Showing papers in "Classical and Quantum Gravity in 1991"

What Is Observable in Classical and Quantum Gravity

Abstract: The problem of the identification of the observable quantities in quantum gravity (or in any diffeomorphism invariant quantum theory) is considered. The author recalls Einstein's 'hole argument' on the impossibility of a priori identifying spacetime points. He argues that only by explicitly taking into account the physical nature of the bodies that form the reference system and their gravitational interactions one can get well defined gauge-invariant (and 'local') observables and a definition of physical spacetime points. A model is considered in which general relativity is coupled to matter: the matter represents the physical reference system. The gauge-invariant physical observables of this theory are displayed.

Self-dual 2-forms and gravity

Abstract: A Palatini-type formulation of gravity coupled to matter and supergravity is given, in which the gravitational variables are a trio of self-dual 2-forms and an SL(2,C) connection. The action is polynomial in all the fields. This framework is shown to be the natural covariantization of Ashtekar's canonical formalism (1988), and is used to find the general vacuum solution of the four initial value constraints associated with spacetime diffeomorphisms in that formalism.

Exact scalar field cosmologies

Abstract: The authors consider exact cosmological solutions of the Einstein gravitational equations with a non-interacting combination of a classical scalar field and isotropic radiation as source. They show how a potential function for the scalar field can be found leading to desired volume behaviour of Robertson-Walker universes, and give a number of exact solutions of the coupled equations. These solutions in general do not obey the 'slow-rolling' approximation usually assumed in inflationary universe models.

Quantum reference systems

Abstract: The author argues that only by taking into account the quantum properties of the bodies that form the reference frames, physical quantum operators can be defined in quantum gravity. The theory of general relativity coupled to matter introduced in a companion paper is considered. Its formal canonical quantization yields two surprising results: the diffeomorphism constraint can be exactly solved; and the Hamiltonian constraint reduces, in the context of a well defined approximation, to a Schrodinger evolution equation. By using the solutions of the quantum constraints of vacuum general relativity recently obtained in the loop representation, and in the context of a 'realistic' local material reference system, he defines a quantum gravitational theory in which the constraints can be solved, the only remaining equation is a regularized Schrodinger equation which expresses the dynamics in the internal clocks, and a class of gauge-invariant physical observables is explicitly displayed.

Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: A Report

Abstract: The formulation of general relativity discovered by Ashtekar (1986, 1987) and the recent results obtained in non-perturbative quantum gravity using loop-space techniques are reviewed. The new formulation is based on the choice of a set of Lagrangian (and Hamiltonian) variables, instead of the spacetime metric. In terms of these new variables, the dynamical equations are remarkably simplified and a structural identity between general relativity and the Yang-Mills theories is revealed. The formalism has proven to be useful in numerous problems in gravitational physics. In quantum gravity, the new formalism has overcome long-standing difficulties and led to unexpected results. A nonperturbative approach to quantum theory has been constructed in terms of the Wilson loops of the Ashtekar connection. This approach, denoted as loop-space representation, has led to the complete solution of the quantum diffeomorphism constraint in terms of knot states, to the discovery of an infinite-dimensional class of solutions to the quantum gravitational dynamics, and to certain surprising indications on the existence of a discrete structure of spacetime around the Planck length. These results are presented in a compact self-contained form. The basic Ashtekar formalism is presented and its applications are outlined. The loop-space representation and the non-perturbative knot states of quantum gravity are described in detail, with particular regard to their physical interpretation and to the information they may provide on the microstructure of spacetime.

Causal Friedmann-Robertson-Walker cosmology

Abstract: The evolution of a Friedmann-Robertson-Walker universe filled with a viscous simple fluid is analysed. At variance with other treatments the authors' approach complies with relativistic causality since dissipative signals travelling at superluminal speeds are forbidden. This is because use is made of the extended thermodynamics theory of irreversible processes instead of the conventional one. As a consequence some novel results arise. In particular, the initial de Sitter phase of the deflationary universe does not occur. Likewise, the generalized second law of thermodynamics is studied within this context.

Topology change in classical and quantum gravity

Abstract: In a first-order formulation, the equations of general relativity remain well defined even in the limit that the metric becomes degenerate. It is shown that there exist smooth solutions to these equations on manifolds in which the topology of space changes. The metric becomes degenerate on a set of measure zero, but the curvature remains bounded. Thus if degenerate metrics play any role in quantum gravity, topology change is unavoidable.

A pure spin-connection formulation of gravity

Abstract: A new action principle, in which the only gravitational variables are an SL(2,C) connection and a scalar density, is derived for general relativity (GR) coupled to matter and for supergravity. In this form, GR appears as a non-metric, generally covariant gauge theory, the metric being reconstructed from the other fields in a solution. A similar non-metric action with a real SO(3,1) connection is also derived, however it involves an independent fourth rank tensor field representing the curvature.

Properties of equations of motion of interacting gauge fields of all spins in (3+1)-dimensions

Abstract: Properties of totally consistent equations of motion of interacting gauge fields of all spins s=0, 1/2, 1, . . ., infinity in 3+1 dimensions are discussed in some detail. The equations under consideration are explicitly general coordinate invariant, possess all necessary higher-spin gauge symmetries, reduce to the standard free massless equations in the linearized approximation and contain Einstein equations with the cosmological term in the spin-2 sector.

Lovelock-Cartan theory of gravity

Abstract: The most general theory of gravity with torsion in D dimensions is discussed The Lagrangian is required to be: (i) a D-form scalar under local Lorentz transformations; (ii) a local polynomial of the vielbein, the spin connection, and their exterior derivatives; (iii) constructed without the Hodge dual (*-operation) Besides the purely metric Lovelock action, there is a series of torsion terms related to the Pontryagin classes in a way analogous to the relation between the Lovelock action and the Euler classes Also, a family of global invariants of the differentiable structure of the manifold, constructed with the torsion, is identified Relaxing the condition of invariance of the Lagrangian under local Lorentz rotations, but still requiring the action to be invariant, another family of torsional actions that generalizes the Chern-Simons theories is found Systematic algorithms for the explicit construction of these actions are provided

Existence and properties of spherically symmetric static fluid bodies with a given equation of state

Abstract: It is shown that for a given equation of state and a given value of the central pressure there exists a unique global solution of the Einstein equations representing a spherically symmetric static fluid body. For the proof a new theorem on singular ordinary differential equations is established which is of interest in its own right. For a given equation of state and central pressure, the fluid will either fill the entire space or be finite in extent with a vacuum exterior. Criteria are given which allow one to decide for certain equations of state which of these two cases occurs. This generalizes well known results in Newtonian theory and is proved by showing that the relativistic model inherits the property of having a finite radius from a Newtonian model. Parameter-dependent families of relativistic solutions are constructed which have a Newtonian limit in a rigorous sense. The relationship between relativistic and Newtonian equations of state is examined by looking at the example of a degenerate Fermi gas.

Bianchi cosmologies: the role of spatial topology

Abstract: The role of spatial topology in the Hamiltonian description of Bianchi models is analysed. It turns out that, in general, the number of degrees of freedom of these models is not uniquely determined by the isometry group but depends in addition on the choice of topology. Consequently, the quantum theory is quite sensitive to this choice. Contrary to one's initial expectation, subtleties arise in the spatially open models-say with topology R3-rather than closed. Finally, it is shown that class B models cannot occur with closed topologies.

Quantum linearization instabilities of de Sitter spacetime. I

Abstract: It is known that all the physical states in linearized gravity are required to be invariant under the continuous isometries of the background spacetime if it is spatially compact. For example, all the physical states in linearized gravity in de Sitter spacetime are required to be SO(4, 1) invariant. A detailed derivation of SO(4, 1) invariance of the physical state is presented. Also, it is found that normal ordering of operators is unnecessary in the constraints which demand SO(4, 1) invariance. Then it is proved that there are no normalizable SO(4, 1)-invariant states other than the vacuum state in the Fock space. This appears to suggest that there would be no dynamics in de Sitter spacetime. A step towards a possible resolution of this paradox will be presented in the sequel to this article.

A DeWitt expansion of the heat kernel for manifolds with a boundary

Abstract: The DeWitt (DW) expansion of the heat kernel GDelta (x, x'; tau ), for a second order elliptic differential operator Delta is extended to manifolds with a boundary. The boundary conditions are satisfied by including an additional term to the usual DW contribution appropriate for manifolds without a boundary, which provides an asymptotic expansion as tau to 0 for x approximately=x'. The extra piece is based on geodesic paths linking x and x', which undergo reflection on the boundary, just as the DW expression is based on the direct geodesic from x to x'. The boundary contributions to Tr(e- tau Delta ), which are an expansion in tau 12/, are found in agreement with previous indirect methods, although the corresponding contributions to GDelta involve non-polynomial functions of tau . Conservation equations for vector and tensor fields obtained from the heat kernel, which include terms restricted to the boundary, are also verified. The results are applied to determine the leading singular behaviour of the Green function for Delta at x=x' in the neighbourhood of the boundary.

The gravitational field of a global monopole

Abstract: The authors present an exact solution to the non-linear equation which describes a global monopole in the flat space. They re-examine the metric and the geodesics outside the global monopole. They show that a global monopole produces a repulsive gravitational field outside the core in addition to a solid anglar deficit. The lensing property of the global monopole and the global monopole-antimonopole annihilation mechanism are studied.

(2,2) vacuum configurations for type IIA superstrings: N=2 supergravity Lagrangians and algebraic geometry

Abstract: The compactification of D=10 type IIA supergravity (the point-field theory limit of the type IIA superstring) on a Calabi-Yau manifold ((2.2) vacuum) is performed. The resulting D=4, N=2 effective supergravity Lagrangian gives rise to an explicit construction of special Kahler manifolds for the (1,1) moduli and dual quaternionic manifolds for the (2.1) moduli. In both cases the scalar manifolds are characterized by a homogeneous holomorphic function of degree two. In the case of the (1.1) moduli, the holomorphic function is given by a cubic polynomial, a result which is valid only in the classical large volume limit (compared to the string scale) of the internal manifold. In contrast, for the (2.1) moduli the holomorphic function, which is related to the 'periods' of a holomorphic three form, is unrestricted and due to non-renormalization theorems, should coincide with the exact (tree-level) string result.

Gravitation and cosmology in (1 + 1) dimensions

Abstract: Various dynamical aspects of a non-trivial theory of gravitation in (1+1) dimensions are investigated. It is shown that many features of Einstein's general relativity for (3+1) dimensions are qualitatively and to some extent quantitatively duplicated in this theory, including gravitational collapse, cosmological solutions and gravitational waves. Contact with Newton's theory is made and post-Newtonian expansions of the theory give relativistic corrections, similar to those of general relativity. Therefore this theory is presented as viable for both pedagogical purposes as well as a laboratory for the testing of new theoretical ideas.

Partial supersymmetry breaking in N=4 supersymmetric quantum mechanics

Abstract: The authors present a simple d=1 model of N=4 supersymmetric quantum mechanics (SQM) in which N=4 supersymmetry can be spontaneously broken down to N=2. When both SU(2) automorphism symmetries of N=4 1D supertranslation algebra are explicitly broken, this superalgebra acquires a constant central charge proportional to the product of two SU(2) breaking parameters. Due to this crucial property, no contradiction arises with Witten's theorem (1989) forbidding the partial spontaneous supersymmetry breaking in SQM models based on standard (having no central charges) 1D supertranslation algebras. For the partial breaking to really come about, it also proves necessary to consider the most general class of N=4, d=1 SQM Hamiltonians involving the terms quartic in fermion fields. They show that after performing a duality transformation these models can be interpreted as resulting from certain U(1) invariant N=2 2D Kahler sigma models via the Scherk-Schwartz-type dimensional reduction (1979).

Conformal symmetries of pp-waves

Abstract: Previous results on Killing and special conformal Killing vectors of pp-waves are generalized by finding the general solution of the conformal Killing equations, together with integrability conditions. The general homothetic and nonspecial conformal Killing vectors are determined. It is shown that nonflat conformally flat pp-waves always admit a G6 of motions and a G1 of proper homothetic motions, but do not, in general, admit special conformal motions. Examples are given of a nonEinstein-vacuum pp-wave with a proper special conformal Killing vector, and a non-conformality-flat pp-wave with a non-special conformal Killing vector. A conformally flat pp-wave, which may be interpreted as an Einstein-Maxwell or Einstein-Klein-Gordon solution, is given and its fifteen conformal Killing vector are explicitly determined.

The Mixmaster cosmology as a dynamical system

Abstract: The Bianchi IX Mixmaster cosmology is treated as a non-linear dynamical system and the entire spectrum of Lyapunov exponents is calculated from a numerical integration of the full Einstein equations. The behaviour of the solutions is studied for times close to the initial and final singularities as well as for times near the point of maximum expansion. It is demonstrated that the methods of numerical integration must be chosen carefully if a vacuum spacetime is to be simulated in a self-consistent manner. In addition it is shown that previous results pointing to chaotic behaviour are due to inaccurate numerical methods and/or the introduction of non-vacuum behaviour.

Asymptotic expansion of the heat kernel for generalized boundary conditions

Abstract: The asymptotic expansion of the heat kernel corresponding to e- tau Delta for a second-order symmetric elliptic differential operator Delta acting on vector fields over a manifold M with a boundary is extended to generalized Neumann boundary conditions. The normal derivative of the vector fields at any point on the boundary delta M is related to the vector field at the same point acted on by a linear operator Lambda which is symmetric with respect to the natural scalar product given by the induced measure on delta M. In this paper Lambda is allowed to be a first-order differential operator defined on vector fields restricted to delta M which is motivated by calculations with open strings. The authors use a method previously developed by them which extends the DeWitt asymptotic expansion to manifolds with a boundary by including geodesic paths undergoing reflection on the boundary. The first two terms in the boundary contributions to the asymptotic expansion are calculated and they involve a non-polynomial dependence on the coefficient of the derivative term in Lambda . The leading terms in the expansion of vector and tensor fields defined by the heat kernel are also obtained. The results are applied to determining the dependence of the functional determinant of Delta on conformal rescalings of the metric in two dimensions.

A cubic interaction of totally symmetric massless representations of the Lorentz group in arbitrary dimensions

Abstract: Analysing the closure conditions for the Poincare algebra with the help of the light-cone formalism, the authors present a complete description of all the interactions of totally symmetric massless representations of the Lorentz group in the cubic approximation on the mass shell in arbitrary dimensions.

The ground-state wavefunction of a radiation-dominated universe

Abstract: The authors propose a minisuperspace construction of the ground-state (no boundary) wavefunction for a radiation-dominated universe. As in the case of a free massless conformally invariant scalar field discussed by Hartle and Hawking (1983), the functional integrations over the gravitational and gauge fields can be performed independently. The gauge part of the wavefunction is semiclassically computed with the help of the (anti-)selfdual solutions of the Euclideanized SO(4)-symmetric Einstein-Yang-Mills (EYM) systems. Implications of their construction are discussed.

Observer fields in Weylian spacetime models

Abstract: Resting upon the Ehlers-Pirani-Schild axiomatic approach to spacetime theory, the author considers Weyl manifolds as reasonable spacetime models. He defines and discuss several concepts associated with a timelike vector field (=observer field=reference frame) on such a spacetime model. In particular, he derives propagation equations for rotation, expansion and shear of V, prove some kinematical vorticity theorems and present a new method how to measure length curvature operationally. As important mathematical tools, he uses the orthogonal decomposition of the Lie derivative and of the covariant derivative; the latter is the Weylian generalization of the Fermi-Walker derivative.

Inflation and singularity prevention in a model for extended-object-dominated cosmology

Abstract: The authors present a geometric procedure which generates a class of cosmological models of deflationary type, i.e. models in which a primordial phase of accelerated expansion evolves smoothly towards the final decelerating state of standard cosmology. They also discuss the possibility that the initial de Sitter geometry, obtained in the context of these models, may be physically interpreted as the consequences of an early phase in which the contribution of finite-size objects becomes dominant.

Homogeneous and isotropic closed cosmologies with a gauge sector

Abstract: The authors study in a systematic way the kinematics and dynamics of closed Friedmann-Robertson-Walker cosmologies in the presence of a gauge sector, with an arbitrary gauge group. For this purpose they complement the SO(4)-invariant ansatz for the metric with SO(4)-symmetric ansatze for the matter fields. The dynamics of the resulting Lagrangian system with a finite number of degrees of freedom is studied for the case of Einstein-Yang-Mills-Higgs theories with an SO(N) gauge group.

Three-dimensional N=4 extended supersymmetric quantum mechanics

Abstract: A description of three-dimensional N=4 extended supersymmetric quantum mechanics is proposed, based on the superfield construction of the action. The main feature of the approach is the unification of three-dimensional bosonic coordinate vector and fermionic spinor of O(3) in one irreducible representation of N=4 supersymmetry algebra.

SU(n)-Einstein-Yang-Mills fields with spherical symmetry

Abstract: For all possible actions of SU(2) on SU(n)-principal bundles over spacetime the corresponding reduced Einstein-Yang-Mills equations are derived. These actions are classified by sets of n integers with sum zero. Only the case where some of these integers have a difference of two leads to interesting equations that may have solutions generalizing the discrete sequence of regular solutions found by Bartnik and McKinnon (1988). For actions when no two integers differ by two the underlying spacetime metric is necessarily Reissner-Nordstrom.

Some exact solutions of the Dirac equation in gravitational fields

Abstract: Some exact solutions of the general covariant Dirac equation in gravitational fields, namely in constant curvature de Sitter spacetime and in Milne's Universe are obtained. The tetrads for both cases are constructed on the basis of global quasi-Cartesian coordinates that allow one to exclude coordinate effects connected with the rotation of the local frame under the transition of the triad from one spacetime point to another. Solutions in the form of a spherical bispinor wave with time amplitude modulation are demonstrated for both cases of gravitational fields and the possibility of analogical solutions in the form of cylindrical and plane spinor waves for the de Sitter spacetime is discussed. All the solutions are obtained by means of the algebraic method of separation of variables.

Gauge invariant perturbations in multi-component fluid cosmologies

Abstract: An alternative approach to the study of cosmological density perturbations in cosmology has been presented. Covariant and gauge invariant quantities were defined that characterize density inhomogeneities in an almost-uniform model universe and propagation equations were derived for these quantities in the case of a general perfect fluid. The author extends this work to the case of an imperfect fluid and derives generalizations of these equations. The author then applies the theory to a multi-component fluid, in particular to the case where the background is described by a spatially flat Friedmann-Lemaitre-Robertson-Walker universe model containing a mixture of non-interacting dust and radiation. Solutions are obtained for the comoving fractional density gradient, the energy flux and the relative velocity in the centre of mass frame which include new modes linking the density gradient to the vorticity.