Table of contents

Classical and Quantum Gravity(CQG) welcomes articles on experimental gravitation. This includes articles on instrumentation that is mostly of interest for gravitational experiments, as long as the relationship to gravitation is clearly explained in the paper. Experimental authors should also take into account that the readership of CQG is broad, consisting of theorists and experimentalists. It is therefore beneficial to include in the introduction a summary that places the findings in a meaningful context for such a readership.

The Editorial Board

Asymptotically anti-de Sitter spacetimes are considered in a general dimension d 4. As one might expect, the boundary conditions at infinity ensure that the asymptotic symmetry group is the anti-de Sitter group (although there is an interesting subtlety if d= 4). Asymptotic field equations imply that, associated with each generator of this group, there is a quantity Qwhich satisfies the expected `balance equation' if there is a flux of physical matter fields across the boundary at infinity and is absolutely conserved in the absence of this flux. Irrespective of the dimension d , all of these quantities vanish if the spacetime under considerations is (globally) anti-de Sitter. Furthermore, this result is required by a general covariance argument. However, it contradicts some of the recent findings based on the conjectured ADS/CFT duality. This and other features of our analysis suggest that, if a consistent dictionary between gravity and conformal field theories does exist in fully non-perturbative regimes, it would have to be more subtle than the one used currently.

If the vacuum is passive for uniformly accelerated observers in anti-de Sitter spacetime (i.e. cannot be used by them to operate a perpetuum mobile ), they will (a) register a universal value of the Hawking-Unruh temperature, (b) discover a TCP symmetry and (c) find that observables in complementary wedge-shaped regions are commensurable (local) in the vacuum state. These results are model independent and hold in any theory which is compatible with some weak notion of spacetime localization.

It is shown that any anisotropic and inhomogeneous cosmological solution to the lowest-order, four-dimensional, dilaton-graviton string equations of motion may be employed as a seed to derive a curved, 3-brane cosmological solution to five-dimensional heterotic M-theory compactified on a Calabi-Yau 3-fold. This correspondence formally relates a weakly coupled string cosmology directly with a strongly coupled one. The asymptotic behaviour of a wide class of spatially homogeneous braneworlds is deduced. Similar solutions may be derived in toroidally compactified massive type IIA supergravity.

A set of boundary conditions defining an undistorted, non-rotating isolated horizonare specified in general relativity. A spacetime representing a black hole which is itself in equilibrium but whose exterior contains radiation admits such a horizon. However, the definition is applicable in a more general context, such as cosmological horizons. Physically motivated, (quasi-)local definitions of the mass and surface gravity of an isolated horizon are introduced and their properties analysed. Although their definitions do not refer to infinity, these quantities assume their standard values in the static black-hole solutions. Finally, using these definitions, the zeroth and first laws of black-hole mechanics are established for isolated horizons.

We give a brief discussion on the limitations involving the expression µ = /8 (G= c= 1), which relates the string linear energy density µ to the conical deficit angle . Then, we establish a new equation between the angular deficit and a combination of the components of the string stress-energy tensor which shows that the angular deficit is determined not only by the amount of proper matter of the string but also, in a Newtonian sense, by its internal gravitational field.

We describe a Riemannian space class where the second Dirac operator arises and prove that the operator is always equivalent to a standard Dirac one. The particle state in this gravitational field is degenerate to some extent and we introduce an additional value in order to describe a particle state completely. Some supersymmetry constructions are also discussed. As an example we study all Riemannian spaces with a five-dimensional motion group and find all metrics for which the second Dirac operator exists. On the basis of our discussed examples we hypothesize about the number of second Dirac operators in Riemannian space.

We present a new point of view on the quantization of the gravitational field, namely we use exclusively the quantum framework of the second quantization. More explicitly, we take as one-particle Hilbert space, the unitary irreducible representation of the Poincarégroup corresponding to a massless particle of helicity 2 and apply the second quantization procedure with Einstein-Bose statistics. The resulting Hilbert space ⁺ ( ) is, by definition, the Hilbert space of the gravitational field. Then we prove that this Hilbert space is canonically isomorphic to a space of the type Ker(Q ) / Im(Q ) where Qis a supercharge defined in an extension of the Hilbert space ⁺ ( ) by the inclusion of ghosts: some fermion ghosts u_µ , _µwhich are vector fields and a bosonic ghost which is a scalar field. This has to be contrasted with the usual approaches where only the fermion ghosts are considered. However, a rigorous proof that this is, indeed, possible seems to be lacking in the literature.

Cylindrically symmetric static manifolds are classified according to their homotheties and metrics. In each case the homothety vector fields and the corresponding metrics are obtained explicitly by solving the homothety equations. It turns out that these metrics admit homothety groups H_m , where m= 4,5,7,11. This classification is then used to identify the cylindrically symmetric static spaces admitting the local homotheties, which are globally prohibited due to their topological construction. Einstein's field equations are then used to identify the physical nature of the spaces thus obtained.

It is proven that the Wahlquist perfect fluid spacetime cannot be smoothly joined to an exterior asymptotically flat vacuum region. The proof uses a power-series expansion in the angular velocity, to a precision of the second order. In this approximation, the Wahlquist metric is a special case of the rotating Whittaker spacetime. The exterior vacuum domain is treated in a like manner. We compute the conditions of matching at the possible boundary surface in both the interior and the vacuum domain. The conditions for matching the induced metrics and the extrinsic curvatures are mutually contradictory.

Conserved quantities are obtained and analysed using new recently proposed models with global scale invariance. Such models allow for non-trivial scalar field potentials and masses for particles, so that the scale symmetry must somehow be broken. We obtain this conclusion by showing that the infrared behaviour of the conserved currents is singular so that there are no conserved charges associated with the global scale symmetry. Nevertheless, the scale symmetry plays a crucial role in determining the structure of the theory and it implies that in some high-field regions the potentials become flat.

We discuss the issue of radiation extraction in asymptotically flat spacetimes within the framework of conformal methods for numerical relativity. Our aim is to show that there exists a well defined and accurate extraction procedure which mimics the physical measurement process. It operates entirely intrinsically within ⁺so that there is no further approximation necessary apart from the basic assumption that the arena should be an asymptotically flat spacetime. We define the notion of a detector at infinity by idealizing local observers in Minkowski space. A detailed discussion is presented for Maxwell fields and the generalization to linearized and full gravity is performed by way of the similar structure of the asymptotic fields.

Non-singularity conditions are established for the Batalin-Fradkin-Vilkovisky (BFV) gauge-fixing fermion which are sufficient for it to lead to the correct path integral for a theory with constraints canonically quantized in the BFV approach. The conditions ensure that the anticommutator of this fermion with the BRST charge regularizes the path integral by regularizing the trace over non-physical states in each ghost sector. The results are applied to the quantization of a system which has a Gribov problem, using a non-standard form of the gauge-fixing fermion.

We study the thermodynamics of four-dimensional Kerr-Newman-AdS black holes both in the canonical and the grand-canonical ensemble. The stability conditions are investigated, and the complete phase diagrams are obtained, which include the Hawking-Page phase transition in the grand-canonical ensemble. In the canonical case, one has a first-order transition between small and large black holes, which disappears for a sufficiently large electric charge or angular momentum. This disappearance corresponds to a critical point in the phase diagram. Via the AdS/CFT conjecture, the obtained phase structure is also relevant for the corresponding conformal field theory living in a rotating Einstein universe, in the presence of a global background U (1) current.

An interesting limit arises when the black holes preserve some supersymmetry. These BPS black holes correspond to highly degenerate zero-temperature states in the dual CFT, which lives in an Einstein universe rotating with the speed of light.

Methods of dynamical systems analysis are used to show rigorously that the presence of a magnetic field orthogonal to the two commuting Killing vector fields in any spatially homogeneous Bianchi type VI₀vacuum solution to Einstein's equation changes the evolution toward the singularity from convergent to oscillatory. In particular, it is shown that the -limit set (for the time direction that puts the singularity in the past) of any of these magnetic solutions contains at least two sequential Kasner points of the Belinskii-Khalatnikov-Lifshitz sequence and the orbit of the transition solution between them. One of the Kasner points in the -limit set is non-flat, which leads to the result that each of these magnetic solutions has a curvature singularity.

The initial data for black hole collisions are constructed using a conformal-imaging approach and a new adaptive mesh refinement technique, a fully threaded tree (FTT). We developed a second-order accurate approach to the solution of the constraint equations on a non-uniformly refined high-resolution Cartesian mesh including second-order accurate treatment of boundary conditions at the black hole throats. FTT-based mesh refinement reduced memory and computer time requirements by several orders of magnitude compared to a uniform grid and allowed the construction of accurate initial data sets on Cartesian meshes. Test computations show quadratic convergence of the solutions with increasing numerical resolution.

A complete qualitative study of the dynamics of string cosmologies is presented for the class of isotopic curvature universes. These models are of Bianchi types I, V and IX and reduce to the general class of Friedmann-Robertson-Walker universes in the limit of vanishing shear isotropy. A non-trivial 2-form potential and cosmological constant terms are included in the system. In general, the 2-form potential and spatial curvature terms are only dynamically important at intermediate stages of the evolution. In many of the models, the cosmological constant is important asymptotically and anisotropy becomes dynamically negligible. There also exist bouncing cosmologies.

We present unique solutions of the Seiberg-Witten monopole equations in which the U (1) curvature is covariantly constant, the monopole Weyl spinor consists of a single constant component and the 4-manifold is a product of two Riemann surfaces of genuses p₁and p₂ . There are p₁ -1 magnetic vortices on one surface and p₂ -1 electric ones on the other, with p₁ +p₂ 2 (p₁= p₂= 1 being excluded). When p₁= p₂ , the electromagnetic fields are self-dual and one also has a solution of the coupled Euclidean Einstein-Maxwell-Dirac equations, with the monopole condensate serving as a cosmological constant. The metric is decomposable and the electromagnetic fields are covariantly constant as in the Bertotti-Robinson solution. The Einstein metric can also be derived from a Kähler potential satisfying the Monge-Ampère equations.

We consider the class of metrics that can be obtained from those of non-extreme black holes by limiting transitions to the extreme state such that the near-horizon geometry expands into a whole manifold. These metrics include, in particular, the Rindler and Bertotti-Robinson spacetimes. The general formula for the entropy of massless radiation valid either for black hole or for acceleration horizons is derived. It is argued that, as a black hole horizon in the limit under consideration turns into an acceleration one, the thermodynamic entropy S_qof quantum radiation is due to the Unruh effect entirely and S_q= 0 exactly. The contribution to the quasilocal energy from a given curved spacetime is equal to zero and the only non-vanishing term stems from a reference metric. In the variation procedure necessary for the derivation of the general first law, the metric on a horizon surface changes along with the boundary one, and the account for gravitational and matter stresses is an essential ingredient of the first law. This law confirms the property S_q= 0. The quantum-corrected geometry of the Bertotti-Robinson spacetime is found and it is argued that backreaction of quantum fields mimics the effect of the cosmological constant _effand can drastically change the character of spacetime depending on the sign of _eff- for instance, turn AdS₂× S₂into dS₂× S₂or Rindler₂× S₂ . The latter two solutions can be thought of as the quantum versions of the cold and ultracold limits of the Reissner-Nordstrom-de Sitter metric.

Black holes, first predicted two centuries ago, are a ubiquitous feature of modern gravitation theory. This volume reports the proceedings of a Spring School in 1998 which tried to give an almost encyclopaedic summary of black hole physics.

The Schwarzschild solution (1915) led on the one hand to an investigation of the gravitational collapse of stellar objects. Those with a mass less than about 1.4 solar masses can collapse to a stable state, a white dwarf, where gravity is balanced by the degeneracy pressure of free electrons. Heavier stars, whose mass is less than about three solar masses, can become neutron stars. Otherwise there is no stable configuration, and classical relativity predicts collapse to a gravitational singularity. The astrophysics of such objects is reviewed succinctly in this volume by Treves and Haardt. (Black holes are of interest in other astrophysical situations, e.g., active galactic nuclei, but that is not covered here, although useful references are given.) Although there are some exact solutions describing idealized models of gravitational collapse, it soon became clear that computers would be needed to describe realistic models, and from this origin has emerged the burgeoning subject of numerical relativity. Research in this area has bifurcated into the study of (a) the gravitational radiation produced by interacting black holes, which (hopefully) will become observable within a decade and (b) critical phenomena occurring in gravitational collapse, a concept discovered first numerically and which has so far eluded a complete theoretical explanation. The numerics of black holes are ably reviewed by Matzner in this volume.

Mathematical studies since 1915 have been no less intense. One can identify a classical (both literal and non-quantum) period before 1974, in which the mathematics of black hole exact solutions was worked out, culminating in the discovery of a striking resemblance between the laws of black hole mechanics and thermodynamics. In 1974 Hawking considered quantum fields propagating in a given classical black hole spacetime and discovered that external observers appear to see thermal radiation at a temperature proportional to Mfor a Schwarzschild black hole of mass M, which implies that the hole has an entropy proportional to M². This raises a number of paradoxes whose resolution cannot be found within a semiclassical approach; a consistent theory of quantum gravity would appear to be necessary.

Three quarters of this book is devoted to that topic. First Kiefer reviews, compactly, black hole thermodynamics, Hawking radiation and its paradoxes, as well as black holes in canonical quantum gravity and in string theory. Next Dijkgraaf reviews in depth the five-dimensional Strominger-Vafa black hole model, which he claims is representative of `a microcosm of the ideas which permeate modern day string theory', looking at, inter alia, the D5-D1 brane system, matrix theory and near-horizon limits in anti-de Sitter string theory. The longest contribution, from D'Auria and Fré, is entitled `BPS black holes in supergravity: duality groups, p-branes, central charges and entropy', which is a reasonable summary of its content. The main goal is to classify Bogomolnyi-Prasad-Sommerfeld black hole solutions of four-dimensional supergravity and to elucidate the group-theoretical structure of black hole entropy.

Although these lectures were addressed to an audience `assumed to be mostly unfamiliar with supergravity and superstrings' (such as your reviewer), few concessions were made. In mitigation, expert guidance through the many review articles at different levels is offered.

For whom is this book intended? As I have indicated, there were two distinct themes running through the School. While everyone should read Kiefer's review, few will wish to pursue both directions. Since less than a quarter of the book is devoted to astrophysical and numerical aspects, readers interested in this material may prefer to consult a library copy. The quantum theoretical contributions are realistically intended for new or prospective research students, or for researchers looking to change fields. They may however be deterred by the price. Although the typesetting is immaculate (thanks to LaTeX) and the paper is good quality the binding does seem somewhat flimsy for £80.