arXiv subjects

M. Rainer

Publications and source records attributed to M. Rainer.

153 records · Page 9Linked to original sources

arXiv1996

Classifying spaces for homogeneous manifolds and their related Lie isometry deformations

Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their continuous deformations is presented: Classifying spaces for homogeneous manifolds and their related Lie isometry deformations. The adjoint representation of n-dimensional real Lie algebras induces a natural topology on their classifying space, which encodes the natural algebraic relationship between different Lie algebras therein. For n>1 this topology is not Hausdorffian. Even more it satisfies only the separation axiom T_0, but not T_1, i.e. there is a constant sequence which has a limit different from the members of the sequence. Such a limit is called a transition. Recently it was found that transitions are the natural generalization and transitive completion of the well-known Inönü-Wigner contractions. For n<5 the relational classifying spaces are constructed explicitly. Calculating their characteristic scalar invariants via triad representations of the characteristic isometry, local homogeneous Riemannian 3-spaces are classified in their natural geometrical relations to each other. Their classifying space is a composition of pieces with different isometry types. Although it is Hausdorffian, different topological transitions to the same limit may induce locally non-Euclidean regions (e.g. at Bianchi tppes VII_0).

gr-qc↗

arXiv1995

Topological Classifying Spaces of Lie Algebras and the Natural Completion of Contractions

The space K^n of all n-dimensional { Lie} algebras has a natural non-Hausdorff topology k^n, which has characteristic limits, called transitions, A -> B, between distinct Lie algebras A and B. The entity of these transitions are the natural transitive completion of the well known Inonu-Wigner contractions and their partial generalizations by Saletan. Algebras containing a common ideal of codimension 1 can be characterized by homothetically normalized Jordan normal forms of one generator of their adjoint representation. For such algebras, transitions A -> B can be described by limit transitions between corresponding normal forms. The topology k^n is presented in detail for n < 5. Regarding the orientation of the algebras as vector spaces has a non-trivial effect for the corresponding topological space K^n_or: There exist both, selfdual points and pairs of dual points w.r.t. orientation reflection.

alg-geom↗

arXiv1995

Dynamics of Dimensions in Factor Space Cosmology

We consider multidimensional cosmological models with a generalized space-time manifold M = R x M_1 ...x M_n, composed from a finite number of factor spaces M_i, i=1,..n. While usually each factor space M_i is considered to be some Riemannian space of integer dimension d_i, here it is, more generally, a fractal space, the dimension of which is a smooth function d_i(t) of time. Hence, besides the scale factor exponents ln a_i and their derivatives, we consider also the dimensions d_i of the factor spaces as classical dynamical variables. The classical equation of motions and the corresponding Wheeler-de Witt equation are set up generally, and the qualitative behaviour of the system is discussed for some specific model with 2 factor spaces.

gr-qc↗

arXiv1995

Inhomogeneous Cosmological Models with Homogeneous Inner Hypersurface Geometry

Space-times which allow a slicing into homogeneous spatial hypersurfaces generalize the usual Bianchi models. One knows already that in these models the Bianchi type may change with time. Here we show which of the changes really appear. To this end we characterize the topological space whose points are the 3-dimensional oriented homogeneous Riemannian manifolds; locally isometric manifolds are considered as same.

gr-qc↗

arXiv1995

A POSSIBLE SOLUTION TO THE PROBLEM OF EXTRA DIMENSIONS

We consider a multidimensional universe with the topology $M= \R\times M_1\times \cdots \times M_n$, where the $M_i$ ($i>1$) are $d_i$-dimensional Ricci flat spaces. Exploiting a conformal equivalence between minimal coupling models and conformal coupling models, we get exact solutions for such an universe filled by a conformally coupled scalar field. One of the solutions can be used to describe trapped unobservable extra dimensions.

gr-qc↗

arXiv1994

Conformal Coupling and Invariance in Different Dimensions

Conformal transformations of the following kinds are compared: (1) conformal coordinate transformations, (2) conformal transformations of Lagrangian models for a D-dimensional geometry, given by a Riemannian manifold M with metric g of arbitrary signature, and (3) conformal transformations of (mini-)superspace geometry. For conformal invariance under this transformations the following applications are given respectively: (1) Natural time gauges for multidimensional geometry, (2) conformally equivalent Lagrangian models for geometry coupled to a spacially homogeneous scalar field, and (3) the conformal Laplace operator on the $n$-dimensional manifold $M of minisuperspace for multidimensional geometry and the Wheeler de Witt equation. The conformal coupling constant xi_c is critically distinguished among arbitrary couplings xi, for both, the equivalence of Lagrangian models with D-dimensional geometry and the conformal geometry on n-dimensional minisuperspace. For dimension D=3,4,6 or 10, the critical number xi_c={D-2}/{4(D-1)} is especially simple as a rational fraction.

gr-qc↗

arXiv1994

Stability Properties of the Starobinsky Cosmological Model

We discuss the instabilities appearing in the cosmological model with a quasi de Sitter phase following from a fourth-order gravity theory. Both the classical equation as well as the quantization in form of a Wheeler - De Witt equation are conformally related to the analogous model with Einstein's theory of gravity with a minimally coupled scalar field. Results are: 1. In the non-tachyonic case, classical fourth-order gravity is not more unstable than Einstein's theory itself. 2. The well-known classically valid conformal relation is also (at least for some typical cases) valid on the level of the corresponding Wheeler - De Witt equations, which turns out to be a non-trivial statement. (to appear in: Proc. Sem. Relativistic Astrophysics Potsdam 1994, Ed .: J. Mücket)

gr-qc↗

arXiv1994

Classical and Quantum Solutions of Conformally Related Multidimensional Cosmological Models

Consider multidim. universes M= R x M_1 x ... x M_n with D = 1+ d_1 .. + d_n, where M_i of dimension d_i are of have constant curvature and compact for i>1. For Lagrangian models L(R,phi) on M which depend only on Ricci curvature R and a scalar field phi, there exists an explicit description of conformal equivalence, with the minimal coupling model and the conformal coupling model as distinguished representatives of a conformal class. For the conformally coupled model we study classical solutions and their relation to solutions in the equivalent minimally coupled model. The domains of equivalence are separated by certain critical values of the scalar field phi. Furthermore the coupling constant xi of the coupling between phi and R is critical at both, the minimal value xi=0 and the conformal value xi_c={D-2}/{4(D-1)}. In different noncritical regions of $xi$ the solutions behave qualitatively different. For vanishing potential of the minimally coupled scalar field we find a multidimensional generalization of Kasner's solution. Its scale factor singularity vanishes in the conformal coupling model. Static internal spaces in the minimal model become dynamical in the conformal one. The nonsingular conformal solution has a particular interesting region, where internal spaces shrink while the external space expands. While the Lorentzian solution relates to a creation of the universe at finite scale, it Euclidean counterpart is an (instanton) wormhole. Solving the Wheeler de Witt equation we obtain the quantum counterparts to the classical solutions. A real Euclidean quantum wormhole is obtained in a special case.

gr-qc↗

arXiv1994

Resolution of simple singularities yielding particle symmetries in a space-time

A finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C^3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The homological intersection graph of this cycles is the Dynkin graph of an ADE Lie group. The deformation of the simple singularity corresponds to ADE symmetry breaking. A 3+1-dimensional topological model of observation is constructed, transforming consistently under SL(2,C), as an evolving 3-dimensional system of world tubes, which connect ``possible points of observation". The existence of an initial singularity for the 4-dimensional space-time is related to its global topological structure. Associating the geometry of ADE singularities to the vertex structure of the topological model puts forward the conjecture on a likewise relation of inner symmetries of elementary particles to local space-time structure.

gr-qc↗