Table of contents

Gravitational waves generated during a first-order electroweak phase transition have a typical frequency which today falls just within the band of the planned space interferometer LISA. Contrary to what happens in the standard model, in its supersymmetric extensions, the electroweak phase transition may be strongly first order providing a mechanism for generating the observed baryon asymmetry in the universe. We show that during the same transition the production of gravitational waves can be rather sizeable. While the energy density in gravitational waves can reach at most h₀² Ω_gw ≃ 10⁻¹⁶ in the minimal supersymmetric standard model, in the next-to-minimal supersymmetric model in some parameter range, h₀² Ω_gw can be as high as 4 × 10⁻¹¹. A stochastic background of gravitational waves of this intensity is within the reach of the planned sensitivity of LISA. Since in the standard model the background of gravitational waves is totally neglegible, its detection would also provide a rather unexpected experimental signal of supersymmetry and a tool to discriminate among supersymmetric models with different Higgs content.

A numerical study of the stability of U(1) × U(1) × Z₂ Dirichlet topological defects (D-walls) has been carried out for a classical scalar field theory. It is found that the D-walls are stable for a wide range of coupling parameters and under perturbations to the wall. However, when two walls are connected by a string, they annihilate via a mechanism analogous to the Langacker–Pi mechanism for monopoles.

Massive spin 2 theories in flat or cosmological (Λ≠0) backgrounds are subject to discontinuities as the masses tend to zero. We show and explain physically why their Newtonian limits do not inherit this behaviour. On the other hand, conventional `Newtonian cosmology', where Λ is a constant source of the potential, displays discontinuities: e.g. for any finite range, Λ can be totally removed.

We try to develop a coherent picture of Liouville theory as a two-dimensional conformal field theory that takes into account the perspectives of the path-integral approach, bootstrap, canonical quantization and the operator approach. To do this, we need to develop further some of these approaches. This includes in particular the construction of general exponential field operators from a set of covariant chiral operators. The latter are shown to satisfy braid relations that allow one to prove the locality of the former.

Recently, several interesting proposals were made modifying the law of gravity on large scales, within a sensible relativistic formulation. This allows a precise formulation of the idea that such a modification might account for galaxy rotation curves, instead of the usual interpretation of these curves as evidence for dark matter. Here we summarize several observational constraints that any such modification must satisfy, and which we believe make more challenging, any interpretation of galaxy rotation curves in terms of new gravitational physics.

Classical general relativity takes place on a manifold with a metric of fixed, Lorentzian, signature. However, attempts to amalgamate general relativity with quantum theory frequently involve manifolds with metrics whose signatures are Lorentzian in some regions and Euclidean in others. (Indeed even more exotic possibilities are discussed frequently.) Most theoretical calculations rely on analyticity arguments to continue variables from the Euclidean to the Lorentzian regime and vice versa. This paper examines models of signature change. It looks at a single second-order quasi-linear partial differential equation on a fixed background, whose principal part is elliptic in one regime and hyperbolic in another, i.e. a mixed problem. It introduces some examples, explains heuristically the concept of a well-posed problem and then discusses the issues involved in constructing a robust numerical algorithm to solve well-posed problems. The paper includes a worked example illustrating the proposed techniques, and a discussion of the role of the potential curvature singularity on the transition hypersurface.

In the canonical approach to Lorentzian quantum general relativity in four spacetime dimensions an important step forward has been made by Ashtekar, Isham and Lewandowski some eight years ago through the introduction of a Hilbert space structure, which was later proved to be a faithful representation of the canonical commutation and adjointness relations of the quantum field algebra of diffeomorphism invariant gauge field theories by Ashtekar, Lewandowski, Marolf, Mourão and Thiemann. This Hilbert space, together with its generalization due to Baez and Sawin, is appropriate for semi-classical quantum general relativity if the spacetime is spatially compact. In the spatially non-compact case, however, an extension of the Hilbert space is needed in order to approximate metrics that are macroscopically nowhere degenerate. For this purpose, in this paper we apply the theory of the infinite tensor product (ITP) of Hilbert Spaces, developed by von Neumann more than sixty years ago, to quantum general relativity. The cardinality of the number of tensor product factors can take the value of any possible Cantor aleph, making this mathematical theory well suited to our problem in which a Hilbert space is attached to each edge of an arbitrarily complicated, generally infinite graph. The new framework opens access to a new arsenal of techniques, appropriate to describe fascinating physics such as quantum topology change, semi-classical quantum gravity, effective low-energy physics etc from the universal point of view of the ITP. In particular, the study of photons and gravitons propagating on fluctuating quantum spacetimes should now be in reach.

We present an exact solution of the averaged Einstein's field equations in the presence of two real scalar fields and a component of dust with spherical symmetry. We suggest that the spacetime found provides the characteristics required by a galactic model that could explain the supermassive central object and the dark matter halo at once, since one of the fields constitutes a central oscillaton surrounded by dust and the other scalar field distributes far from the coordinate centre and can be interpreted as a halo. We show the behaviour of the rotation curves all along the background. Thus the solution could be a first approximation of a 'long exposition photograph' of a galaxy.

We consider the motion of a particle described by an action that is a functional of the Frenet–Serret (FS) curvatures associated with the embedding of its worldline in Minkowski space. We develop a theory of deformations tailored to the FS frame. Both the Euler–Lagrange equations and the physical invariants of the motion associated with the Poincaré symmetry of Minkowski space, the mass and the spin of the particle, are expressed in a simple way in terms of these curvatures. The simplest non-trivial model of this form, with the Lagrangian depending on the first FS (or geodesic) curvature, is integrable. We show how this integrability can be deduced from the Poincaré invariants of the motion. We go on to explore the structure of these invariants in higher-order models. In particular, the integrability of the model described by a Lagrangian, that is, a function of the second FS curvature (or torsion), is established in a three-dimensional ambient spacetime.

We extend the Vecchia–Veneziano–Witten model of quantum chromodynamics in the chiral limit and for large colour number N_c, by introducing an effective dilaton–gluon coupling from which we derive both the axion and dilaton potentials. Furthermore, using a string-inspired model, we determine a new interquark potential as a perturbative series in terms of the interquark distance r. Our potential goes beyond Dick's potential (Dick R 1999 Eur. Phys. J. C 6 701) and shares the same features as those of the Bian–Huang–Shen potential (V_BHS) which depends only on odd powers of r (Bian J G, Huang T and Shen Q X 1994 Commun. Theor. Phys. 21 333).

Some interesting consequences of spacelike matter shells are presented, in particular the possibility of travelling through Cauchy horizons and violating the strong cosmic censorship hypothesis. These show that the weak energy condition does not guarentee cosmic censorship.

We consider covariant metric theories of coupled gravity-matter systems satisfying the following two conditions: first, it is assumed that, by a hyperbolic reduction process, a system of first-order symmetric hyperbolic partial differential equations can be deduced from the matter field equations. Second, gravity is supposed to be coupled to the matter fields by requiring that the Ricci tensor is a smooth function of the basic matter field variables and the metric. It is shown then that the 'time' evolution of these types of gravity-matter systems preserves the symmetries of initial data specifications.

We give a careful general relativistic and (1+3)-covariant analysis of cosmological peculiar velocities induced by matter density perturbations in the presence of a cosmological constant. In our quasi-Newtonian approach, constraint equations arise to maintain zero shear of the non-comoving fundamental worldlines which define a Newtonian-like frame, and these lead to the (1+3)-covariant dynamical equations, including a generalized Poisson-type equation. We investigate the relation between peculiar velocity and peculiar acceleration, finding the conditions under which they are aligned. In this case we find (1+3)-covariant relativistic generalizations of well-known Newtonian results.

Using the quantum Hamiltonian for a gravitational system with boundary, we find the partition function and derive the resulting thermodynamics. The Hamiltonian is the boundary term required by functional differentiability of the action for the Lorentzian general relativity. In this model, states of quantum geometry are represented by spin networks. We show that the statistical mechanics of the model reduces to that of a simple non-interacting gas of particles with spin. Using both canonical and grand canonical descriptions, we investigate two temperature regimes determined by the fundamental constant in the theory, m. In the high temperature limit (kT ≫ m), the model is thermodynamically stable. For low temperatures (kT ≪ m) and for macroscopic areas of the bounding surface, the entropy is proportional to the area (with logarithmic correction), providing a simple derivation of the Bekenstein–Hawking result. By comparing our results to known semiclassical relations we are able to fix the fundamental scale m. Also in the low temperature, macroscopic limit, the quantum geometry on the boundary forms a 'condensate' in the lowest energy level (j = 1/2).

We consider the class of space–time defects investigated by Puntigam and Soleng. These defects describe space–time dislocations and disclinations (cosmic strings) and are in close correspondence with the actual defects that arise in crystals and metals. It is known that in such materials dislocations and disclinations require a small and large amount of energy, respectively, to be created. The present analysis is carried out in the context of the teleparallel equivalent of general relativity (TEGR). We evaluate the gravitational energy of these space–time defects in the framework of the TEGR and find that there is an analogy between defects in space–time and in continuum material systems: the total gravitational energy of space–time dislocations and disclinations (considered as idealized defects) is zero and infinite, respectively.

This paper gives the construction and complete classification of all three-dimensional spherical manifolds and orders them by decreasing volume, in the context of multi-connected universe models with positive spatial curvature. It discusses which spherical topologies are likely to be detectable by crystallographic methods using three-dimensional catalogues of cosmic objects. The expected form of the pair separation histogram is predicted (including the location and height of the spikes) and is compared to computer simulations, showing that this method is stable with respect to observational uncertainties and is well suited for detecting spherical topologies.

The response of the ultrarelativistic particle with spin in a Schwarzschild field to the gravitomagnetic components as measured by the comoving observer is investigated. The dependence of the particle's spin–orbit acceleration on the Lorentz γ-factor and the spin orientation is studied. The concrete circular ultrarelativistic orbit of radius r = 3m is considered as a partial solution of the Mathisson–Papapetrou equations and as the corresponding high-energy quantum state of the Dirac particle. Numerical estimates for protons and electrons near black holes are given. A tendency of gravitational and electromagnetic interactions to approach in quantitative terms at ultrarelativistic velocities is discussed.

A large class of vacuum spacetimes is constructed in dimension 4 + 1 from hyperboloidal initial data sets which are not small perturbations of empty space data. These spacetimes are future geodesically complete, smooth up to their future null infinity ⁺ and extend as vacuum spacetimes through their Cauchy horizon. Dimensional reduction gives non-vacuum spacetimes with the same properties in 3 + 1 dimensions.

The effect of a non-zero cosmological constant on the power spectrum of background gravitational waves is investigated. The power spectra of models with and without a non-zero cosmological constant are compared, and measurable differences between the models are identified. In particular, the power spectrum for non-zero cosmological constant is 0.3–0.6 times lower than the models with a zero constant for large-scale gravitational waves, yet identical at the super-horizon scale.

We consider spacetimes which in addition to admitting an isolated horizon also admit Killing horizons with or without an event horizon. We show that an isolated horizon is a Killing horizon provided either (1) it admits a stationary neighbourhood, or (2) it admits a neighbourhood with two independent, commuting Killing vectors. A Killing horizon is always an isolated horizon. For the case when an event horizon is definable, all conceivable relative locations of isolated horizon and event horizons are possible. The corresponding conditions are given.

It is shown that a quantum-induced (inflationary) brane Universe occurs in the bulk 5D AdS black hole in accordance with AdS/CFT correspondence. Brane stress tensor is induced by quantum effects of dual CFT and the brane crosses the horizon of AdS black hole. Quantum-corrected Hubble constant, Hawking temperature and entropy are found on the brane (and at the horizon). The similarity between CFT entropy at the horizon and FRW equations is extended on the quantum level. This suggests the way to understand cosmological entropy bounds in quantum gravity.

Properties of the domain wall (kink) solution in the five-dimensional Randall–Sundrum model are examined both analytically and numerically. The configuration is derived by the bulk Higgs mechanism. We focus on (1) the convergence property of the solution, (2) the stablity of the solution, (3) the non-singular property of the Riemann curvature and (4) the behaviours of the warp factor and the Higgs field. It is found that the bulk curvature changes the sign around the surface of the wall. We also present some exact solutions for two simple cases: (a) the no-potential case, (b) the cosmological term-dominated case. Both solutions have the (naked) curvature singularity. We can regard the domain wall solution as a singularity resolution of the exact solutions.

A general system of equations is derived, using the 1 + 3 orthonormal tetrad formalism, describing the influence of a plane-fronted parallel gravitational wave on a warm relativistic two-component plasma. We focus our attention on phenomena that are induced by higher order terms in the gravitational wave amplitude. In particular, it is shown that the parametric excitations of ion-acoustic waves take place due to these higher order gravitational nonlinearities. The implications of the results are discussed.

We study the quasi-normal modes (QNM) of scalar, electromagnetic and gravitational perturbations of black holes in general relativity whose horizons have toroidal, cylindrical or planar topology in an asymptotically anti-de Sitter (AdS) spacetime. The associated QNM frequencies describe the decay in time of the corresponding test field in the vicinities of the black hole. In terms of the AdS/CFT conjecture, the inverse of the frequency is a measure of the dynamical timescale of approach to thermal equilibrium of the corresponding conformal field theory.

We obtain ADM and Komar surface integrals for the energy density, tension and angular momentum density of stationary p-brane solutions of Einstein's equations. We use them to derive a Smarr-type formula for the energy density and thence a first law of black brane mechanics. The intensive variable conjugate to the worldspace p-volume is an 'effective' tension that equals the ADM tension for uncharged branes, but vanishes for isotropic boost-invariant charged branes.

The conformal Killing equations for the most general (non-plane wave) conformally flat pure radiation field are solved to find the conformal Killing vectors. As expected 15 independent conformal Killing vectors exist, but in general the metric admits no Killing or homothetic vectors. However for certain special cases a one-dimensional group of homotheties or motions may exist and in one very special case, overlooked by previous investigators, a two-dimensional homethety group exists. No higher dimensional groups of motions or homotheties are admitted by these metrics.

We present an argument that for a large class of possible dynamics, a canonical quantization of gravity will satisfy the Bekenstein–Hawking entropy–area relation. This result holds for temperatures low compared to the Planck temperature and for boundaries with areas large compared to the Planck area. We also relate our description, in terms of a grand canonical ensemble, to previous geometric entropy calculations using area ensembles.

A rank-n tensor on a Lorentzian manifold whose contraction with n arbitrary causal future-directed vectors is non-negative is said to have the dominant property. These tensors, up to sign, are called causal tensors, and we determine their general mathematical properties in arbitrary dimension N. Then, we prove that rank-2 tensors which map the null cone on itself are causal tensors. Previously it has been shown that, to any tensor field A on a Lorentzian manifold there is a corresponding 'superenergy' tensor field T{A} (defined as a quadratic sum over all Hodge duals of A) which always has the dominant property. Here we prove that, conversely, any symmetric rank-2 tensor with the dominant property can be written in a canonical way as a sum of N superenergy tensors of simple forms. We show that the square of any rank-2 superenergy tensor is proportional to the metric in dimension N ≤ 4, and that the square of the superenergy tensor of any simple form is proportional to the metric in arbitrary dimension. Conversely, we prove in arbitrary dimension that any symmetric rank-2 tensor T whose square is proportional to the metric must be a causal tensor and, up to sign, the superenergy of a simple p-form, and that the trace of T determines the rank p of the form. This generalizes, with respect to both the dimension N and the rank p, the classical algebraic Rainich conditions, which are necessary and sufficient conditions for a metric to originate algebraically in some physical field. Furthermore, it has the important geometric interpretation that the set of superenergy tensors of simple forms is precisely the set of tensors which leave the null cone invariant and preserve its time orientation. It also means that all involutory Lorentz transformations can be represented as superenergy tensors of simple forms, and that any rank-2 superenergy tensor is the sum of at most N conformally involutory Lorentz transformations. Non-symmetric null cone preserving maps are shown to have a symmetric part with the dominant property and are classified according to the null eigenvectors of the skew-symmetric part. We therefore obtain a complete classification of all conformal Lorentz transformations and singular null cone preserving maps on any Lorentzian manifold of any dimension.

In this erratum, I would like to correct some errors due to incomplete variation of the H Kalb-Ramond field sector in the anomaly cancellation term in the Lagrangian of my previous two papers about the first-order and high-order effective string field theory in the early universe [2, 3]. With this aim, only, I present the Lagrangian, the correct Euler-Lagrange variation of this and the correct field equations.

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