Showing papers in "Classical and Quantum Gravity in 1996"

The Gravitational Hamiltonian, action, entropy and surface terms

Abstract: We give a derivation of the gravitational Hamiltonian starting from the Einstein - Hilbert action, keeping track of all surface terms. This derivation can be applied to any spacetime that asymptotically approaches a static background solution. The surface term that arises in the Hamiltonian can be taken as the definition of the `total energy', even for spacetimes that are not asymptotically flat. (In the asymptotically flat case, it agrees with the usual ADM energy.) We also discuss the relation between the Euclidean action and the Hamiltonian when there are horizons of infinite area (e.g. acceleration horizons) as well as the usual finite area black hole horizons. Acceleration horizons seem to be more analogous to extreme than nonextreme black holes, since we find evidence that their horizon area is not related to the total entropy.

Semiclassicality and decoherence of cosmological perturbations

Abstract: Transition to the semiclassical behaviour and the decoherence process for inhomogeneous perturbations generated from the vacuum state during an inflationary stage in the early Universe are considered in both the Heisenberg and the Schrodinger representations to show explicitly that both approaches lead to the same prediction: the equivalence of these quantum perturbations to classical perturbations having stochastic Gaussian amplitudes and belonging to the quasi-isotropic mode. This equivalence and the decoherence are achieved once the exponentially small (in terms of the squeezing parameter ) decaying mode is neglected. In the quasi-classical limit , the perturbation mode functions can be made real by a time-independent phase rotation; this is shown to be equivalent to a fixed relation between squeezing angle and phase for all modes in the squeezed-state formalism. Though the present state of the gravitational wave background is not a squeezed quantum state in the strict sense and the squeezing parameters loose their direct meaning due to interaction with the environment and other processes, the standard predictions for the rms values of the perturbations generated during inflation are not affected by these mechanisms (at least, for scales of interest in cosmological applications). This stochastic background still occupies a small part of phase space.

Temperley - Lieb Recoupling Theory and Invariants of 3-Manifolds

Abstract: I used to keep the entire spin networks literature in a small folder on my shelf. The recent explosion of interest in the subject has made this impossible; more is probably now written every week than the contents of my folder in 1991. This brings the need to consolidate the subject and collect together results and formulae in one coherent place. Kauffman and Lins' book is a splendid contribution, as the well thumbed reference copy can testify. The renewed interest in the subject came from two areas. Firstly there was the q-revolution which spawned the representation theory of the q-deformation of SU(2) and its application to 3-manifold theory by combinatorial techniques. Secondly, spin networks came to be applied in quantum field theories in places where discrete methods began to seem more fundamental. There are many overlaps between these, and many of the tools developed for specific 3-manifold purposes have, in fact, a much wider application. Perhaps recognizing this, the authors made the first half of the book a review of the recoupling theory of SU(2), leaving the application to 3-manifolds strictly to the second part. The development of the recoupling theory makes no reference to SU(2) itself, as the invariant theory can be developed entirely by the means of diagrams. This is a continuation of the theme of the spin networks and strand networks developed by Penrose, extended to the q-deformed case. The use of diagrams was originally conceived as an aid to understanding algebraic formulae; now the connection runs deeper. We now understand that a planar diagram for the recoupling of representations of SU(2) is an indispensible part of the theory. Without it, the fine detail of the formulae become impossible to keep track of in a coherent way. Thus basing the theory on diagrams is probably the most useful and certainly the quickest way of grasping the subject. The book starts with the Jones - Wenzl projectors, which are the diagrammatic version of the projection onto irreducible representations of SU(2). Then it quickly moves on to calculating the value of q-spin networks with these irreducible representations on the edges. Particular networks are calculated in detail, namely the diagram and the tetrahedron. This then allows the properties of the 6j symbol to be developed. Having all the formulae in one place, and they have proved remarkably reliable, makes this book invaluable, as the literature is plagued by slightly different versions of the same formulae caused by differences in convention. One unfortunate departure from the conventions established in the physics literature is that the curly bracket symbol for the 6j symbol has been assigned a different normalization, which makes the formulae related to associativity simple but not symmetrical under permutations of the tetrahedron. Most of the recoupling theory is developed for general q, and the particular case of q as a root of unity. However, the reader interested only in classical spin networks need not be lost; one can put q = 1 everywhere and still enjoy reading the book. The calculations of q-spin network evaluations are long but tantalizingly similar to the formulae for the classical q = 1 case. General formulae are known only for the classical case. Chapter 8 describes the chromatic method for q = 1 due to Penrose, leaving the generalization to the q-deformation as an unsolved problem. The combinatorics of the chromatic method are somewhat delicate to describe, and the informal method of presentation, which serves the book well elsewhere, is stretched to the limit at this point. The second part of the book is much more specialized. There are reviews of the Turaev - Viro invariant, the Witten - Reshetikhin - Turaev invariant and the `shadow world' description of graph invariants due to Kirillov and Resheatikhin. All of these have been condensed and simplified from their original presentations, but some of the theory, for example, of the topology, has been omitted. The culmination of the book is a series of tables of the WRT invariants for some 3-manifolds, based on the formulae given in the first part of the book.

Quantization of point particles in (2+1)-dimensional gravity and spacetime discreteness

Abstract: By investigating the canonical commutation rules for gravitating quantized particles in a (2 + 1)-dimensional world, it is found that these particles live on a spacetime lattice. The spacetime lattice points can be characterized by three integers. Various representations are possible, the details depending on the topology chosen for energy - momentum space. We find that an topology yields a physically most interesting lattice within which first quantization of Dirac particles is possible. An topology also gives a lattice, but does not allow first quantized particles.

Gravitational waveforms from inspiralling compact binaries to second-post-Newtonian order

Abstract: The two independent 'plus' and 'cross' polarization waveforms associated with the gravitational waves emitted by inspiralling, non-spinning, compact binaries are presented, ready for use in the data analysis of signals received by future laser interferometer gravitational-wave detectors such as LIGO and VIRGO. The computation is based on a recently derived expression of the gravitational field at the second-post-Newtonian approximation of general relativity beyond the dominant (Newtonian) quadrupolar field. The use of these theoretical waveforms to make measurements of astrophysical parameters and to test the nature of relativistic gravity is discussed.

Making anti-de Sitter black holes

Abstract: It is known from the work of Banados et al that a spacetime with event horizons (much like the Schwarzschild black hole) can be obtained from (2 + 1)-dimensional anti-de Sitter space through a suitable identification of points. We point out that this can be done in 3 + 1 dimensions as well. In this way we obtain black holes with event horizons that are tori or Riemann surfaces of genus higher than one. They can have either one or two asymptotic regions (with non-standard topology). Locally, the spacetime is isometric to anti-de Sitter space.

Hyperbolic reductions for Einstein's equations

Abstract: We consider the problem of reducing initial value problems for Einstein's field equations to initial value problems for hyperbolic systems, a problem of importance for numerical as well as analytical investigations of gravitational fields. The main steps and the most important objectives in designing hyperbolic reductions are discussed. Various reductions which have already been studied in the literature or which can easily be derived from previous discussions of the field equations are pointed out and some of their specific features are indicated. We propose new reductions based on the use of the Bianchi equation for the conformal Weyl tensor. These reductions involve symmetric hyperbolic systems of propagation equations and allow a number of different gauge conditions. They use unknowns in a most economic way, supplying direct and non-redundant information about the geometry of the time slicing and the four-dimensional spacetime. Some of this information is directly related to concepts of gravitational radiation. All these reductions can be extended to include the conformal field equations. Those which are based on the ADM representation of the metric can be rewritten in flux conserving form.

LISA: laser interferometer space antenna for gravitational wave measurements

Abstract: LISA (laser interferometer space antenna) is designed to observe gravitational waves from violent events in the Universe in a frequency range from to which is totally inaccessible to ground-based experiments. It uses highly stabilized laser light (Nd:YAG, ) in a Michelson-type interferometer arrangement. A cluster of six spacecraft with two at each vertex of an equilateral triangle is placed in an Earth-like orbit at a distance of 1 AU from the Sun, and behind the Earth. Three subsets of four adjacent spacecraft each form an interferometer comprising a central station, consisting of two relatively adjacent spacecraft (200 km apart), and two spacecraft placed at a distance of from the centre to form arms which make an angle of with each other. Each spacecraft is equipped with a laser. A descoped LISA with only four spacecraft has undergone an ESA assessment study in the M3 cycle and the full six-spacecraft LISA mission has now been selected as a cornerstone mission in the ESA Horizon 2000-plus programme.

Global extensions of spacetimes describing asymptotic final states of black holes

Abstract: We consider a globally hyperbolic, stationary spacetime containing a black hole but no white hole. We assume, further, that the event horizon, , of the black hole is a Killing horizon with compact cross-sections. We prove that if surface gravity is non-zero and constant throughout the horizon one can globally extend such a spacetime so that the image of is a proper subset of a regular bifurcate Killing horizon in the enlarged spacetime. The necessary and sufficient conditions are given for the extendibility of matter fields to the enlarged spacetime. These conditions are automatically satisfied if the spacetime is static (and hence `t'-reflection symmetric) or stationary-axisymmetric with `t - ' reflection isometry and the matter fields respect the reflection isometry. In addition, we prove that a necessary and sufficient condition for the constancy of the surface gravity on a Killing horizon is that the exterior derivative of the twist of the horizon Killing field vanishes on the horizon. As a corollary of this, we recover a result of Carter that constancy of surface gravity holds for any black hole which is static or stationary-axisymmetric with the `t - ' reflection isometry. No use of Einstein's equation is made in obtaining any of the above results. Taken together, these results support the view that any spacetime representing the asymptotic final state of a black hole formed by gravitational collapse may be assumed to possess a bifurcate Killing horizon or a Killing horizon with vanishing surface gravity.

Ten Physical Applications of Spectral Zeta Functions

Abstract: The book starts with a nice historical introduction to the subject of zeta functions, zeta function regularization and its applications in physics. The author emphasizes some of the advantages of the zeta function regularization method. In particular, the zeta function regularization can conveniently be applied in a curved spacetime. Anybody who wants to get a quick overview of the state of the art in the zeta function regularization method is recommended to read this introduction. Apart from the introduction the main virtue of this book is to provide help to any physics student and/or researcher who encounters zeta functions in his/her work. For this purpose the book provides the reader first of all with many technical details about zeta functions and some of their most important properties. Crucial amongst these properties is the existence of the so-called zeta function regularization theorem. This theorem is discussed and explained in the first part of the book. At a more advanced level, the author emphasizes the importance of a non-trivial extension of the celebrated Chowla--Selberg formula. This extension was invented by the author and plays an important role in some of the physics applications discussed later in the book. The second and main part of the book is devoted to illustrate the zeta function regularization method by giving ten physical applications. These applications cover such a wide range of topics as the Casimir effect, Kaluza--Klein compactification, two- and three-dimensional quantum gravity, extended objects like strings and membranes, critical behaviour of a field theory at non-zero temperature and topological mass generation. It is needless to say that all these applications give an impressive overview of the universality and power of the zeta function regularization method. I like in particular the discussion of the Casimir effect which is done in quite some detail. Apart from the technical discussion the author also explains why the Casimir effect only received so much attention many years after its discovery. Finally, I should note that it is interesting that the book discusses an application to the theory of membranes and, more generally p-branes, which has received such renewed interest in the past year. In short, I can recommend this book to anybody who encounters zeta functions in research and/or education. Undoubtedly, this book will be of great help in taking away some of the confusion on this topic.

The covariant approach to LRS perfect fluid spacetime geometries

Abstract: The dynamics of perfect fluid spacetime geometries which exhibit local rotational symmetry (LRS) are reformulated in the language of a 1 + 3 `threading' decomposition of the spacetime manifold, where covariant fluid and curvature variables are used. This approach presents a neat alternative to the orthonormal frame formalism. The dynamical equations reduce to a set of differential relations between purely scalar quantities. The consistency conditions are worked out in a transparent way. We discuss their various subcases in detail and focus in particular on models with higher symmetries within the class of expanding spatially inhomogeneous LRS models, via a consideration of functional dependences between the dynamical variables.

Junctions and thin shells in general relativity using computer algebra: I. The Darmois-Israel formalism

Abstract: We present the GRjunction computer algebra program which allows the study of non-null boundary surfaces and thin shells in general relativity. Implementing the Darmois - Israel thin-shell formalism requires a careful selection of definitions and algorithms to ensure that results are generated in a straightforward way. We have used the package to correctly reproduce a wide variety of examples from the literature. In this paper GRjunction is used to perform two new calculations: joining two Kerr solutions with differing masses and angular momenta along a thin shell in the slow rotation limit, and the calculation of the stress - energy of a Curzon wormhole. The Curzon wormhole has the interesting property that shells located at radius R < 2m have regions which satisfy the weak energy condition.

The spherically symmetric collapse of a massless scalar field

Abstract: We report on a numerical study of the spherically symmetric collapse of a self-gravitating massless scalar field. Earlier results of Choptuik (1992, 1994) are confirmed. The field either disperses to infinity or collapses to a black hole, depending on the strength of the initial data. For evolutions where the strength is close to but below the strength required to form a black hole, we argue that there will be a region close to the axis where the scalar curvature and field energy density can reach arbitrarily large levels, and which is visible to distant observers.

The final fate of spherical inhomogeneous dust collapse

Abstract: We examine, in a general manner, the role played by the initial density and velocity distributions in the gravitational collapse of a spherically symmetric inhomogeneous dust cloud. Such a collapse is described by the Tolman - Bondi metric which has two free functions: the `mass function' and the `energy function', which are determined by the initial density and velocity profiles of the cloud. The collapse can end in either a black hole or a naked singularity, depending on the initial parameters characterizing these profiles. In the marginally bound case, we find that the collapse ends in a naked singularity if the leading non-vanishing derivative of the density at the centre is either the first one or the second one. If the first two derivatives are zero, and the third derivative non-zero, the singularity could either be naked or covered, depending on a quantity determined by the third derivative and the central density. If the first three derivatives are zero, the collapse ends in a black hole. In particular, the classic result of Oppenheimer and Snyder, that homogeneous dust collapse leads to a black hole, is recovered as a special case. Analogous results are found when the cloud is not marginally bound, and also for the case of a cloud starting from rest. A condition on the initial density profile is given for the singularity to be globally naked. We also show how the strength of the naked singularity depends on the density and velocity distribution. Our analysis generalizes and simplifies the earlier works of Christodoulou and Newman by dropping the assumption of evenness of density functions. It turns out that relaxing this assumption allows for a smooth transition from the naked singularity phase to the black hole phase, and also allows for the occurrence of strong curvature naked singularities.

Reality conditions inducing transforms for quantum gauge field theory and quantum gravity

Abstract: The algebraic form of the Hamiltonian or Hamiltonian constraint of various (field) theories simplifies considerably if one uses certain complex-valued phase space variables. We show, for a general theory, that if we prescribe first a canonical complexification and second representation of the canonical commutation relations in which the real connection is diagonal, then there is only one choice of a holomorphic representation which incorporates the correct reality conditions and keeps the Hamiltonian (constraint) algebraically simple. We derive a canonical algorithm to obtain this holomorphic representation and in particular explicitly compute it for quantum gravity in terms of a Wick rotation transform.

The gravitational Hamiltonian in the presence of non-orthogonal boundaries

Abstract: This paper generalizes earlier work on Hamiltonian boundary terms by omitting the requirement that the spacelike hypersurfaces intersect the timelike boundary orthogonally. The expressions for the action and Hamiltonian are calculated and the required subtraction of a background contribution is discussed. The new features of a Hamiltonian formulation with non-orthogonal boundaries are then illustrated in two examples.

Finite-time response of inertial and uniformly accelerated Unruh - DeWitt detectors

Abstract: We study the response of inertial and uniformly accelerated Unruh - DeWitt detectors in the Minkowski vacuum when they are coupled to the quantum field for a finite time interval. A finite-time detector will respond even on an inertial trajectory due to transient effects. Also, the response will depend on the manner in which the detector is switched on and off. We study the response for smooth as well as abrupt switching of the detector. The detectors are switched on and off with window functions whose width, T, determines the effective timescale for which the detector is coupled to the field. We work out in detail the response of inertial and uniformly accelerated detectors for Gaussian, exponential and rectangular window functions and also obtain a general formula for the response of these detectors when a window function is specified. The and limits are discussed in detail and several subtleties in the limiting procedure are clarified.

Classical and quantum gravity in dimensions: I. A unifying approach

Abstract: We provide a concise approach to generalized dilaton theories with and without torsion and coupling to Yang - Mills fields. Transformations on the space of fields are used to trivialize the field equations locally. In this way their solution becomes accessible within a few lines of calculation only. In this first of a series of papers we set the stage for a thorough global investigation of classical and quantum aspects of more or less all available 2D gravity - Yang - Mills models.

A causal order for spacetimes with Lorentzian metrics: proof of compactness of the space of causal curves

Abstract: We recast the tools of `global causal analysis' in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call , and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime which is free of causal cycles, one may define a causal curve simply as a compact connected subset of which is linearly ordered by . Our definitions all make sense for arbitrary metrics (and even for certain metrics which fail to be invertible in places). Using this feature, we prove for a general metric the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. Our results have application in a recent positive-energy theorem, and may also prove useful in the study of topology change. We have tried to make our treatment self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry.

Generalized functions and distributional curvature of cosmic strings

Abstract: A new method is presented for assigning distributional curvature, in an invariant manner, to a spacetime of low differentiability, using the techniques of Colombeau's `new generalized functions'. The method is applied to show that the scalar curvature density of a cone is equivalent to a delta function. The same is true under small enough perturbations.

The complete spectrum of the area from recoupling theory in loop quantum gravity

Abstract: We compute the complete spectrum of the area operator in the loop representation of quantum gravity, using recoupling theory. This result extends previous derivations, which did not include the `degenerate' sector, and agrees with the recently computed spectrum of the connection-representation area operator.

Classical and quantum gravity in 1 + 1 dimensions. II: The universal coverings

Abstract: A set of simple rules for constructing the maximal (e.g. analytic) extensions for any metric with a Killing field in an (effectively) two-dimensional spacetime is formulated. The application of these rules is extremely straightforward, as is demonstrated at various examples and illustrated with numerous figures. Despite the resulting simplicity we also comment on some subtleties concerning the concept of Penrose diagrams. Most noteworthy among these, maybe, is that (smooth) spacetimes which have both degenerate and non-degenerate (Killing) horizons do not allow for globally smooth Penrose diagrams. Physically speaking this obstruction corresponds to an infinite relative red/blueshift between observers moving across the two horizons. – The present work provides a further step in the classification of all global solutions of the general class of two-dimensional gravity-Yang-Mills systems introduced in Part I [1], comprising, e.g., all generalized (linear and nonlinear) dilaton theories. In Part I we constructed the local solutions, which were found to always have a Killing field; in this paper we provide all universal covering solutions (the simply connected maximally extended spacetimes). A subsequent Part III [2] will treat the diffeomorphism inequivalent solutions for all other spacetime topologies. Part II is kept entirely self-contained; a prior reading of Part I is not necessary.

Gravitational theory in Weyl integrable spacetime

Abstract: The spacetime manifold is extended to a Weyl integrable spacetime (WIST). It is shown that this structure is a natural way to geometrize a long-range scalar field in the theory of gravitation. Several configurations are described in WIST (vacuum, dust, electromagnetic and external scalar field distribution). The general mechanism for studying all these cases and a special description of the analysis of the perfect fluid case are given.

Models of regular Schwarzschild black holes satisfying weak energy conditions

Abstract: We prove the existence of regular Schwarzschild black holes satisfying the weak energy conditions everywhere by presenting two explicit models. One of these models is explicitly seen to be complete (and therefore regular) by giving a maximal extension across the horizons.

Inflationary models driven by adiabatic matter creation

Abstract: The flat inflationary dust universe with matter creation proposed by Prigogine and co-workers is generalized and its dynamical properties are re-examined. It is shown that the starting point of these models depends critically on a dimensionless parameter , closely related to the matter creation rate . For bigger or smaller than unity flat universes can emerge, respectively, either like a big-bang FRW singularity or as a Minkowski spacetime at . The case corresponds to a de Sitter-type solution, a fixed point in the phase diagram of the system, supported by the matter creation process. The curvature effects have also been investigated. The inflating de Sitter is a universal attractor for all expanding solutions regardless of the initial conditions as well as of the curvature parameter.

Solution-generating transformations and the string effective action

Abstract: We study exhaustively the solution-generating transformations (dualities) that occur in the context of the low-energy effective action of superstring theory. We first consider target-space duality (“T duality”) transformations in absence of vector fields. We find that for one isometry the full duality group is (SO ↑ (1,1)) 3 × D4, the discrete part (D4) being nonAbelian. We, then, include non-Abelian Yang–Mills fields and find the corresponding generalization of the T duality transformations. We study the � ′ corrections to these transformations and show that the T duality rules considerably simplify if the gauge group is embedded in the holonomy group. Next, in the case in which there are Abelian vector fields, we consider the duality group that includes the transformation introduced by Sen that rotates among themselves components of the metric, axion and vector field. Finally we list the duality symmetries of the Type II theories with one isometry.

Testing the equivalence principle: why and how?

Abstract: Part of the theoretical motivation for improving the present level of testing of the equivalence principle is reviewed. The general rationale for optimizing the choice of pairs of materials to be tested is presented. A simplified rationale is introduced based on a trichotomy of competing classes of theoretical models.

A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds

Abstract: We prove that a certain class of conformal data (open in the set of all conformal data) maps to solutions of the Einstein constraint equations with nonconstant mean curvature. The method of proof is constructive, and it should apply to larger classes of conformal data as well.

The geometry of quantum spin networks

Abstract: The discrete picture of geometry arising from the loop representation of quantum gravity can be extended by a quantum deformation of the observable algebra. Operators for area and volume are extended to this theory and, partly, diagonalized. The eigenstates are expressed in terms of q-deformed spin networks. The q-deformation breaks some of the degeneracy of the volume operator so that trivalent spin networks have non-zero volume. These computations are facilitated by use of a technique based on the recoupling theory of , which simplifies the construction of these and other operators through diffeomorphism invariant regularization procedures.

Chaos in static axisymmetric spacetimes: I. Vacuum case

Abstract: We study the motion of a test particle in static axisymmetric vacuum spacetimes and discuss two criteria for strong chaos to occur: (i) a local instability measured by the Weyl curvature, and (ii) a tangle of a homoclinic orbit, which is closely related to an unstable periodic orbit in general relativity. We analyse several static axisymmetric spacetimes and find that the first criterion is a sufficient condition for chaos, at least qualitatively. Although some test particles which do not satisfy the first criterion show chaotic behaviour in some spacetimes, these can be accounted for by the second criterion.