Table of contents

We explore the role of a generalized Cremmer-Scherk mechanism of mass generation by coupling the antisymmetric 3-form gauge field to an Abelian gauge potential carrying weak hypercharge. The theory admits a class of exact self-gravitating solutions in the spontaneously broken phase in which dual fields acquire massive perturbative modes. Despite the massive nature of these fields they admit non-perturbative progressive longitudinal modes that together with pp-type gravitational waves travel in a direction of a line source at the speed of light.

In this paper we review some of the recent mathematical progress concerning the initial value problem formulation of general relativity. It is not our intention, however, to give an exhaustive presentation of all recent results on this topic, but rather to discuss some of the most promising mathematical techniques, which have been advanced in connection with the general Cauchy problem, in the absence of any special symmetries. Moreover, for the sake of simplicity and coherence, we restrict ourselves to the Einstein vacuum equations in the asymptotically flat regime. Our main goal is to discuss the main mathematical methods behind the various local existence and uniqueness results, as well as those used in the proof of global nonlinear stability of the Minkowski space. We also present an outline of a somewhat different and more transparent approach obtained by the authors in collaboration with Christodoulou. This relies, instead of the maximal foliation used by Christodoulou and Klainerman, on a double-null foliation. The new approach is fully adapted to domains of dependence and thus allows one to provide, directly, without having to rely on interior estimates, a proof of stability of `null infinity' for large asymptotically flat initial data sets.

We consider a Lorentzian manifold (,g) with an observer field (timelike vector field) V. Along each lightlike geodesic the redshift z and the angular diameter distance D are then well defined functions. (Instead of the angular diameter distance one may equivalently use the corrected or uncorrected luminosity distance.) (,g,V) is said to admit an isotropic Hubble law at p if all past-oriented lightlike geodesics issuing from p yield the same z-D relation. For infinitesimally short lightlike geodesics (i.e. to within a linear approximation with respect to the variable D) it is well known that an isotropic Hubble law holds at all points p if and only if V is freely falling and shear-free. Here we derive the necessary and sufficient conditions for an isotropic Hubble law beyond the linear regime. To that end we expand the z-D relation in a Taylor series (Kristian-Sachs series) and we investigate the validity of an isotropic Hubble law order by order. In particular, we prove that (,g,V) admits an isotropic Hubble law of third order at every point p if and only if (,g,V) is either redshift-free or a Robertson-Walker model (locally around every point p ).

We study the late-time evolution of a class of exact anisotropic cosmological solutions of Einstein's equations, namely spatially homogeneous cosmologies of Bianchi type VII₀ with a perfect fluid source. We show that, in contrast to models of Bianchi type VII_h which are asymptotically self-similar at late times, Bianchi VII₀ models undergo a complicated type of self-similarity breaking. This symmetry breaking affects the late-time isotropization that occurs in these models in a significant way: if the equation of state parameter satisfies (4/3) the models isotropize as regards the shear but not as regards the Weyl curvature. Indeed, these models exhibit a new dynamical feature that we refer to as Weyl curvature dominance: the Weyl curvature dominates the dynamics at late times. By viewing the evolution from a dynamical systems perspective we show that, despite the special nature of the class of models under consideration, this behaviour has implications for more general models.

A simple example is given to show that the gauge equivalence classes of physical states in Chern-Simons theory are not in one-to-one correspondence with those of Einstein gravity in three spacetime dimensions. The two theories are therefore not equivalent. It is shown that including singular metrics into general relativity has more, and in fact a quite counter-intuitive, impact on the theory than one naively expects.

A diagramatic heat kernel expansion technique is presented. The method is especially well suited to the small-derivative expansion of the heat kernel, but it can also be used to reproduce the results obtained by the approach known as covariant perturbation theory. The new technique gives an expansion for the heat kernel at coincident points. It can also be used to obtain the derivative of the heat kernel, and this is useful for evaluating the expectation values of the stress-energy tensor.

We present a formulation of the coupling of vector multiplets to N = 2 supergravity which is symplectic covariant (and thus is not based on a prepotential) and uses superconformal tensor calculus. We do not start from an action, but from the combination of the generalized Bianchi identities of the vector multiplets in superspace, a symplectic definition of special Kähler geometry, and the supersymmetric partners of the corresponding constraints. These involve the breaking to super-Poincaré symmetry, and lead to on-shell vector multiplets.

This symplectic approach gives the framework to formulate vector multiplet couplings using a weaker defining constraint for special Kähler geometry, which is an extension of older definitions of special Kähler manifolds for some cases with only one vector multiplet.

We consider the problem of evolving the state of a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation is computed; it does not correspond to a unitary transformation on the Fock space. This means that functional evolution of the quantum state as originally envisioned by Tomonaga, Schwinger and Dirac is not a viable concept. Nevertheless, we demonstrate that functional evolution of the quantum state can be satisfactorily described using the formalism of algebraic quantum field theory. We discuss possible implications of our results for canonical quantum gravity.

We present new results concerning the existence of static, electrically charged, perfect fluid spheres that have a regular interior and are arbitrarily close to a maximally charged black-hole state. These configurations are described by exact solutions of Einstein's field equations. A family of these solutions had already been found (de Felice et al 1995 Mon. Not. R. Astron. Soc.277 L17) but here we generalize that result to cases with different charge distributions within the spheres and show, in an appropriate parameter space, that the set of such physically reasonable solutions has a non-zero measure. We also perform a perturbation analysis and identify the solutions which are stable against adiabatic radial perturbations. We then suggest that the stable configurations can be considered as classic models of charged particles. Finally, our results are used to show that a conjecture of Kristiansson et al (1998 Gen. Rel. Grav.30 275) is incorrect.

We study an anisotropic simplicial minisuperspace model with a cosmological constant and a massive scalar field. After obtaining the classical solutions we then compute the semiclassical approximation of the no-boundary wavefunction of the Universe along a steepest-descent contour passing through classical Lorentzian solutions. The oscillatory behaviour of the resulting wavefunction is consistent with the prediction of classical Lorentzian spacetime for the late Universe.

Recently, a set of tools has been developed with the purpose of studying quantum gravity. Until now, there have been very few attempts to put these tools into a rigorous mathematical framework. This is the case, for example, for the so-called path bundle of a manifold. It is well known that this topological principal bundle plays the role of a universal bundle for the reconstruction of principal bundles and their connections. The path bundle is canonically endowed with parallel transport and, associated with it, important types of derivatives have been considered by several authors: the Mandelstam derivative, the connection derivative and the loop derivative. Here we shall give a unified viewpoint for all of these derivatives by developing a differentiable calculus on differentiable spaces. In particular, we shall show that the loop derivative is the curvature of a canonically defined 1-form that we shall call the universal connection 1-form.

Plane-wave Cauchy horizons serve as an important testing ground for a previously developed Cauchy horizon stability conjecture. The conjecture uses test fields to predict the stability of Cauchy horizons in general relativistic spacetimes. Three cases are considered: single plane waves with (a) aligned test fields and (b) colliding test fields; and (c) non-singular interaction regions in colliding-wave spacetimes. A restricted form of the stability conjecture is proven for case (a). The conjecture is shown to agree with arbitrary exact back-reaction solutions in this case, as long as no Weyl-tensor singularities are introduced on an initial Cauchy surface. Four examples of case (b) are analysed; in three the conjecture agrees with exact back-reaction solutions, and in the fourth the conjecture correctly indicates instability, but misjudges the type of singularity formed. A possible reason for the disagreement is discussed. Two examples of case (c) are analysed, in which test null dust is added to the interaction regions of a colliding electromagnetic spacetime (Bell-Szekeres) and a colliding gravitational wave spacetime (Chandrasekhar-Xanthopoulos). The conjecture is shown to agree with known exact back-reaction solutions.

A canonical action describing the interaction of chiral gauge fields in D = 6 Minkowski spacetime is constructed. In a particular partial gauge fixing it reduces to the action found by Perry and Schwarz. The additional gauge symmetries are used to show the off-shell equivalence of the dimensional reduction to D = 5 Minkowski spacetime of the chiral gauge field canonical action and the Born-Infeld canonical action describing an interacting D = 5 Abelian vector field. Its extension to improve the on-shell equivalence arguments of dual D-brane actions to off-shell ones is discussed.

Einstein's field equations with a variable speed of light c = c(t), gravitational coupling G = G(t) and decaying vacuum energy density = (t) are considered as describing matter creation in a cosmological framework. The particle creation rate is uniquely determined by the variation rates of c, G and . By considering simple phenomenological evolution laws for the `constants', exact analytical and numerical solutions of the gravitational field equations are obtained for a homogeneous and isotropic Friedmann-Robertson-Walker spacetime leading to a self-consistent cosmological model describing matter and entropy generation in the very early Universe.

Using the formalism of Kramer for the stationary, cylindrically symmetric and rigidly rotating perfect fluid, two models are derived explicitly. The solutions are regular on the axis and the energy conditions are satisfied for a certain range of the angular velocity. In the limit of no rotation they are reduced to the Kramer (p = (-1)µ) and Evans (µ = 5p+constant) static cylindrically symmetric metrics.

A generic ansatz is introduced which provides families of exact solutions to the equations of motion and constraints for null strings in Bianchi type I cosmological models. This is achieved irrespective of the form of the metric. Within classes of dilaton cosmologies a backreaction mapping relation is established where the null string leads to more or less anisotropic members of the family. The equations of motion and constraints for the generic model are cast in first-order form and integrated both analytically and numerically.

It was shown recently that the metric functions which describe a spherically symmetric spacetime with vanishing radial pressure can be explicitly integrated. We investigate the nakedness and curvature strength of the shell-focusing singularity in that spacetime. If the singularity is naked, the relation between the circumferential radius and the Misner-Sharp mass is given by R2y₀m with (1/3)< 1 along the first radial null geodesic from the singularity. The is closely related to the curvature strength of the naked singularity. For example, for the outgoing or ingoing null geodesic, if the strong curvature condition (SCC) of Tipler holds, then must be equal to 1. We define the `gravity-dominance condition' (GDC) for a geodesic. If GDC is satisfied for the null geodesic, both SCC and the limiting focusing condition (LFC) of Królak holds for = 1 and y₀1, not SCC but only LFC holds for ½ <1, and neither holds for (1/3)<<½, for the null geodesic. On the other hand, if GDC is satisfied for the timelike geodesic r = 0, both SCC and LFC are satisfied for the timelike geodesic, irrespective of the value of . Several examples are also discussed.

We study boson stars in Brans-Dicke (BD) gravity and use them to illustrate some of the properties of three different mass definitions: the Schwarzschild mass, the Keplerian mass and the tensor mass. We analyse the weak-field limit of the solutions and show that only the tensor mass leads to a physically reasonable definition of the binding energy. We examine numerically strong-field = -1 solutions and show how, in this extreme case, the three mass values and the conserved particle number behave as a function of the central boson field amplitude. The numerical studies imply that for = -1, solutions with extremal tensor mass also have extremal particle number. This is a property that a physically reasonable definition of the mass of a boson star must have, and we prove analytically that this is true for all values of . The analysis supports the conjecture that the tensor mass uniquely describes the total energy of an asymptotically flat solution in BD gravity.