4.2 Two-Killing vector reductions

The next natural step is to consider two-dimensional spacelike isometry groups. A local approach that parallels the one given for the one-Killing vector reduction was developed by Geroch in [97] for the abelian case (corresponding to commuting Killing fields). The global aspects for these models, in the case of considering commutative, connected and two dimensional isometry groups with effective and proper action, have been discussed in detail in [38Jump To The Next Citation Point, 66]. In the following we will simply refer to them as two-Killing vector reductions despite the fact that they do not correspond to the most general situation.

The classification of the smooth, effective, proper actions by isometries of a commutative, connected, two-dimensional Lie group on a connected, smooth 3-manifold has been studied in the literature both for the compact [176, 177] and non-compact [38] topologies. Of all the possible choices of group action and spatial topology only two have been considered with sufficient detail from the quantum point of view:

In the restricted case when the group orbits are hypersurface orthogonal we have the polarized models (also known as linearly polarized models). Otherwise we have general polarization (or non-polarized) models. Historically, two-Killing vector reductions were introduced to explore some concrete problems; in particular, the original motivation by Einstein and Rosen [84, 29] was to use cylindrical symmetry as a simplifying assumption to explore the existence of gravitational waves (see, however, [131]). Gowdy considered the U(1) × U(1) model as a first step to study inhomogeneous cosmologies [102Jump To The Next Citation Point, 103Jump To The Next Citation Point]. As we are mainly interested in the quantization of midisuperspaces we discuss next only those classical issues of direct use in the quantum treatment of these models, in particular their Hamiltonian description.

The first Hamiltonian analysis of the Einstein–Rosen waves was carried out by Kuchař in the early seventies [141Jump To The Next Citation Point]. However, a complete treatment incorporating the appropriate surface terms had to wait until the nineties [13Jump To The Next Citation Point]. A work that was very influential in getting the final Hamiltonian formulation was [15Jump To The Next Citation Point], in which Ashtekar and Varadarajan study the Hamiltonian formalism of asymptotically-flat 2+1 dimensional GR coupled to matter fields satisfying positive energy conditions. They find that, due to the peculiarities associated with the definition of asymptotic flatness in the 2+1 setting, the Hamiltonian of those systems is always bounded both from below and from above. This is a very important result for the class of midisuperspaces whose 2+1 description can be performed in terms of non-compact Cauchy surfaces. As we pointed out before, this is the case for certain one-Killing symmetry reductions and especially for Einstein–Rosen waves [13Jump To The Next Citation Point]. Before we proceed further we want to point out that the asymptotic analysis mentioned above has been extended to discuss the structure of the null infinity in the 2+1 description of one-Killing symmetry reductions of 3+1 GR in [5] and the behavior of Einstein–Rosen waves at null infinity has been studied in detail in [6].

In the case of the Einstein–Rosen waves the two Killing vector fields correspond to translations and rotations. The translational Killing field has a non-vanishing norm whereas the rotational one vanishes at the symmetry axis. The main steps to arriving at the Hamiltonian formulation of the Einstein–Rosen waves can be summarized as follows:

After all these steps the Hamiltonian is obtained in closed form as a function of the free Hamiltonian H0 for an axially-symmetric massless scalar field evolving in an auxiliary, 2+1 dimensional Minkowskian background

( ) H = 2 1 − exp(− H0 ∕2) . (3 )
The fact that this Hamiltonian is a function of a free one can be used to study the exact classical dynamics of this system. It is very important to point out that the Minkowskian metric (induced by the metric in the asymptotic region) plays an auxiliary role. Although the Hamilton equations are non-linear, it is possible to achieve a remarkable simplification by introducing a redefined time variable of the form T := exp (− H0∕2 )t where t is the asymptotic inertial time. When this is done the scalar field that encodes the gravitational degrees of freedom of the model must simply satisfy the free field equation for an axisymmetric field. This fact allows us to quantize the model by using a Fock space. In particular the exact unitary evolution operator can be written in closed form and, hence, closed form expressions can be written for many interesting physical objects such as two-point functions.

We want to mention that some generalizations of the Einstein–Rosen waves to a class of cylindrical spacetimes endowed with angular momentum have been considered by Manojlović and Mena in [156], where the authors found boundary conditions that ensure that the resulting reduced system has consistent Hamiltonian dynamics. This work lead Mena [160] to consider a definition of cylindrical spacetimes that is less restrictive than that usually employed in the literature. The key idea is to remove the symmetry axis from the spacetime and, as a consequence, relax the regularity conditions in the fields when they approach this spacetime boundary. Little is presently known about the quantization of these systems so we will not consider them further.

The Hamiltonian formalism for the Gowdy models has been developed by many authors both in vacuum [159Jump To The Next Citation Point, 189Jump To The Next Citation Point] and coupled to scalar fields for all the possible topologies [22Jump To The Next Citation Point]. The Hamiltonian analysis for the T3 topology can be seen in [159Jump To The Next Citation Point, 189Jump To The Next Citation Point]. An interesting technical point that is relevant here concerns gauge fixing. In this model the natural gauge fixing condition gives rise to a deparametrization because the fixing is not complete. Actually, after this partial gauge fixing, two first class constraints remain. One of them is linear in the momentum canonically conjugate to a variable that can be interpreted as time. As a consequence of this it is possible to reinterpret the system as one described by a time dependent Hamiltonian. The other constraint remains as a condition on physically acceptable configurations. A thorough discussion of these issues can be found in [22Jump To The Next Citation Point]. This last paper carefully extends the Hamiltonian analysis to the other possible spatial topologies. In particular, it discusses the constraints that must be included in the Hamiltonian formulation to take into account the regularity conditions on the metric in the symmetry axis for the S2 × S1 and S3 topologies. The main difficulty that arises now is the vanishing of some of the (rotational) Killing vector fields at some spacetime points. In the case of the S2 ×  S1 topology only one of the two Killings vanish so the problem can be dealt with by using the non-vanishing Killing to perform a first Geroch reduction and carefully deal with the second by imposing suitable regularity conditions for the fields in the 2+1 dimensional formulation. The S3 case is harder to solve because both Killing fields vanish somewhere. Nevertheless the problem can be successfully tackled by excising the axes from the spacetime manifold and imposing suitable regularity conditions on the fields when they approach the artificial boundary thus introduced. An interesting feature that shows up in the Hamiltonian analysis for the S2 × S1 and S3 topologies is the presence of the polar constraints induced by the regularity conditions. The final description for these topologies is somehow similar to the one corresponding to the T3 case, in particular the fact that the dynamics of the system can be described with a time dependent Hamiltonian that clearly shows how the initial and final singularities appear. The main difference is the absence of the extra constraint present in the T3 model. This is a consequence of the details of the deparametrization process.

Other classical aspects related to Gowdy models coupled with matter (concerning integrability or the obtention of exact solutions) have been covered in detail in [62Jump To The Next Citation Point, 61, 63Jump To The Next Citation Point, 57, 56]; in particular, the identification of a complete set of Dirac observables for the Einstein–Rosen and the Gowdy T3 midisuperspace was obtained in [202, 205, 140] (and in [117] by using Ashtekar variables). The relation of two-Killing reductions and sigma and chiral models have been considered by many authors [9, 118, 158]. We want to mention also an interesting paper by Romano and Torre [192Jump To The Next Citation Point] where they investigate the possibility of developing an internal time formalism for this type of symmetry reduction. They also show there that the Hamiltonian of these models corresponds to that of a parametrized field theory of axially symmetric harmonic maps from a 3-dimensional flat spacetime to a 2-dimensional manifold endowed with a constant negative curvature metric (though in the compact cases the presence of constraints must be taken into account).


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