Quantum cosmology in Ashtekar variables with non-minimally coupled scalar-tensor theory

Using non-minimally coupled scalar-tensor theory in homogeneous and isotropic cosmological model, quantum cosmology has been developed for Ashtekar variables. The wave function has been evaluated by solving the Wheeler-Dewitt (WD) equation and also using path integral formulation. Semi-classical limit using WKB approximation has also been discussed. Finally, the quantum Bohmian trajectories has been studied in detail.     

Canonical forms for axial symmetric space-times

The general canonical forms for axial symmetric space-times are investigated. Special forms such as static, stationary and cylindrical are considered. Our results are based only upon the symmetries assumed, not upon the field equations; thus they are applicable to vacuum, electromagnetic and matter field problems. In particular, we find a term that was missing in previous work. There can be in the general canonical form a metric coefficient between the axis of symmetry and the angle about this axis. This metric coefficient survives even for static or stationary space-times.     

Splitting theorems for spatially closed space-times

A Lorentzian splitting theorem is obtained for spatially closed spacetimes. The proof employs and extends some recent results of Bartnik and Gerhardt concerning the existence and rigid uniqueness of compact maximal hypersurfaces in spatially closed space-times. A splitting theorem for spatially closedtime-periodic space-times, which generalizes a result first considered by Avez, is derived as a corollary.     

A Case for Lorentzian Relativity

The Lorentz transformation (LT) is explained by changes occurring in the wave characteristics of matter as it changes inertial frame. This explanation is akin to that favoured by Lorentz, but informed by later insights, due primarily to de Broglie, regarding the underlying unity of matter and radiation. To show the nature of these changes, a massive particle is modelled as a standing wave in three dimensions. As the particle moves, the standing wave becomes a travelling wave having two factors. One is a carrier wave displaying the dilated frequency and contracted ellipsoidal form described by the LT, while the other (identified as the de Broglie wave) is a modulation defining the dephasing of the carrier wave (and thus the failure of simultaneity) in the direction of travel. The superluminality of the de Broglie wave is thus explained, as are several other mysterious features of the optical behaviour of matter, including the physical meaning of the Schrödinger equation and the relevance to scattering processes of the de Broglie wave vector. Consideration is given to what this Lorentzian approach to relativity might mean for the possible existence of a preferred frame and the origin of the observed Minkowski metric.     

Irreducible mass for the Tomimatsu-Sato space-times

A global definition of irreducible mass for the odd- T-S metrics is investigated. We find that its expression in terms of the source parameters is the same for all the members of the family and reduces to the formula that holds in the Kerr case (=1). As a consequence, we show that processes withm _ir = const no longer imply zero variations of the horizon's area for >1.     

Curvature collineations for type-N Robinson-Trautman space-times

The metric of type-N Robinson-Trautman space-times is generated by a real functionP satisfying certain field equations. Canonical forms forP are obtained under the assumption that at least one curvature collineation exists. In order to give an example of the improper subgroup structure of a group of curvature collineations all the curvature collineations are determined for the space-times corresponding to one of the two canonical forms.     

Field equations for spatially homogeneous space-times

It is shown that there is a natural tetrad for each Bianchi type cosmological model in which the metric and field equations take a simple and convenient form.     

On the problem of reality conditions for the Ashtekar formulation

In this paper we discuss the reality conditions for Lorentzian and Euclidean gravity in the Ashtekar formulation by introducing a double conformal transformation.We generalize Marugan‘s results and demonstrate that the values of the double conformal factor have to be either real or double complex numbers.Either Lorentzian or Enclidean gravitational theory is up to the different values of the double conformal factor.Furthermore,the reality conditions of Lorentzian and Euclidean gravitational theory can be expressed in a unified way be use of the double complex function method.     

Stability of geodesic incompleteness for Robertson-Walker space-times

Let (M, g) be a Lorentzian warped product space-timeM=(a, b)×H, g = –dt ² fh, where –a<b+, (H, h) is a Riemannian manifold andf: (a, b)(0, ) is a smooth function. We show that ifa>– and (H, h) is homogeneous, then the past incompleteness of every timelike geodesic of (M,g) is stable under smallC ⁰ perturbations in the space Lor(M) of Lorentzian metrics forM. Also we show that if (H,h) is isotropic and (M,g) contains a past-inextendible, past-incomplete null geodesic, then the past incompleteness of all null geodesics is stable under smallC ¹ perturbations in Lor(M). Given either the isotropy or homogeneity of the Riemannian factor, the background space-time (M,g) is globally hyperbolic. The results of this paper, in particular, answer a question raised by D. Lerner for big bang Robertson-Walker cosmological models affirmatively.Partially supported by a grant from the Research Council of the Graduate School of the University of Missouri-Columbia.Partially supported by a grant from the Research Council of the Graduate School of the University of Missouri-Columbia and NSF grant No. MCS77-18723(02).     

A Newman-Penrose-type formalism for space-times with torsion

The familiar Newman-Penrose formalism, in which the curvature of space-time is represented in terms of spin coefficients, is here extended to include the possibility of an asymmetric connection. It is hoped that this approach will be useful in dealing with certain problems in Einstein-Cartan theory, and also in other theories of gravitation that include torsion.     

A new bundle completion for parallelizable space-times

The Schmidt [9]b-boundaryM, for completing a space-timeM, has several desirable features. It is uniquely determined by the space-time metric in an elegant geometrical manner. The completed space-time is¯M=M M, where¯M= ⁺ M/O ⁺ and ⁺ M is the Cauchy completion (with respect to a toplogical metric induced by the Levi-Cività connection) of a component of the orthonormal frame bundle having structure groupO ⁺. ThenM consists of the endpoints of incomplete curves inM that have finite horizontal lifts in ⁺ M, and ifM= we say thatM isb-complete. It turns out thatM isb-complete if and only ifO ⁺ M is complete. This criterion for space-time completeness is stronger than geodesic completeness and Beem [1] has shown that this remains so even for the restricted class of globally hyperbolic space-times. Clarice [3] has shown that for such space-times the curvature becomes unbounded as theb-boundary is approached.Now ifM, then ⁺ M may contain degenerate fibers; thus the quotient topology for¯M is non-Hausdorff and precludes a manifold structure. Precisely this has been demonstrated by Bosshard [2] for Friedmann space-time, casting doubt on the physical significance of the completion. The only neighborhood of the Friedmann singularity is the whole of¯M, and in the closed model initial and final singularities are identified inM. Similarly, Johnson [7] showed that the completion of Schwarzschild space-time is non-Hausdorff because of degenerate ibers in¯O ⁺ M.Here we introduce a modification of the Schmidt procedure that appears to be useful in avoiding fiber degeneracy and in promoting a Hausdorff completion. The modification is to introduce an explicit vertical component into the metric forO ⁺ M by reference to a standard section, that is, to a parallelizationpMO ⁺ M We prove some general properties of thisp-completion and examine the particular case of a Friedmann space-time where there is a fairly natural choice of parallelization.