Table of contents

New exact solutions of the Einstein–Maxwell field equations that describe pp-waves are presented.

We study braneworlds in six-dimensional Einstein–Gauss–Bonnet gravity. The Gauss–Bonnet term is crucial for the equations to be well-posed in six dimensions when non-trivial matter on the brane is included (the induced-gravity term also involved is not significant for their structure), and the matching conditions of the braneworld are known. We show that the energy–momentum of the brane is always conserved, independently of any regular bulk energy–momentum tensor, in contrast to the situation of the five-dimensional case.

We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings.

Many years ago Ehlers and Kundt showed that a spacetime M is an Einstein space if and only if the sectional curvatures of any pair of orthogonal non-null 2-spaces at any point of M are equal. This paper generalizes this result by first showing a very straightforward relation between the sectional curvatures of such orthogonal pairs of 2-spaces and the trace-free part of the Ricci tensor and then by establishing for each algebraic (Segre) type of the energy–momentum tensor precisely which orthogonal pairs of non-null 2-spaces have the same sectional curvature. The results are described in a manifold theoretic sense and are tabulated for each Segre type.

We obtain expressions for the mass and angular momenta of rotating black holes in anti-de Sitter backgrounds in four, five and higher dimensions. We verify explicitly that our expressions satisfy the first law of thermodynamics, thus allowing an unambiguous identification of the entropy of these black holes with of the area. We find that the associated thermodynamic potential equals the background-subtracted Euclidean action multiplied by the temperature. Our expressions differ from many given in the literature. We find that in more than four dimensions, only our expressions satisfy the first law of thermodynamics. Moreover, in all dimensions we show that our expression for the mass coincides with that given by the conformal conserved charge introduced by Ashtekar, Magnon and Das. We indicate the relevance of these results to the AdS/CFT correspondence.

Gödel-type metrics are introduced and used in producing charged dust solutions in various dimensions. The key ingredient is a (D − 1)-dimensional Riemannian geometry which is then employed in constructing solutions to the Einstein–Maxwell field equations with a dust distribution in D dimensions. The only essential field equation in the procedure turns out to be the source-free Maxwell's equation in the relevant background. Similarly the geodesics of this type of metric are described by the Lorentz force equation for a charged particle in the lower dimensional geometry. It is explicitly shown with several examples that Gödel-type metrics can be used in obtaining exact solutions to various supergravity theories and in constructing spacetimes that contain both closed timelike and closed null curves and that contain neither of these. Among the solutions that can be established using non-flat backgrounds, such as the Tangherlini metrics in (D − 1)-dimensions, there exists a class which can be interpreted as describing black-hole-type objects in a Gödel-like universe.

Ultrarelativistic circular orbits of spinning particles in a Schwarzschild field described by the Mathisson–Papapetrou equations are considered. The preliminary estimates of the possible synchrotron electromagnetic radiation of highly relativistic protons and electrons on these orbits in the gravitational field of a black hole are presented.

We introduce a Dirac–Born–Infeld action to a self-dual N = 1 supersymmetric vector multiplet in three dimensions. This action is based on the supersymmetric generalized self-duality in odd dimensions developed originally by Townsend, Pilch and van Nieuwenhuizen. Even though such a self-duality had been supposed to be very difficult to generalize to a supersymmetrically interacting system, we show that the Dirac–Born–Infeld action is actually compatible with supersymmetry and self-duality in three dimensions, even though the original self-duality receives corrections by the Dirac–Born–Infeld action. The interactions can be further generalized to arbitrary (non)polynomial interactions. As a by-product, we also show that a third-rank field strength leads to a more natural formulation of self-duality in 3D. We also show an interesting role played by the third-rank field strength leading to supersymmetry breaking, in addition to accommodating a Chern–Simons form.

We show the existence of an infinite family of finite-time singularities in isotropically expanding universes which obey the weak, strong and dominant energy conditions. We show what new type of energy condition is needed to exclude them ab initio. We also determine the conditions under which finite-time future singularities can arise in a wide class of anisotropic cosmological models. New types of finite-time singularity are possible which are characterized by divergences in the time rate of change of the anisotropic-pressure tensor. We investigate the conditions for the formation of finite-time singularities in a Bianchi-type VII₀ universe with anisotropic pressures and construct specific examples of anisotropic sudden singularities in these universes.

With the help of a generalized Raychaudhuri equation non-expanding null surfaces are studied in an arbitrary dimensional case. The definition and basic properties of non-expanding and isolated horizons known in the literature in the four- and three-dimensional cases are generalized. A local description of the horizon's geometry is provided. The zeroth law of black-hole thermodynamics is derived. The constraints have a similar structure to that of the four-dimensional spacetime case. The geometry of a vacuum isolated horizon is determined by the induced metric and the rotation 1-form potential, local generalizations of the area and the angular momentum typically used in the stationary black-hole solutions case.

It is proven that under mild physical assumptions, an isolated stationary relativistic perfect fluid consists of a finite number of cells fibred by invariant annuli or invariant tori. For axially symmetric circular flows, it is shown that the fluid consists of cells fibred by rigidly rotating annuli or tori.

In this paper we develop a method of finding the static axisymmetric spacetime corresponding to any given set of multipole moments. In addition to an implicit algebraic form for the general solution, we also give a power series expression for all finite sets of multipole moments. As conjectured by Geroch we prove in the special case of axisymmetry, that there is a static spacetime for any given set of multipole moments subject to a (specified) convergence criterion. We also use this method to confirm a conjecture of Hernández-Pastora and Martín concerning the monopole–quadrupole solution.

Experimental evidence suggests that we live in a spatially flat, accelerating universe composed of roughly one-third of matter (baryonic + dark) and two-thirds of a negative-pressure dark component, generically called dark energy. The presence of such energy not only explains the observed accelerating expansion of the universe but also provides the remaining piece of information connecting the inflationary flatness prediction with astronomical observations. However, despite its good observational indications, the nature of dark energy still remains an open question. In this paper we explore a geometrical explanation for such a component within the context of braneworld theory without mirror symmetry, leading to a geometrical interpretation for dark energy as a warp in the universe given by the extrinsic curvature. In particular, we study the phenomenological implications of the extrinsic curvature of a Friedmann–Robertson–Walker universe in a five-dimensional constant curvature bulk, with signatures (4,1) or (3,2), as compared with the x-matter (XCDM) model. From the analysis of the geometrically modified Friedmann's equations, the deceleration parameter and the weak energy condition, we find a consistent agreement with the presently known observational data on inflation for the de Sitter bulk, but not for the anti-de Sitter case.

We express stress tensor correlators using the Schwinger–Keldysh formalism. The absence of off-diagonal counterterms in this formalism ensures that the +− and −+ correlators are free of primitive divergences. We use dimensional regularization in position space to explicitly check this at one loop order for a massless scalar on a flat space background. We use the same procedure to show that the ++ correlator contains the divergences first computed by 't Hooft and Veltman for the scalar contribution to the graviton self-energy.

In a recent paper by Thomas Jurke, it was proved that the asymptotic behaviour of a solution to the polarized Gowdy equation in the expanding direction is of the form αln t + β + t^−1/2ν + O(t^−3/2), where α and β are constants and ν is a solution to the standard wave equation with zero mean value. Furthermore, it was proved that α, β and ν uniquely determine the solution. Here we wish to point out that given α, β and ν with the above properties, one can construct a solution to the polarized Gowdy equation with the above asymptotics. In other words, we show that α, β and ν constitute data at the moment of infinite expansion. We then use this fact to make the observation that there are polarized Gowdy spacetimes such that in the areal time coordinate, the quotient of the maximum and the minimum of the mean curvature on a constant time hypersurface is unbounded as time tends to infinity.

It is shown that spatially homogeneous solutions of the Einstein equations coupled to a nonlinear scalar field and other matter exhibit accelerated expansion at late times for a wide variety of potentials V. These potentials are strictly positive but tend to zero at infinity. They satisfy restrictions on V'/V and V''/V' related to the slow-roll approximation. These results generalize Wald's theorem for spacetimes with positive cosmological constant to those with accelerated expansion driven by potentials belonging to a large class.

We study S-duality transformations that mix the Riemann tensor with the field strength of a 3-form field. The dual of an (A)dS spacetime—with arbitrary curvature—is seen to be flat Minkowski spacetime if the 3-form field has vanishing field strength before the duality transformation. It is discussed whether matter could couple to the dual metric, related to the Riemann tensor after a duality transformation. This possibility is supported by the facts that the Schwarzschild metric can be obtained as a suitable contraction of the dual of a Taub-NUT-AdS metric, and that metrics describing FRW cosmologies can be obtained as duals of theories with matter in the form of torsion.

This paper uses the conformal Einstein equations and the conformal representation of spatial infinity introduced by Friedrich to analyse the behaviour of the gravitational field near null and spatial infinity for the development of initial data which are, in principle, non-conformally flat and time asymmetric. The paper is the continuation of the investigation started in Class. Quantum Grav.21 (2004) 5457–92, where only conformally flat initial data sets were considered. For the purposes of this investigation, the conformal metric of the initial hypersurface is assumed to have a very particular type of non-smoothness at infinity in order to allow for the presence of non-Schwarzschildean stationary initial data sets in the class under study. The calculation of asymptotic expansions of the development of these initial data sets reveals—as in the conformally flat case—the existence of a hierarchy of obstructions to the smoothness of null infinity which are expressible in terms of the initial data. This allows for the possibility of having spacetimes where future and past null infinity have different degrees of smoothness. A conjecture regarding the general structure of the hierarchy of obstructions is presented.

In this paper, we study M-theory compactifications on manifolds of G₂ structure. By computing the gravitino mass term in four dimensions, we derive the general form for the superpotential which appears in such compactifications and show that beside the normal flux term there is a term which appears only for non-minimal G₂ structure. We further apply these results to compactifications on manifolds with weak G₂ holonomy and make a couple of statements regarding the deformation space of such manifolds. Finally, we show that the superpotential derived from fermionic terms leads to the potential that can be derived from the explicit compactification, thus strengthening the conjectures we make about the space of deformations of manifolds with weak G₂ holonomy.

In this paper, we address the problem of the dynamics in three-dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space—corresponding to the quantization of the infinite-dimensional kinematical configuration space of the theory—to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canonical quantization of three-dimensional gravity and spin-foam models. We emphasize two main properties of the result: first that no cut-off in the kinematical degrees of freedom of the theory is introduced (in contrast to standard 'lattice' methods), and second that no ill-defined sum over spins ('bubble' divergences) are present in the spin-foam representation.

The long-standing difficulty in general relativity of classifying the dynamics of cosmological models, e.g. as chaotic, is directly related to the gauge freedom intrinsic to relativistic spacetime theories: in general the invariance under diffeomorphisms makes any analysis of dynamical evolution dependent on the particular choice of time slicing one uses. We show here that the speciality index, a scalar dimensionless curvature invariant that has been mainly used in numerical relativity as an indicator of the special or non-special Petrov-type character of a spacetime, is a time-independent quantity (a pure number) at each Kasner step of the Belinski–Khalatnikov–Lifshitz (BKL) map approximating the mixmaster cosmology. Thus the BKL dynamics can be characterized in terms of the speciality index, i.e. in terms of curvature invariants directly related to observables. Possible applications for the associated mixmaster dynamics are discussed.

The divergence of the constraint quantities is a major problem in computational gravity today. Apparently, there are two sources for constraint violations. The use of boundary conditions which are not compatible with the constraint equations inadvertently leads to 'constraint violating modes' propagating into the computational domain from the boundary. The other source for constraint violation is intrinsic. It is already present in the initial-value problem, i.e. even when no boundary conditions have to be specified. Its origin is due to the instability of the constraint surface in the phase space of initial conditions for the time evolution equations. In this paper, we present a technique to study in detail how this instability depends on gauge parameters. We demonstrate this for the influence of the choice of the time foliation in the context of the Weyl system. This system is the essential hyperbolic part in various formulations of the Einstein equations.

When constructing general relativity (GR), Einstein required 4D general covariance. In contrast, we derive GR (in the compact, without boundary case) as a theory of evolving three-dimensional conformal Riemannian geometries obtained by imposing two general principles: (1) time is derived from change; (2) motion and size are relative. We write down an explicit action based on them. We obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation, but also all the equations used in York's conformal technique for solving the initial-value problem. This shows that the independent gravitational degrees of freedom obtained by York do not arise from a gauge fixing but from hitherto unrecognized fundamental symmetry principles. They can therefore be identified as the long-sought Hamiltonian physical gravitational degrees of freedom.

We investigate the quasinormal modes for gravitational perturbations of rotating black holes in four-dimensional anti-de Sitter (AdS) spacetime. The study of the quasinormal frequencies related to these modes is relevant to the AdS/CFT correspondence. Although results have been obtained for Schwarzschild and Reissner–Nordstrom AdS black holes, quasinormal frequencies of Kerr-AdS black holes are computed for the first time. We solve the Teukolsky equations in AdS spacetime, providing a second-order and a Padé approximation for the angular eigenvalues associated with the Teukolsky angular equation. The transformation theory and the Regge–Wheeler–Zerilli equations for Kerr-AdS are obtained.

Evidence for free precession has been observed in the radio signature of several pulsars. Freely precessing pulsars radiate gravitationally at frequencies near the rotation rate and twice the rotation rate, which for rotation frequencies greater than ∼10 Hz is in the LIGO band. In older work, the gravitational wave spectrum of a precessing neutron star has been evaluated to first order in a small precession angle. Here, we calculate the contributions to second order in the wobble angle, and we find that a new spectral line emerges. We show that for reasonable wobble angles, the second-order line may well be observable with the proposed advanced LIGO detectors for precessing neutron stars as far away as the galactic centre. Observation of the full second-order spectrum permits a direct measurement of the star's wobble angle, oblateness and deviation from axisymmetry, with the potential to significantly increase our understanding of neutron star structure.

We study the Robertson–Walker minisuperspace model in histories theory, motivated by the results that emerged from the histories approach to general relativity. We examine, in particular, the issue of time reparametrization in such systems. The model is quantized using an adaptation of reduced state space quantization. We finally discuss the classical limit, the implementation of initial cosmological conditions and estimation of probabilities in the histories context.

We establish a new self-consistent system of equations for the gravitational and electromagnetic fields. The procedure is based on a non-minimal nonlinear extension of the standard Einstein–Hilbert–Maxwell action. General properties of a three-parameter family of non-minimal linear models are discussed. In addition, we show explicitly that a static spherically symmetric charged object can be described by a non-minimal model, second order in the derivatives of the metric, when the susceptibility tensor is proportional to the double-dual Riemann tensor.

Using Cartan's equivalence method for point transformations, we obtain from first principles the conformal geometry associated with third-order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a family of Lorentzian metrics, the conformal group in three and four dimensions and the so-called normal metric connection. A special feature of this connection is that the non-vanishing components of its torsion depend on one relative invariant, the (generalized) Wünschmann invariant. We show that the above-mentioned construction naturally contains the null surface formulation of general relativity.

Claus Kiefer presents his book, Quantum Gravity, with his hope that '[the] book will convince readers of [the] outstanding problem [of unification and quantum gravity] and encourage them to work on its solution'. With this aim, the author presents a clear exposition of the fundamental concepts of gravity and the steps towards the understanding of its quantum aspects.

The main part of the text is dedicated to the analysis of standard topics in the formulation of general relativity. An analysis of the Hamiltonian formulation of general relativity and the canonical quantization of gravity is performed in detail. Chapters four, five and eight provide a pedagogical introduction to the basic concepts of gravitational physics. In particular, aspects such as the quantization of constrained systems, the role played by the quadratic constraint, the ADM decomposition, the Wheeler-de Witt equation and the problem of time are treated in an expert and concise way. Moreover, other specific topics, such as the minisuperspace approach and the feasibility of defining extrinsic times for certain models, are discussed as well.

The ninth chapter of the book is dedicated to the quantum gravitational aspects of string theory. Here, a minimalistic but clear introduction to string theory is presented, and this is actually done with emphasis on gravity. It is worth mentioning that no hard (nor explicit) computations are presented, even though the exposition covers the main features of the topic. For instance, black hole statistical physics (within the framework of string theory) is developed in a pedagogical and concise way by means of heuristical arguments.

As the author asserts in the epilogue, the hope of the book is to give 'some impressions from progress' made in the study of quantum gravity since its beginning, i.e., since the end of 1920s. In my opinion, Kiefer's book does actually achieve this goal and gives an extensive review of the subject.