Internal Time Formalism for Spacetimes with Two Killing Vectors

The Hamiltonian structure of spacetimes with two commuting Killing vector fields is analyzed for the purpose of addressing the various problems of time that arise in canonical gravity. Two specific models are considered: (i) cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes, which respectively involve open and closed universe boundary conditions. For each model canonical variables which can be used to identify points of space and instants of time, {\it i.e.}, internally defined spacetime coordinates, are identified. To do this it is necessary to extend the usual ADM phase space by a finite number of degrees of freedom. Canonical transformations are exhibited that identify each of these models with harmonic maps in the parametrized field theory formalism. The identifications made between the gravitational models and harmonic map field theories are completely gauge invariant, that is, no coordinate conditions are needed. The degree to which the problems of time are resolved in these models is discussed.Comment: 36 pages, Te

On the area of the symmetry orbits in T2T^2 symmetric spacetimes

We obtain a global existence result for the Einstein equations. We show that in the maximal Cauchy development of vacuum $T^2$ symmetric initial data with nonvanishing twist constant, except for the special case of flat Kasner initial data, the area of the $T^2$ group orbits takes on all positive values. This result shows that the areal time coordinate $R$ which covers these spacetimes runs from zero to infinity, with the singularity occurring at R=0.Comment: The appendix which appears in version 1 has a technical problem (the inequality appearing as the first stage of (52) is not necessarily true), and since the appendix is unnecessary for the proof of our results, we leave it out. version 2 -- clarifications added, version 3 -- reference correcte

The Gowdy T3 Cosmologies revisited

We have examined, repeated and extended earlier numerical calculations of Berger and Moncrief for the evolution of unpolarized Gowdy T3 cosmological models. Our results are consistent with theirs and we support their claim that the models exhibit AVTD behaviour, even though spatial derivatives cannot be neglected. The behaviour of the curvature invariants and the formation of structure through evolution both backwards and forwards in time is discussed.Comment: 11 pages, LaTeX, 6 figures, results and conclusions revised and (considerably) expande

Why Solve the Hamiltonian Constraint in Numerical Relativity?

The indefinite sign of the Hamiltonian constraint means that solutions to Einstein's equations must achieve a delicate balance--often among numerically large terms that nearly cancel. If numerical errors cause a violation of the Hamiltonian constraint, the failure of the delicate balance could lead to qualitatively wrong behavior rather than just decreased accuracy. This issue is different from instabilities caused by constraint-violating modes. Examples of stable numerical simulations of collapsing cosmological spacetimes exhibiting local mixmaster dynamics with and without Hamiltonian constraint enforcement are presented.Comment: Submitted to a volume in honor of Michael P. Ryan, Jr. Based on talk given at GR1

Mixmaster Behavior in Inhomogeneous Cosmological Spacetimes

Numerical investigation of a class of inhomogeneous cosmological spacetimes shows evidence that at a generic point in space the evolution toward the initial singularity is asymptotically that of a spatially homogeneous spacetime with Mixmaster behavior. This supports a long-standing conjecture due to Belinskii et al. on the nature of the generic singularity in Einstein's equations.Comment: 4 pages plus 4 figures. A sentence has been deleted. Accepted for publication in PR

Global existence problem in T3T^3-Gowdy symmetric IIB superstring cosmology

We show global existence theorems for Gowdy symmetric spacetimes with type IIB stringy matter. The areal and constant mean curvature time coordinates are used. Before coming to that, it is shown that a wave map describes the evolution of this system

Global Foliations of Vacuum Spacetimes with T2T^2 Isometry

We prove a global existence theorem (with respect to a geometrically- defined time) for globally hyperbolic solutions of the vacuum Einstein equations which admit a $T^2$ isometry group with two-dimensional spacelike orbits, acting on $T^3$ spacelike surfaces.Comment: 38 pages, 0 figures, LaTe

Complete quantization of a diffeomorphism invariant field theory

In order to test the canonical quantization programme for general relativity we introduce a reduced model for a real sector of complexified Ashtekar gravity which captures important properties of the full theory. While it does not correspond to a subset of Einstein's gravity it has the advantage that the programme of canonical quantization can be carried out completely and explicitly, both, via the reduced phase space approach or along the lines of the algebraic quantization programme. This model stands in close correspondence to the frequently treated cylindrically symmetric waves. In contrast to other models that have been looked at up to now in terms of the new variables the reduced phase space is infinite dimensional while the scalar constraint is genuinely bilinear in the momenta. The infinite number of Dirac observables can be expressed in compact and explicit form in terms of the original phase space variables. They turn out, as expected, to be non-local and form naturally a set of countable cardinality.Comment: 32p, LATE

Numerical Investigation of Cosmological Singularities

Although cosmological solutions to Einstein's equations are known to be generically singular, little is known about the nature of singularities in typical spacetimes. It is shown here how the operator splitting used in a particular symplectic numerical integration scheme fits naturally into the Einstein equations for a large class of cosmological models and thus allows study of their approach to the singularity. The numerical method also naturally singles out the asymptotically velocity term dominated (AVTD) behavior known to be characteristic of some of these models, conjectured to describe others, and probably characteristic of a subclass of the rest. The method is first applied to the unpolarized Gowdy T$^3$ cosmology. Exact pseudo-unpolarized solutions are used as a code test and demonstrate that a 4th order accurate implementation of the numerical method yields acceptable agreement. For generic initial data, support for the conjecture that the singularity is AVTD with geodesic velocity (in the harmonic map target space) < 1 is found. A new phenomenon of the development of small scale spatial structure is also observed. Finally, it is shown that the numerical method straightforwardly generalizes to an arbitrary cosmological spacetime on $T^3 \times R$ with one spacelike U(1) symmetry.Comment: 37 pp +14 figures (not included, available on request), plain Te