On Bianchi Type VI $$_0$$ 0 Spacetimes with Orthogonal Perfect Fluid Matter
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nnales Henri Poincar´ e
On Bianchi Type VI0 Spacetimes with Orthogonal Perfect Fluid Matter Hans Oude Groeniger Abstract. We study the asymptotic behaviour of Bianchi type VI0 spacetimes with orthogonal perfect fluid matter satisfying Einstein’s equations. In particular, we prove a conjecture due to Wainwright about the initial singularity on such backgrounds. Using the expansion-normalized variables of Wainwright–Hsu, we demonstrate that for a generic solution the initial singularity is vacuum dominated, anisotropic and silent. In addition, by employing known results on Bianchi backgrounds, we obtain convergence results on the asymptotics of solutions to the Klein–Gordon equation on all backgrounds of this type, except for one specific case.
1. Introduction The subject of this study is the asymptotic behaviour of a certain class of spatially homogeneous cosmological models. Our interest is twofold. First, we prove a conjecture, made over twenty years ago in [11], regarding the initial singularity of Bianchi type VI0 spacetimes with orthogonal perfect fluid (OPF) matter which satisfy Einstein’s equations. Resolving this conjecture allows for the study of more complicated cosmological models, for which those of the type above appear as limit cases. Second, by combining the asymptotic behaviour that we find with known results on Bianchi models from [8], we are able to obtain results regarding the asymptotics of solutions to the Klein–Gordon equation on this type of model. This fills a gap in the unified treatment of the Klein–Gordon equation on Bianchi backgrounds of [8]. In order to reduce the complexity of Einstein’s equations one often demands a high degree of symmetry to be present in the spacetime as well as in the matter model. Typically, this comes in the form of isotropy or spatial homogeneity. As a prime example, the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) spacetimes are both isotropic and spatially homogeneous. Dropping
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H. O. Groeniger
Ann. Henri Poincar´e
the requirement of isotropy, but retaining that of spatial homogeneity, one finds the Kantowski-Sachs spacetimes and the Bianchi spacetimes. For a Bianchi spacetime, there is a three-dimensional group of isometries acting transitively and freely on spacelike hypersurfaces, while for the Kantowski-Sachs spacetimes the situation is similar but the action is not free. The Bianchi spacetimes may be classified by the structure constants of the corresponding Lie algebras, by the so-called Bianchi-Sch¨ ucking-Behr approach, as described in e.g. Section 2 of [1]. According to this classification, one distinguishes the class A types I, II, VI0 , VII0 , VIII and IX, and the class B types III, IV, V, VIh , VIIh , for a parameter h. As mentioned, the subject of this study are solutions of Bianchi type VI0 with perfect fluid matter with velocity vector orthogonal to the group orbits. For solutions of type I, II and VII0 a similar treatment—on which this one is largely based—is already available, see Sections 8–10 of [9]. Using the orthonormal frame formalism due
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