the Cauchy hypersurface Σ is 2 R × S (notice, however, that this is not the only possibility [67, 197, 200]). In this case the SO(3) symmetry group acts without fixed points (there is no center of symmetry). The spherically-symmetric metric on the Cauchy slices R × S2 is given by Λ2(t,r)dr2 + R2(t,r)dΩ2 (where dΩ2 denotes the metric of the unit 2-sphere). This metric depends only on two functions Λ (t,r) and R (t,r). The radial coordinate r ∈ R is such that the r → ± ∞ limits correspond to the two different spatial infinities of the full Schwarzschild extension and t denotes a “time” coordinate.

The first attempt to study spherically-symmetric models in GR from the Hamiltonian ADM point of view goes back to the paper by Berger, Chitre, Moncrief and Nutku [37Jump To The Next Citation Point]. Here the authors considered vacuum gravity and also coupled to other fields such as massless scalars. The problem with this approach, as pointed out by Unruh in [210Jump To The Next Citation Point], was that they did not reproduce the field equations. The cause for this was identified also by Unruh: a boundary term needed to guarantee the differentiability of the Hamiltonian was missing in the original derivation. It must be pointed out that the paper [37Jump To The Next Citation Point] predates the classic one by Regge and Teitelboim [190] where the role of surface terms in the correct definition of the Hamiltonian framework for GR is discussed in detail. We also want to mention that [37] was the starting point for an interesting series of articles by Hájíček on Hawking radiation [106, 104, 105].

The spacetime slicings chosen in the first studies of spherically-symmetric models covered only the static regions of the extended Schwarzschild spacetime (the Kruskal extension). This means that, in practice, they only considered the Schwarzschild geometry outside the event horizon. This problem was tackled by Lund [153] who used a different type of slicing that, however, did not cover the whole Kruskal spacetime with a single foliation. An interesting issue that was explored in this paper had to do with the general problem of finding a canonical transformation leading to constraints that could give rise to a generalized Schrödinger representation (as was done by Kuchař in the case of cylindrical symmetry [141Jump To The Next Citation Point]). One of the conclusions of this analysis was that this was, in fact, impossible, i.e., there is no “time variable” such that the constraints are linear in its canonically conjugate momentum. This negative conclusion was, however, sidestepped by Kuchař in [143Jump To The Next Citation Point] by cleverly using a less restrictive setting in which he considered foliations going from one of the asymptotic regions of the full Kruskal extension to the other. This paper by Kuchař [143Jump To The Next Citation Point] is in a sense the culmination of the continued effort to understand the quantization of Schwarzschild black holes in the traditional geometrodynamical setting. It must be said, however, that it was predated by the analysis performed by Thiemann and Kastrup [130Jump To The Next Citation Point, 201Jump To The Next Citation Point] on the canonical treatment of Schwarzschild black holes in the Ashtekar formalism. In [130Jump To The Next Citation Point] the authors found a pair of canonical variables that coordinatize the reduced phase space for spherically-symmetric black holes consisting of two phase-space variables M and T where M is the black hole mass and T is its conjugate variable that can be interpreted as “time” (more precisely the difference of two time variables associated with the two spatial asymptotic regions of an eternal black hole). This description of the reduced phase space precisely coincides with the ones given by Kuchař [141Jump To The Next Citation Point]. We want to mention also that an interesting extension of Kuchař’s work appears in [214]. In this paper, Varadarajan gave a non-singular transformation from the usual ADM phase-space variables on the phase space of Schwarzschild black holes to a new set of variables corresponding to Kruskal coordinates. In this way it was possible to avoid the singularities appearing in the canonical transformations used by Kuchař.

The Hamiltonian formulations obtained by these methods provide a precise geometrical description of the reduced phase for vacuum spherically-symmetric GR. In particular an exact parametrization of the reduced phase space is achieved. At this point it is just fitting to quote Kuchař [143Jump To The Next Citation Point]:

Primordial black holes, despite all the care needed for their proper canonical treatment, are dynamically trivial.

A possible way to have spherically-symmetric gravitational models with local degrees of freedom and avoid this apparent triviality consists in coupling matter to gravity. It must be said, nevertheless, that for some types of matter couplings the reduced phase space of spherically-symmetric systems is finite dimensional. This is so, for example, in the case of adding infinitesimally spherical thin shells. The Hamiltonian analysis of the massive and the null-dust shell cases has been extensively studied in the literature [90, 33Jump To The Next Citation Point, 151, 107Jump To The Next Citation Point, 109Jump To The Next Citation Point]. The presence of additional null shells has been also analyzed [110Jump To The Next Citation Point, 111Jump To The Next Citation Point].

It is perhaps more surprising to realize that this finite-dimensional character is also a property of spherically-symmetric Einstein–Maxwell spacetimes with a negative cosmological constant, for which the gauge symmetries exclude spherically-symmetric local degrees of freedom in the reduced phase space. In this case canonical transformations of the Kuchař type can be used [152Jump To The Next Citation Point] to obtain the reduced phase-space Hamiltonian formulation for the system. Once matter in the form of massless scalar fields is coupled to gravity, the reduced system is a (1+1)-dimensional field theory and some of the techniques developed by Kuchař cannot be applied. In particular, Romano has shown [191] that the coupled Einstein–Klein–Gordon system does not have a suitable extrinsic time variable. As we mentioned above, the Hamiltonian formulation for the gravity-scalar field model was clarified by Unruh in [210]. Recently, some simplifications have been obtained by using flat slice Painlevé–Gullstrand coordinates [123]. Other types of matter that can be coupled to gravity giving rise to infinite-dimensional reduced phase spaces are those including collapsing dust clouds [223Jump To The Next Citation Point, 220Jump To The Next Citation Point].

To end this section, we should mention that another interesting way to gain insights into the quantization of more realistic gravity models, such as the collapse of spherically-symmetric matter in 3+1 dimensions, is to consider two-dimensional dilaton gravity as in the Callan, Giddings, Harvey and Strominger (CGHS) model [53] – and similar ones that admit a phase-space description close to the 3+1 spherically-symmetric spacetimes. These systems are usually exactly solvable (both classically and quantum mechanically) and hence can be used to study the consequences of quantizing gravity and matter. From a technical point of view these models are close to spherical symmetry because they can be treated by using the same type of canonical transformations introduced by Kuchař in [143Jump To The Next Citation Point]. They lead to descriptions that are quite close to the ones obtained for the vacuum Schwarzschild case [211, 49, 95].


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