Vacuum Plane-wave Research Articles

Stability of spatially homogeneous plane wave spacetimes. I

The problem of ascertaining the stability of a given spatially homogeneous solution of Einstein's equations to small metric perturbations was examined by Barrow and Sonoda (1986), and their method was discussed by Siklos (1988). In this paper, one of the points mentioned by Siklos is investigated further: the role of the constraint equation. It is shown that the constraint equation can lead to problems aside from those (e.g. linearization stability) usually considered. A particularly simple example is used to illustrate these points, namely the homogeneous vacuum plane wave metric, which for a certain range of values of its parameters admits spatially homogeneous hypersurfaces. One way of avoiding the problems associated with the constraint equation is simply to ignore it. In most cases, this will not affect the outcome.

On the gravitational interaction of null fields

The authors show that when two null fields with planar symmetry collide, they generate a source which behaves as a non-perfect fluid, and that an infinite energy density always develops at a finite time after the instant of collision. Since the Klein-Gordon equation for a massless scalar field is satisfied in the region of interaction, and since the scalar field satisfies the required junction conditions across the null boundaries, the null colliding fields can be interpreted as scalar waves. The four-parameter family of solutions of the Einstein equations they present also includes most of the known solutions for the collision of gravitational plane waves in vacuum, or coupled with stiff matter.

The automorphism groups for Bianchi universe models and computer-aided invariant classification of metrics

An implementation of the time-dependent automorphism formalism for Bianchi cosmologies, in the computer algebra system SHEEP, is described. Its application to the particular case of Bianchi cosmologies admitting vacuum plane-wave solutions is given in relation to the equivalence problem. Using this approach, it is shown that the invariant characterisations of these metrics allow many properties of these spacetimes to be obtained in a straightforward manner. In particular, the invariant characterisations are used to determine the subclasses of spatially homogeneous vacuum plane-wave solutions which simultaneously admit distinct foliations with different Bianchi symmetries. The subclasses of the general vacuum plane-wave solution which admit a Bianchi group on a spacelike hypersurface are evaluated. These results are found to be expressible graphically, in the form of a phase plane which is physically interpretable as describing the amplitude and phase of polarisation of the plane wave. As a final application, it is shown that the Lifshitz-Khalatnikov Bianchi VIh solution is the only vacuum plane-wave solution for which the gravitational wave is linearly polarised.

Isotropic singularities and isotropization in a class of Bianchi type-VI h cosmologies

The evolution of a class of exact spatially homogeneous cosmological models of Bianchi type VI h is discussed. It is known that solutions of type VI h cannot approach isotropy asymptotically at large times. Indeed the present class of solutions become asymptotic to an anisotropic vacuum plane wave solution. Nevertheless, for these solutions the initial anisotropy can decay, leading to a stage of finite duration in which the model is close to isotropy. Depending on the choice of parameters in the solution, this quasi-isotropic stage can commence at the initial singularity, in which case the singularity is of the type known as “isotropic” or “Friedmann-like.” The existence of this quasi-isotropic stage implies that these models can be compatible in principle with the observed universe.