M. Yu. Konstantinov

Topological transitions and large-scale structure of space-time in multi-dimensional theory of gravity

The possibility of building a theory of topological transitions within the framework of multidimensional gravitation theories is discussed. It is shown that consideration of a four-dimensional space-time as a submanifold in the space of a large number of measurements provides a real possibility of constructing a theory of the large-scale structure of a four-dimensional physical space-time and, in particular, a theory of topological transitions. The fundamental principles underlying the construction of the theory are expounded.

A Simple Model of Space-Time Metric Signature Change

The problem of describing the space-time metric signature change is discussed. A simple model with the space-time metric signature determined by a nonlinear scalar field is suggested. It is demonstrated that both classical and quantum descriptions of the metric signature change are possible within the framework of the examined model, and the most characteristic features and differences of the classical and quantum descriptions are briefly discussed.

Faster-than-Light Signals and Causality

The problems of causality are analyzed in terms of space-time models which admit the propagation of signals with superrelativistic velocities. It is shown that there is no violation in causality if the propagation of faster-than-light signals is described by general-covariant equations and occurs along invariant curves, as it is in some well-known models.

Numerical investigation of multidimensional cosmological models based on the Weyl integrable geometry

The dynamics of multidimensional cosmological models based on the Weyl integrable geometry are investigated by means of numerical methods. Models are considered in space and in the presence of matter, the latter modeled by an ideal liquid and a nonminimal scalar field. Sufficient conditions are obtained under which cosmological singularity is absent and the scenario of dynamic dimensional reduction is realized.

Singularities in the classical theory of topological transitions

Within the framework of classical (nonquantum) theory of topological transitions, the problem of singularities is discussed; this is one of the basic obstacles to transition to a quantum description. The features of the solution of this problem for a gravitational field and the fields of the sources are considered. In the first case, the singularity problem may be solved by constructing a Lagrangian that is regular in the vicinity of the topological transition. For gravitational-field sources this method is inapplicable, and therefore it is necessary either to use a mechanism analogous to the mechanism of spontaneous symmetry violation or to introduce additional boundary conditions which ensure regularity of the Lagrangian and the field equations.

Scalar-tensor approach to the derivation of a theory of topological transitions

The derivation of a classical theory of gravitation whose solutions explicitly contain a description of topological transitions is discussed. Toward this goal there is a discussion of a scalar-tensor formalism based on the representation of a certain subclass of spacelike hypersurfaces as surfaces of a constant level of a smooth function on a four-dimensional manifold. The solutions of a theory of this type, along with the Lorentzian structure of space-time and the topology of space-like hypersurfaces, determine a frame of reference, but the nature of the topological transitions does not depend on the choice of a frame of reference. This independence proves the correctness of this new approach. Two versions of a scalar-tensor theory of topological transitions are considered as examples. One version reduces to Einsteins theory of gravitation in a regular region of space-time, while the other is a nontrivial modification of the Brans-Dicke theory.

Strong gravitational fields and geometry in the large

The problem of strong gravitational fields (μ∼1) can be neither formulated invariantly nor solved in a local manner; it belongs to geometry in the large and requires the discussion of a complete atlas of maps. At μ∼1 a complicated topology of space-time is possible. Requirements for a solution with a complete atlas of maps and a physical example, a rigorous discussion of which has led to new results, are discussed.

Classical models of a topological transition in cosmology

Some applications of a scheme, based on the theory of spherical rearrangements, for constructing classical models of topological transitions in the general theory of relativity to cosmological models are examined. In particular, models of creation (annihilation) of open and closed universes, the transition of an open universe into a closed one, and coalescence of universes are described.

SIMPLEST CLASSICAL MODELS OF TOPOLOGICAL TRANSITIONS

It is shown that the simplest classical models of topological transitions have scalar singularity of curvature with a point carrier that is a source of spacetime incompleteness. It is also shown that, close to topological transition, the condition of energy dominance is violated, while the asymptotic behavior of the curvature tensor (increase in curvature on approaching topological transition) and the energy-momentum tensor (violation of the energy-dominance condition) is a common property of the given models and is determined overall by the type of topological transition.

A possible approach to the description of topological transitions in the quantum theory of gravitation

ConclusionsOne of the problems generated in attempts of describing topological transitions in the quantum theory of gravitation is the introduction of variables describing the topology of space-time into the state functional of the quantum theory. The considerations above justify the assumption that this problem can be solved by a procedure similar to the Feynman procedure of summation over history, if as variables describing the topology of space-time one chooses the system FΛ of characteristic functions of a nerve ΛU of some locally finite covering U of space-time. In this case the characteristic functions of the nerve ΛU of the covering U play for a given manifold topology the same role as local coordinates for given points on the manifold, and the sequence of the form (11) can be considered as the analog of trajectories of classical mechanics. Finally, the terms Sjtop appearing in equality (24) for the action on the sequence (11) can be considered as a manifestation of some nonlocal (topological) interaction.A more detailed treatment of the problems related to the description of topological transitions, particularly the description of the space of states, the determination and study of the amplitude, as well as formulation of the validity conditions of the transformation, is outside the scope of the present paper, and will be provided elsewhere.