Extensions and applications of 1+3 decomposition methods in general relativistic cosmological modelling
Abstract
1+3 "threading" decomposition methods of the pseudo-Riemannian spacetime manifold (M, g) and all its geometrical objects and dynamical relations with respect to an invariantly defined preferred timelike reference congruence u/c have been useful tools in general relativistic cosmological modelling for more than three decades. In this thesis extensions of the 1+3 decomposition formalism are developed, partially in fully covariant form, and partially on the basis of choice of an arbitrary Minkowskian orthonormal reference frame, the timelike direction of which is aligned with u/c.
After introductory remarks, in Chapter 2 first an exposition is given of the general 1+3 covariant dynamical equations for the fluid matter and Weyl curvature variables, which arise from the Ricci and second Bianchi identities for the Riemann curvature tensor of (M, g, u/c). New evolution equations are then derived for all spatial derivative terms of geometrical quantities orthogonal to u/c. The latter are used to demonstrate in 1+3 covariant terms that the spatial constraints restricting relativistic barotropic perfect fluid spacetime geometries are preserved along the integral curves of u/c. The integrability of a number of different special subcases of interest can easily be derived from this general result. In Chapter 3, 1+3 covariant representations of two classes of well-known cosmological models with a barotropic perfect fluid matter source are obtained. These are the families of the locally rotationally symmetric (LRS) and the orthogonally spatially homogeneous (OSH) spacetime geometries, respectively. Subcases arising from either dynamical restrictions or the existence of higher symmetries are systematically discussed. For example, models of purely "magnetic" Weyl curvature and, in the LRS case, a transparent treatment of tilted spatial homogeneity can be obtained. The 1+3 covariant discussion of the OSH models requires completion. Chapter 4 reviews the complementary 1+3 orthonormal frame (ONF) approach and extends it to include the second Bianchi identities, which provide dynamical relations for the physically interesting Weyl curvature variables. Then, possible choices of local coordinates within the 1+3 ONF framework are introduced, taking both the 1+3 threading and the ADM 3+1 slicing perspectives. The 1+3 ONF method is employed in Chapter 5 to investigate the integrability of the dynamical equations describing "silent" irrotational dust spacetime geometries, for which the "magnetic" part of the Weyl curvature is required to vanish. Evidence is obtained that these equations may not be consistent in the generic case, but that only either algebraically special or spatially homogeneous classes of solutions may be covered. Furthermore, this chapter uses the extended 1+3 ONF dynamical equations to describe LRS models with an imperfect fluid matter source and contrasts the perfect fluid subcase with the results obtained in Chapter 3. In Chapter 6, a brief detour is taken into considering those classical theories of gravitation in which the Lagrangean density of the gravitational field is assumed to be proportional to a general differentiable function f(R) in the Ricci curvature scalar. The generalisations of the relativistic 1+3 covariant dynamical equations to the f(R) case are derived and a few examples of applications are commented on. Finally, Chapter 7 investigates in detail features of the dynamical evolution of the cosmological density parameter in anisotropic inflationary models of Bianchi Type-I and Type-V and points out important qualitative changes as compared to the idealised standard FLRW situation. A related analysis employing the same spacetime geometries addresses the occurrence of restrictions on the permissible functional form of the inflationary expansion length scale parameter S as a consequence of the so-called reality condition for Einstein-Scalar-Field configurations. Again, the effect of the (exact) anisotropic perturbations on the FLRW case is thoroughly studied and found to have significant effects. Both cases can be treated as examples of structural instability. This thesis ends with concluding remarks and an appendix section containing the conventions employed and mathematical relations relevant to derivations given in various chapters.- Publication:
-
Ph.D. Thesis
- Pub Date:
- November 1996
- Bibcode:
- 1996PhDT........25V
- Keywords:
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- 04.20.-q;
- 98.80.Hw;
- 98.80.Dr;
- 04.20.Jb