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Causal, Homogeneous, and Multidimensional Structures of Manifolds and Geometries Applied in Mathematical Cosmology

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Causal, Homogeneous, and Multidimensional Structures of Manifolds and Geometries Applied in Mathematical Cosmology Causal, homogeneous, and multidimensional structures of manifolds and geometries applied in mathematical cosmology Martin Rainer Institut f¨ur Mathematik Universit¨at Potsdam, PF 601553 D-14415 Potsdam, Germany August 2000 vorgelegt an der Math.-Nat. Fakult¨at der Universit¨at Potsdam zur Habilitation im Fachgebiet “Mathematische Kosmologie” Meinen Eltern Table of contents 1. Introduction 1 2. Causal structures 7 2.1. Local cone structures 8 2.2. Causality on topological manifolds 12 2.3. Causal index sets and diffeomorphisms 16 3. Cosmological application I: quantum general relativity 17 3.1. Algebraic QFT on manifolds 17 3.1.1. General axioms of QFT on a differentiable manifold 17 3.1.2. Axioms of QFT on a manifold with cone causality 18 3.2. Quantum geometry 19 4. Homogeneous structures 21 4.1. Homogeneous manifolds 21 4.2. Local homogeneous geometries 23 4.2.1. Classifying spaces of local isometries 24 4.2.2. Zariski dual topology and Lie algebra cohomology 31 4.2.3. Classifying spaces of local homogeneous geometries 32 5. Cosmological application II: rigidity of isometries 43 6. Multidimensional structures 44 6.1. Effective σ–model for pure multidimensional geometry 49 6.2. σ–model with extra scalars and p + 2-forms 52 6.3. Target space structure 54 6.4. Special coordinate gauges on M0 55 7. Cosmological application III: Multidimensional solutions 57 7.1. Solutions with Abelian target-space 59 7.2. Orthobrane solutions with (E)V = 0 60 7.3. Spherically symmetric p-branes 62 7.4. Black holes with EM branes 67 7.4.1. Solutions with orthobranes 67 2 2 7.4.2. Solutions with degenerate brane charges Qe = Qm 71 7.5. Spatially homogeneous solutions 73 7.5.1. Multi-component perfect fluid cosmology 77 7.5.2. General multidimensional dynamics 79 7.5.3. Integrable 3-component model: Classical solutions 83 7.5.4. Classical wormholes 94 7.5.5. Reconstruction of potentials 97 7.5.6. Solutions to the quantized model 100 7.6. The Einstein frame in cosmology 102 7.6.1. Generating solutions in the Einstein frame 103 7.6.2. Solutions in original form 106 7.6.3. Solutions in the Einstein frame 108 7.7. Multidimensional m-component cosmology 110 7.7.1. Am Toda chain solution 114 7.7.2. Example in Einstein frame 118 7.8. 2-dimensional dilaton gravity 121 7.8.1. Reduction of inhomogeneous cosmology to dimension 2 121 7.8.2. Example: dilaton gravity from 5-dimensional Einstein gravity 123 7.8.3. Spherically symmetric model coupled to matter 125 7.8.4. Static solutions and the horizon problem 126 8. Discussion 129 Acknowledgements 136 References 137 Thesen zur Dissertation (German) 149 1. Introduction Topological and differentiable (real) manifolds are nowadays present in almost all mathe- matical models of natural phenomena or engineering problems containing some differential equation. Often the manifold under consideration carries an additional structure char- acteristic for the model, and its intended range of validity. Causal, homogeneous, and multidimensional structures shall be the topic below. They find their natural applica- tions within the range of current mathematical cosmology. The latter today includes not only classical relativistic cosmology but in particular also its extensions towards quantum geometry and to dimensions different than 1 + 3. Accordingly, already one third of the International Seminar on Current Topics in Mathematical Cosmology, 1998 in Potsdam, was dedicated to these modern extensions [1]. It is well known that a geometry given by a differentiable connection on a manifold need not be related to any metric, although vice versa any differentiable metric defines a unique metric compatible connection, namely the Levi-Civita connection. Similarly, below we define rather general causal structures on (real) manifolds which do not imply the existence of a metric. Even a non-differentiable topological manifold may carry a causal structure. However, in this case there is no definition neither of a connection 1-form, nor of a curvature tensor, nor of a metric. Even if the manifold is differentiable, a differentiable causal structure does not imply the existence of a (conformal) metric (of any signature). Moreover, given a causal structure and a metric of corresponding signature, both need not necessarily be compatible with each other. Simple, simply laced singularities (often also known as ADE singularities according to the Cartan type of corresponding Lie algebras) provide a local model for a rather general class of structures on manifolds which might also be called pseudo-causal. Relativity and cosmology usually refer to the conical singularity as local model for causality. Therefore, unless stated otherwise, here we will refer to causal structures which are given as local cone structures. A Hausdorff (i.e. T2-strongly separating) topological manifold M without boundary is topologically homogeneous per definition, since the neighborhoods of all points are modeled over the same standard vector space. Note however that a homogeneous Cr- structure on a closed manifold need not be Cr+1-homogeneous, while the vice versa is always true. The homogeneity of a manifold M can be expressed by the existence of a transitive group action on the manifold. A homogeneous manifold is characterized locally by its homeomorphism group. If M is homogeneous, any two points are connected by a local homeomorphism, and the (local) homeomorphism group HomM acts on the entire man- ifold. (Note: Here we do not consider global homeomorphisms which are a topic in its own.) If M contains boundary points, one can always achieve topological homogeneity by restricting the manifold to its interior. However, if the manifold carries more structure, homeomorphisms have to preserve not only the local topology but also the additional structure. So, a Cr-differential structure on a manifold restricts the homeomorphism to Cr-diffeomorphisms. Similarly, a causal structure on a manifold restricts its homeomorphisms to those which preserve the causal 1 structure. Likewise, a diffeomorphism of a Riemannian manifold (M, g) is a homeomor- phism of the latter, iff it is an isometry. In general with any given structure on a manifold, the structure-preserving homeo- morphisms form a subgroup of the homeomorphism group. A structure s on a manifold M is homogeneous, iff the structure preserving homeomorphisms group Hom(M, s) acts transitively on M. The cosmological application of causal topological manifolds refers to the requirement of quantum geometric gravity to implement the local topological content of general relativ- ity in absence of a background geometry. Homogeneous and multidimensional geometries both relate to symmetries present either in the large scale cosmological structure or due to unification of general relativity with other fundamental symmetries in higher dimension. Dynamically the different symmetries reflect themselves in fundamental forces: gravity resulting from general relativity, and other forces from other symmetries. Rather than trying to give an exhaustive discussion of causal, homogeneous, and mul- tidimensional structures in mathematical cosmology, below the focus will be on some selected aspects of these structures which are of particular relevance in current research. Topologically defined causal structures are of importance when the geometry itself is subject to quantization, i.e. for canonical quantum gravity in particular. Homogeneity structures are not only suggested by phenomenological symmetries in cosmology, but the construction of a classifying space of local homogeneous geometries in terms of scalar algebraic invariants, and the relation to a similar classifying space of the isometries gives systematic insight about the stability of a particular isometry under deformation of the geometry. Multidimensional and at least partially homogeneous geometries are essential for the existence of elegant effective mathematical models, like the effective sigma-model of multidimensional geometry, which result in clarification, generalization, and prediction of structures and results inherent in the geometrical content of modern unification ap- proaches such as M-theory, and also may be sufficiently tractable in order to be a basis for further investigations on higher dimensional geometry, like their canonical quantization via midisuperspaces, or the imprint of features of the higher-dimensional geometry in 3+1 dimensions (”windows towards extra-dimensions”), e.g. in the Hawking temperature of black holes extended by p-branes. Classical general relativity includes traditionally clas- sification and analysis of solutions to the field equations of Einstein-Hilbert actions with various matter terms on 4-dimensional Lorentzian manifolds within some Cr-category of differentiability, where most results are given for r ≥ 2 and often C∞-smoothness is as- sumed. Traditional mathematical cosmology is targeted to investigate those solutions, which model a hypothetical large scale structure of our 3 + 1-dimensional universe un- der more or less realistic assumptions about its matter content. Roughly those models may be divided in those which are spatially homogeneous, and those which are spatially in homogeneous. With an appropriate choice of time coordinates, homogeneous spatial 3-manifolds result here as hypersurfaces of constant time. In traditional classical general relativity and cosmology, the causal structure is given a priori by the Lorentzian signature of the geometry. Continuous efforts towards a con- sistent canonical quantization of the geometry underlying general relativity,
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