Showing papers in "Classical and Quantum Gravity in 1994"

Twenty years of the Weyl anomaly

Abstract: In 1973 two Salam proteges (Derek Capper and the author) discovered that the conformal invariance under Weyl rescalings of the metric tensor displayed by classical massless-field systems in interaction with gravity no longer survives in the quantum theory. Since then these Weyl anomalies have found a variety of applications in black-hole physics, cosmology, string theory and statistical mechanics. We give a nostalgic review.

The warp drive: hyper-fast travel within general relativity.

Abstract: It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the \warp drive" of science ction. However, just as it happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here.

Black Hole Evaporation without Information Loss

Abstract: An approach to black hole quantization is proposed wherein it is assumed that quantum coherence is preserved. After giving our motivations for such a quantization procedure we formulate the background field approximation, in which particles are divided into `hard' particles and `soft' particles. The background spacetime metric depends both on the in-states and on the out-states. A consequence of our approach is that four-geometries describing gravitational collapse will show the same topological structure as flat Minkowski space. We present some model calculations and extensive discussions. In particular, we show, in the context of a toy model, that the S-matrix describing soft particles in the hard particle background of a collapsing star is unitary; nevertheless, part of the spectrum of particles is shown to be approximately thermal. We also conclude that there is an interesting topological (and signature) constraint on manifolds underlying conventional functional integrals.

Extended gravity theories and the Einstein--Hilbert action

Abstract: I discuss the relation between arbitrarily high-order theories of gravity and scalar--tensor gravity at the level of the field equations and the action. I show that (2n+4)-order gravity is dynamically equivalent to Brans--Dicke gravity with an interaction potential for the Brans--Dicke field and n further scalar fields. This scalar--tensor action is then conformally equivalent to the Einstein--Hilbert action with n+1 scalar fields. This clarifies the nature and extent of the conformal equivalence between extended gravity theories and general relativity with many scalar fields.

Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories

Abstract: We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the `timelessness' of the hypothetical, fundamental generally covariant quantum field theory with the `evidence' of the flow of time. By using the algebraic formulation of quantum theory, we propose a unifying perspective on these problems, based on the hypothesis that in a generally covariant quantum theory the physical time flow is not a universal property of the mechanical theory, but rather it is determined by the thermodynamical state of the system (`thermal time hypothesis'). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita--Takesaki theorem, which allows us to derive a time flow, namely a one-parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time flow, its classical limit, and we relate it to various characteristic theoretical facts, such as the Unruh temperature and the Hawking radiation. We point out the existence of a state-independent notion of `time', given by the canonical one-parameter subgroup of outer automorphisms provided by the co-cycle Radon--Nikodym theorem.

Actions for gravity, with generalizations: A Review

Abstract: The search for a theory of quantum gravity has for a long time been almost fruitless. A few years ago, however, Ashtekar found a reformulation of Hamiltonian gravity, which thereafter has given rise to a new promising quantization project: the canonical Dirac quantization of Einstein gravity in terms of Ahtekar's new variables. This project has already produced interesting results, although many important ingredients are still needed before we can say that the quantization has been successful. Related to the classical Ashtekar-Hamiltonian, there have been discoveries regarding new classical actions for gravity in (2+1) and (3+1) dimensions, and also generalizations of Einstein's theory of gravity. In the first type of generalization, one introduces infinitely many new parameters, similar to the conventional Einstein cosmological constant, into the theory. These generalizations are called `neighbours of Einstein's theory' or `cosmological constants generalizations', and the theory has the same number of degrees of freedom, per point in spacetime, as the conventional Einstein theory. The second type is a gauge group generalization of Ashtekar's Hamiltonian, and this theory has the correct number of degrees of freedom to function as a theory for a unification of gravity and Yang--Mills theory. In both types of generalizations, there are still important problems that are unresolved: e.g. the reality conditions, the metric-signature condition, the interpretation, etc. In this review, I will try to clarify the relations between the new and old actions for gravity, and also give a short introduction to the new generalizations. The new results/treatments in this review are: (1) a more detailed constraint analysis of the Hamiltonian formulation of the Hilbert--Palatini Lagrangian in (3+1) dimensions; (2) the canonical transformation relating the Ashtekar- and the ADM-Hamiltonian in (2+1) dimensions is given; (3) there is a discussion regarding the possibility of finding a higher-dimensional Ashtekar formulation. There are also two clarifying figures (at the beginning of sections 2 and 3, respectively) showing the relations between different action-formulations for Einstein gravity in (2+1) and (3+1) dimensions.

The timelessness of quantum gravity: I. The evidence from the classical theory

Abstract: The issue of time is addressed. It is argued that time as such does not exist but that instants, defined as complete relative configurations of the universe, do. It is shown how the classical mechanics (both non-relativistic and generally relativistic) of a complete universe can be expressed solely in terms of such relative configurations. Time is therefore a redundant concept, as are external inertial frames of reference (so that Machian ideas about the relativity of motion are fully implemented). Although time plays no role in kinematics, it can be recovered as an effective concept associated with any classical history of the universe. In the case of classical mechanics, this operationally defined time is identical to the astronomers' ephemeris time. In the case of general relativity it is shown how local proper time is a kind of local ephemeris time. It is argued that because general relativity is timeless in a deep and precise sense, the standard representation of the theory as a theory of curved spacetime disguises important aspects of its structure and that just these aspects may be the most important for the quantum form of the theory. This issue and the effective recovery of time from a genuinely timeless quantum theory are addressed in a following companion paper.

The Universality of vacuum Einstein equations with cosmological constant

Abstract: It is shown that for a wide class of analytic Lagrangians, which depend only on the scalar curvature of a metric and a connection, the application of the so called `Palatini formalism', i.e. treating the metric and the connection as independent variables, leads to `universal' equations. If the dimension n of spacetime is greater than two these universal equations are vacuum Einstein equations with cosmological constant for a generic Lagrangian and are suitably replaced by other universal equations at degenerate points. We show that degeneracy takes place in particular for conformally invariant Lagrangians and we prove that their solutions are conformally equivalent to solutions of Einstein's equations. For two-dimensional spacetimes we find instead that the universal equation is always the equation of constant scalar curvature; in this case the connection is a Weyl connection, containing the Levi-Civita connection of the metric and an additional vector field ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their degenerate points.

Non-linear charged black holes

Abstract: Very strong electromagnetic fields are characterized by the appearance of non-linearities. In order to explore the consequences of such non-linearities we have solved exactly the combined gravitational non-linear electromagnetic field equations for a static and spherical symmetric spacetime, where the source of the curvature is a point electric charge. We have chosen a non-linear Lagrangian density in such a way that the Maxwell, the Born--Infeld, and the Heisenberg--Euler theories are included as particular cases. The general solution can represent a charged black hole with the same characteristics of the Reissner--Nordstro m one (two horizons and timelike singularity), or it can have only one horizon and a spacelike singularity, as in the Schwarzschild solution.

Light propagation in arbitrary spacetimes and the gravitational lens approximation

Abstract: We reformulate the transport equation which determines the size, shape and orientation of infinitesimal light beams in arbitrary spacetimes. The behaviour of such light beams near vertices and conjugate points is investigated, with special attention to the singular behaviour of the optical scalars. We then specialize the general transport equation to the case of an approximate metric of an inhomogeneous universe, which is a Friedmann metric `on average' with superposed isolated weak matter inhomogeneities. In a series of well defined approximations, the equations of gravitational lens theory are derived. Finally, we derive a relative optical focusing equation which describes the focusing of light beams relative to the case that the beam is unaffected by matter inhomogeneities in the universe, from which it follows immediately that no beam can be focused less than one which is unaffected by matter clumps, before it propagates through its first conjugate point.

Hamiltonian formalism and gauge invariance for linear perturbations in inflation

Abstract: A Hamiltonian treatment of gauge-invariant cosmological perturbations is presented. In this framework the gauge-invariant perturbations as well as their evolution equations are obtained with minimal work, thanks to the use of a Hamilton--Jacobi technique which takes advantage of the gauge invariance of general relativity. This formalism is applied to perturbations around spatially flat and closed Friedmann--Robertson--Walker universes with the matter being given by a scalar field, both for the gauge-invariant `scalar' perturbations and gravitational waves.

The timelessness of quantum gravity: II. The appearance of dynamics in static configurations

Abstract: A strategy for quantization of general relativity is considered in the context of the `timelessness' of classical general relativity discussed in the preceding companion paper. The Wheeler--DeWitt equation (WDE) of canonical quantum gravity is interpreted as being like a time-independent Schrodinger equation for one fixed energy, the solution of which simply gives, once and for all, relative probabilities for each possible static relative configuration of the complete universe. Each such configuration is identified with a possible instant of experienced time. These instants are not embedded in any kind of external or internal time and, if experienced, exist in their own right. The central question is then: Whence comes the appearance of the passage of time, dynamics, and history? The answer proposed here is that these must all be `coded', in the form of what appear to be mutually consistent `records', in the individual static configurations of the universe that are actually experienced. Such configurations are called time capsules and suggest a new, many-instants, interpretation of quantum mechanics. Mott's explanation of why -particles make straight tracks in Wilson cloud chambers shows that the time-independent Schrodinger equation can concentrate its solution on time capsules. This demonstrates how the appearance of dynamics and history can arise in a static situation. If it can be shown that solutions of the Wheeler--DeWitt equation are spontaneously and generically concentrated on time capsules, this opens up the possibility of an explanation of time at a very deep level: the timeless wavefunction of the universe concentrates the quantum mechanical probability on static configurations that are time capsules, so that the situations which have the highest probability of being experienced carry within them the appearance of time and history. It is suggested that the inescapable asymmetry of the configuration space of the universe could play an important role in bringing about such concentration on time capsules and be the ultimate origin of the arrow of time.

On the topology of stationary black holes

Abstract: We prove that the domain of outer communication of a stationary, globally hyperbolic spacetime satisfying the null energy condition must be simply connected. Under suitable additional hypotheses, this implies, in particular, that each connected component of a cross section of the event horizon of a stationary black hole must have spherical topology.

Nöther's symmetries and exact solutions in flat non-minimally coupled cosmological models

Abstract: Adopting the method described in previous papers, we search for Nother's symmetries in point-like Friedman--Robertson--Walker (FRW) Lagrangians derived for general non-minimally coupled gravitational theories. We obtain exact solutions for flat models capable of producing inflation and recovering the Einstein regime at the present time (i.e. the scalar field and the coupling become constants directly related to the Newton constant ).

Unfolded representation for relativistic equations in 2 + 1 anti-de Sitter space

Abstract: We show how free equations of massless and massive fields in 2 + 1-dimensional anti-de Sitter space can be reformulated in a form of certain zero-curvature conditions This `unfolded formulation' makes it trivial to construct explicit solutions of field equations as well as evaluation of relevant Casimir operators of the anti-de Sitter group, in a coordinate-independent way The three-dimensional theories under consideration exemplify in the simplest fashion general features of the unfolded formulations of relativistic equations developed previously for massless higher-spin theories in 3 + 1 dimensions, which are expected to have a wide area of applicability

Perfect Fluid Sources in 2+1 Dimensions

Abstract: We show that a class of metrics of Einstein theory with perfect fluid sources are also the solutions of the Deser--Jackiw--Templeton (DJT) theory with a cosmological constant. Because of this analogy we interpret a recently found black hole solution of the DJT equations as a rotating fluid solution of the Einstein theory in three dimensions.

The physical basis for infra-red divergences in inflationary quantum gravity

Abstract: We consider tree-order graviton scattering amplitudes on a -dimensional de Sitter background in conformally flat coordinates. Infra-red divergences which cannot be absorbed using conventional techniques are shown to arise because conformal factors from the vertices are not compensated either by propagators or by external wavefunctions. The physical problem at tree order is therefore late interaction times rather than small spatial coordinate momenta, despite a mathematical problem at small momenta in the naive mode expansion of the propagator. Even in loops, concern over small spatial momenta is physically irrelevant because the chaotic conditions likely to prevail after the Big Bang could not have resulted in the simultaneous onset of inflation over a patch extending much beyond the Hubble radius. This motivates our proposal for a propagator which can be used to compute expectation values well inside the de Sitter patch of a plausible initial state.

Analytic example of critical behaviour in scalar field collapse

Abstract: An exact one parameter () family of solutions representing scalar field collapse is shown to exhibit a type of critical behaviour which has been discussed by Choptuik The three possible evolutions are outlined For supercritical evolution (when black holes form) it is shown that a quantity related to the mass of the black hole exhibits a power law dependence on , for near critical evolution The solution supports the conjecture that black hole formation initially occurs at infinitesimal mass

Conical singularities and torsion

Abstract: Various metrics with different kinds of conical singularities, including one which does not seem to have appeared in the literature previously, are obtained by identification of flat space. A consideration of holonomy, both linear and affine in the terminology of Kobayashi and Nomizu, suggests an intepretation of the metrics in terms of torsion.

Geodesic structure of the (2 + 1)-dimensional BTZ black hole

Abstract: Null and time-like geodesics around a 2+1 black hole are determined. Complete geodesics of both types exist in the rotating black-hole background, but not in the spinless case. Upper and lower bounds for the radial size of the orbits are given in all cases and the possibility of passing from one black-hole exterior spacetime to another is discussed using Penrose diagrams. An analysis of particle motions by means of effective potentials and orbit graphs is also included.

Spin coefficient form of the new laws of black hole dynamics

Abstract: General laws of black hole dynamics, some of which are analogous to the laws of thermodynamics, have recently been found for a general definition of a black hole in terms of a future outer trapping horizon, a hypersurface foliated by marginal surfaces of a certain type. This theory is translated here into spin coefficient language. Most of the following results assume an energy condition. Second law: the area form of a future outer trapping horizon is generically increasing, otherwise constant. First law: the rate of change of the area form is given by an energy flux and the trapping gravity. Zeroth law: the total trapping gravity of a compact outer marginal surface has an upper bound, attained if and only if the trapping gravity is constant. Topology law: a compact future outer marginal surface has spherical topology. Signature law: an outer trapping horizon is generically spatial, otherwise null. Trapping law: spatial surfaces sufficiently close to a compact future outer marginal surface are trapped if they lie inside the trapping horizon. Confinement law: if the interior and exterior of a future outer trapping horizon are disjoint, an observer inside the horizon cannot get outside.

A formulation of the virial theorem in general relativity

Abstract: A relativistic generalization of the classical virial theorem is obtained for any stationary and asymptotically flat spacetime. This formulation is derived within the 3+1 formalism of general relativity. It may be useful as a consistency check of numerical solutions of the Einstein equations.

Distributional energy--momentum tensor of the Kerr--Newman spacetime family

Abstract: Using the Kerr--Schild decomposition of the metric tensor that employs the algebraically special nature of the Kerr--Newman spacetime family, we calculate the energy--momentum tensor. The latter turns out to be a well defined tensor distribution with disc-like support.

Electric and magnetic Weyl tensors: classification and analysis

Abstract: A Weyl tensor of Petrov type I can be decomposed into two parts, an electric and a magnetic part, by any observer with 4-velocity vector u. It is shown here that when a metric is such that there exists an observer who sees the metric's Weyl tensor as purely electric or purely magnetic, then the Weyl tensor is of Petrov type I in the Arianrhod--McIntosh classification (and thus its four principal null directions are linearly dependent). It is also shown that an observer exists for whom the Weyl tensor is either purely electric or magnetic if and only if the Weyl tensor is of Petrov type I and the invariant I of the Weyl tensor is real. The magnetic and electric cases are distinguished by the sign of I. In the electric and magnetic cases, the spanning vectors of the principal null directions at each point are u and two other vectors picked out by the geometry; this combines and simplifies results of Trumper and Narain. The results here are formulated in terms of invariants, and are thus easily amenable to computer classification of metrics. Spacetime examples are discussed, and new theoretical results for the Petrov type D subcase are presented.

Global structure of a black hole cosmos and its extremes

Abstract: We analyze the global structure of a family of Einstein-Maxwell solutions parametrized by mass, charge and cosmological constant. In a qualitative classification there are: (i) generic black-hole solutions, describing a Wheeler wormhole in a closed cosmos of spatial topology ; (ii) generic naked-singularity solutions, describing a pair of `point' charges in a closed cosmos; (iii) extreme black-hole solutions, describing a pair of `horned' particles in an otherwise closed cosmos; (iv) extreme naked-singularity solutions, in which a pair of point charges forms and then evaporates, in a way which is not even weakly censored; and (v) an ultra-extreme solution. We discuss the properties of the solutions and of various coordinate systems, and compare with the Kastor-Traschen multi-black-hole solutions.

First order Regge calculus

Abstract: A first order form of Regge calculus is defined in the spirit of Palatini's action for general relativity. The extra independent variables are the interior dihedral angles of a simplex, with conjugate variables the areas of the triangles. There is a discussion of the extent to which these areas can be used to parametrize the space of edge lengths of a simplex.

Spherically symmetric anisotropic fluid ICKV spacetimes

Abstract: Spherically symmetric spacetimes representing an anisotropic fluid and admitting a proper inheriting conformal Killing vector (ICKV) are studied and all such spacetimes are found. It is shown that the ICKV all lie in the (t,r)-plane except in the case of those solutions that are conformally flat, in which case there exist three angular-dependent ICKV. The physical properties of these solutions are investigated (with particular attention focused on static spacetimes), and, in the context of these solutions, examples are given which can be interpreted as stellar models (with reasonable physical properties), as models of magnetic fields in a plasma, and as models of viscous heat-conducting fluids. It is shown that global monopole solutions cannot admit a proper ICKV.

Particle-like solutions to topologically massive gravity

Abstract: The solution of topologically massive gravity with cosmological constant is reduced, for spacetimes with two commuting Killing vectors, to a special-relativistic dynamical problem. This approach is applied to the construction of a class of exact sourceless, horizonless solutions asymptotic to the BTZ extreme black holes.

Semiclassical limits of simplicial quantum gravity

Abstract: We consider the simplicial state sum model of Ponzano and Regge as a path integral for quantum gravity in three dimensions. We examine the Lorentzian geometry of a single 3-simplex and of a simplicial manifold, and interpret an asymptotic formula for 6j-symbols in terms of this geometry. This extends Ponzano and Regge's similar interpretation for Euclidian geometry. We give a geometric interpretation of the stationary points of this state sum, by showing that, at these points, the simplicial manifold may be mapped locally into flat Lorentzian or Euclidian space. This lends weight to the interpretation of the state sum as a path integral, which has solutions corresponding to both Lorentzian and Euclidian gravity in three dimensions.

The cosmic no-hair theorem and the non-linear stability of homogeneous Newtonian cosmological models

Abstract: The validity of the cosmic no-hair theorem is investigated in the context of Newtonian cosmology with a perfect fluid matter model and a positive cosmological constant. It is shown that if the initial data, for an expanding cosmological model of this type, is subjected to a small perturbation then the corresponding solution exists globally in the future and the perturbation decays in a way which can be described precisely. It is emphasized that no linearization of the equations or special symmetry assumptions are needed. The result can also be interpreted as a proof of the non-linear stability of the homogeneous models. In order to prove the theorem we write the general solution as the sum of a homogeneous background and a perturbation. As a by-product of the analysis it is found that there is an invariant sense in which an inhomogeneous model can be regarded as a perturbation of a unique homogeneous model. A method is given for associating uniquely to each Newtonian cosmological model with compact spatial sections a spatially homogeneous model which incorporates its large-scale dynamics. This procedure appears very naturally in the Newton--Cartan theory which we take as the starting point for Newtonian cosmology.