Table of contents

Dirac's method for variations of a brane embedded in co-dimension one is demonstrated. The variation in the location of the brane invokes a rest frame formulation of the 'sandwiched' brane action. We first demonstrate the necessity of this method by re-deriving Snell's law. Second, we apply the method to a general N-dimensional brane embedded in co-dimension one bulk in the presence of gravity. We re-derive the brane equations: (i) the Israel junction condition, (ii) the energy/momentum conservation on the brane and (iii) a geodetic-type equation for the brane.

We performed a careful numerical analysis of the late tail behaviour of waves propagating in the Schwarzschild spacetime. Specifically the scalar monopole, the electromagnetic dipole and the gravitational axial quadrupole waves have been investigated. The results obtained agree with a falloff 1/t^2l+3 for the general initial data and 1/t^2l+4 for the initially static data.

We discuss the asymptotic dynamical evolution of spatially inhomogeneous brane-world cosmological models close to the initial singularity. By introducing suitable scale-invariant dependent variables and a suitable gauge, we write the evolution equations of the spatially inhomogeneous G₂ brane cosmological models with one spatial degree of freedom as a system of autonomous first-order partial differential equations. We study the system numerically, and we find that there always exists an initial singularity, which is characterized by the fact that spatial derivatives are dynamically negligible. More importantly, from the numerical analysis we conclude that there is an initial isotropic singularity in all these spatially inhomogeneous brane cosmologies for a range of parameter values which include the physically important cases of radiation and a scalar field source. The numerical results are supported by a qualitative dynamical analysis and a calculation of the past asymptotic decay rates. Although the analysis is local in nature, the numerics indicate that the singularity is isotropic for all relevant initial conditions. Therefore this analysis, and a preliminary investigation of general inhomogeneous (G₀) models, indicates that it is plausible that the initial singularity is isotropic in spatially inhomogeneous brane-world cosmological models and consequently that brane cosmology naturally gives rise to a set of initial data that provide the conditions for inflation to subsequently take place.

We study transverse asymptotically flat spacetimes without horizons that arise from brane matter sources. We assume that asymptotically there is a spatial translation Killing vector that is tangent to the brane. Such spacetimes are characterized by a tension, analogous to the ADM mass, which is a gravitational charge associated with the asymptotic spatial translation Killing vector. Using spinor techniques, we prove that the purely gravitational contribution to the spacetime tension is positive definite.

We analyse the physical implications of the energy and matching conditions that take part in the construction of spherically symmetric models for stars and voids. We find an important condition on the radial pressures that can be of relevance, for instance, in the modellization of voids in the universe. In this way we show, among other results, that some models for the universe could not possess viable voids. The condition is also applied in order to construct charged radiating stellar models.

A modelling code has been developed that allows the thermal noise in a pendulum suspended by multiple wires to be predicted. The code has been applied, in particular, to allow optimization of the design of the suspension for the main mirrors in GEO 600, the German/United Kingdom gravitational wave detector. Results from one-, two- and four-wire suspensions are presented. For the GEO 600 detector it was concluded that a four-wire (two-loop) suspension was optimum.

A Schwarzschild black hole being thermodynamically unstable, corrections to its entropy due to small thermal fluctuations cannot be computed. However, a thermodynamically stable Schwarzschild solution can be obtained within a cavity of any finite radius by immersing it in an isothermal bath. For these boundary conditions, classically there are either two black-hole solutions or no solution. In the former case, the larger mass solution has a positive specific heat and hence is locally thermodynamically stable. We find that the entropy of this black hole, including first-order fluctuation corrections, is given by: , where S_BH = A/4 is its Bekenstein–Hawking entropy and R is the radius of the cavity. We extend our results to four-dimensional Reissner–Nordström black holes, for which the corresponding expression is: . Finally, we generalize the stability analysis to Reissner–Nordström black holes in arbitrary spacetime dimensions, and compute their leading order entropy corrections. In contrast to previously studied examples, we find that the entropy corrections in these cases have a different character.

In this paper, we investigate the asymptotic nature of the quasinormal modes for 'dirty' black holes—generic static and spherically symmetric spacetimes for which a central black hole is surrounded by arbitrary 'matter' fields. We demonstrate that, to the leading asymptotic order, the (imaginary) spacing between modes is precisely equal to the surface gravity, independent of the specifics of the black-hole system. Our analytical method is based on locating the complex poles in the first Born approximation for the scattering amplitude. We first verify that our formalism agrees, asymptotically, with previous studies on the Schwarzschild black hole. The analysis is then generalized to more exotic black-hole geometries. We also extend considerations to spacetimes with two horizons and briefly discuss the degenerate-horizon scenario.

A new metric is obtained by applying a complex coordinate transformation to the static metric of the self-gravitating Born–Infeld monopole. The behaviour of the new metric is typical of a rotating charged source, but this source is not a spherically symmetric Born–Infeld monopole with rotation. We show that the structure of the energy–momentum tensor obtained with this new metric does not correspond to the typical structure of the energy–momentum tensor of Einstein–Born–Infeld theory induced by a rotating spherically symmetric source. This also shows that the complex coordinate transformations have the interpretation given by Newman and Janis only in spacetime solutions with linear sources.

This work is a natural continuation of our recent study in quantizing relativistic particles. There it was demonstrated that, by applying a consistent quantization scheme to the classical model of a spinless relativistic particle as well as to the Berezin–Marinov model of a 3 + 1 Dirac particle, it is possible to obtain a consistent relativistic quantum mechanics of such particles. In the present paper, we apply a similar approach to the problem of quantizing the massive 2 + 1 Dirac particle. However, we stress that such a problem differs in a nontrivial way from the one in 3 + 1 dimensions. The point is that in 2 + 1 dimensions each spin polarization describes different fermion species. Technically this fact manifests itself through the presence of a bifermionic constant and of a bifermionic first-class constraint. In particular, this constraint does not admit a conjugate gauge condition at the classical level. The quantization problem in 2 + 1 dimensions is also interesting from the physical viewpoint (e.g., anyons). In order to quantize the model, we first derive a classical formulation in an effective phase space, restricted by constraints and gauges. Then the condition of preservation of the classical symmetries allows us to realize the operator algebra in an unambiguous way and construct an appropriate Hilbert space. The physical sector of the constructed quantum mechanics contains spin-1/2 particles and antiparticles without an infinite number of negative-energy levels, and exactly reproduces the one-particle sector of the 2 + 1 quantum theory of a spinor field.

We derive the Dirac equation in the Euclidean version of the Newman–Penrose formalism and show that it splits into two sets of equations, particle and anti-particle equations, under the swapping symmetry and these equations are coupled, respectively, with the self-dual and anti-self-dual parts of the gauge in the gravity. We also solve it for Eguchi–Hanson and Bianchi VII₀ gravitational instanton metrics. The solutions are obtained for the Bianchi VII₀ gravitational instanton metric as exponential functions by using complex variable ξ and for the Eguchi–Hanson gravitational instanton metric as the product of two hypergeometric functions. In addition, we discuss the regularity and the swapping symmetry of the solutions and show that the topological index of the Dirac equation is zero for both of these metrics.

In this paper, by using the quasi-normal frequencies for a non-rotating BTZ black hole derived by Cardos and Lemos, also via Bohr–Sommerfeld quantization for an adiabatic invariant, , where E is the energy of system and ω(E) is vibrational frequency, we are led to an equally spaced mass spectrum. The result for the area of the event horizon is which is not equally spaced, in contrast with the area spectrum of a black hole in higher dimension.

By embedding Einstein's original formulation of general relativity into a broader context, we show that a dynamic covariant description of gravitational stress–energy emerges naturally from a variational principle. A tensor T^G is constructed from a contraction of the Bel tensor with a symmetric covariant second degree tensor field Φ and has a form analogous to the stress–energy tensor of the Maxwell field in an arbitrary spacetime. For plane-fronted gravitational waves helicity-2 polarized (graviton) states can be identified carrying non-zero energy and momentum.

We present results of 3D numerical simulations using a finite difference code featuring fixed mesh refinement (FMR), in which a subset of the computational domain is refined in space and time. We apply this code to a series of test cases including a robust stability test, a nonlinear gauge wave and an excised Schwarzschild black hole in an evolving gauge. We find that the mesh refinement results are comparable in accuracy, stability and convergence to unigrid simulations with the same effective resolution. At the same time, the use of FMR reduces the computational resources needed to obtain a given accuracy. Particular care must be taken at the interfaces between coarse and fine grids to avoid a loss of convergence at higher resolutions, and we introduce the use of 'buffer zones' as one resolution of this issue. We also introduce a new method for initial data generation, which enables higher order interpolation in time even from the initial time slice. This FMR system, 'Carpet', is a driver module in the freely available Cactus computational infrastructure, and is able to endow generic existing Cactus simulation modules ('thorns') with FMR with little or no extra effort.

The behaviour near the initial state of the anisotropy parameter of the arbitrary type, homogeneous and anisotropic Bianchi models is considered in the framework of the braneworld cosmological models. The matter content on the brane is assumed to be an isotropic perfect cosmological fluid obeying a barotropic equation of state. To obtain the value of the anisotropy parameter at an arbitrary moment an evolution equation is derived, describing the dynamics of the anisotropy as a function of the volume scale factor of the universe. The general solution of this equation can be obtained in an exact analytical form for the Bianchi I and V types and in a closed form for all other homogeneous and anisotropic geometries. The study of the values of the anisotropy in the limit of small times shows that for all Bianchi-type spacetimes filled with a non-zero pressure cosmological fluid, obeying a linear barotropic equation of state, the initial singular state on the brane is isotropic. This result is obtained by assuming that in the limit of small times the asymptotic behaviour of the scale factors is of Kasner-type. For braneworlds filled with dust, the initial values of the anisotropy coincide in both braneworld and standard four-dimensional general relativistic cosmologies.

We cast the Reissner Nordström solution in a particular coordinate system which shows dynamical evolution from initial data. The initial data for the E < M case are regular. This procedure enables us to treat the metric as a collapse to a singularity. It also implies that one may assume Wald axioms to be valid globally in the Cauchy development, especially when Hadamard states are chosen. We can thus compare the semiclassical behaviour with the spherical dust case, looking upon the metric as well as state-specific information as evolution from initial data. We first recover the divergence on the Cauchy horizon obtained earlier. We point out that the semiclassical domain extends right up to the Cauchy horizon. This is different from the spherical dust case where the quantum gravity domain sets in before. We also find that the backreaction is not negligible near the central singularity, unlike the dust case. Apart from these differences, the Reissner Nordstrom solution has a similarity with dust in that it is stable over a considerable period of time. The features appearing in dust collapse mentioned above were suggested to be applicable within general spherical symmetry. The Reissner Nordstrom background (along with the quantum state) generated from initial data, is shown not to reproduce them.

Sum-over-histories quantization of particle-like theory in curved space is discussed. It is reviewed that the propagator and the related Green function satisfy the Schrödinger equation and wave equation with a Laplace-like operator, respectively. The exact dependence of the operator on the choice of measure is shown. Modifications needed for a manifold with a boundary are then introduced, and the exact form of the equation for the propagator is derived. It is shown that the Laplace-like operator contains some distributional terms localized on the boundary. These terms define boundary conditions for the propagator. Such a choice of boundary conditions is explained as a consequence of a measurement of particles on the boundary. Finally, the interaction with sources inside the domain and sources on the boundary are also discussed.

We study radial perturbations of general relativistic stars with elastic matter sources. We find that these perturbations are governed by a second-order differential equation which, along with the boundary conditions, defines a Sturm–Liouville-type problem that determines the eigenfrequencies. Although some complications arise compared to the perfect fluid case, leading us to consider a generalization of the standard form of the Sturm–Liouville equation, the main results of Sturm–Liouville theory remain unaltered. As an important consequence we conclude that the mass–radius curve for a one-parameter sequence of regular equilibrium models belonging to some particular equation of state can be used in the same well-known way as in the perfect fluid case, at least if the energy density and the tangential pressure of the background solutions are continuous. In particular, we find that the fundamental mode frequency has a zero for the maximum mass stars of the models with solid crusts considered in paper I of this series.

The global properties of static perfect-fluid cylinders and their external Levi-Civita fields are studied both analytically and numerically. The existence and uniqueness of global solutions is demonstrated for a fairly general equation of state of the fluid. In the case of a fluid admitting a non-vanishing density for zero pressure, it is shown that the cylinder's radius has to be finite. For incompressible fluid, the field equations are solved analytically for nearly Newtonian cylinders and numerically in fully relativistic situations. Various physical quantities such as proper and circumferential radii, external conicity parameter and masses per unit proper/coordinate length are exhibited graphically.

We look for the necessary conditions allowing the universe isotropization in the presence of a minimally coupled and massive scalar field with a perfect fluid. We conclude that it arises only when the universe is scalar field dominated, leading to flat spacelike sections and accelerated expansion, and examine the case of a SUGRA theory.

A conjectured connection to quantum gravity has led to a renewed interest in highly damped black-hole quasinormal modes (QNMs). In this paper we present simple derivations (based on the WKB approximation) of conditions that determine the asymptotic QNMs for both Schwarzschild and Reissner–Nordström black holes. This confirms recent results obtained by Motl and Neitzke, but our analysis fills several gaps left by their discussion. We study the Reissner–Nordström results in some detail, and show that, in contrast to the asymptotic QNMs of a Schwarzschild black hole, the Reissner–Nordström QNMs are typically not periodic in the imaginary part of the frequency. This leads to the charged black hole having peculiar properties which complicate an interpretation of the results in the context of quantum gravity.

Present knowledge of higher-derivative terms in string effective actions is, with a few exceptions, restricted to the NS–NS sector, a situation which prevents the development of a variety of interesting applications for which the RR terms are relevant. We here provide the formalism as well as efficient techniques to determine the latter directly from string-amplitude calculations. As an illustration of these methods, we compute the dependence of the type-IIB action on the 3- and 5-form RR field strengths at four-point, genus-one, order-(α')³ level. We explicitly verify that our results are in accord with the S-duality invariance of type-IIB string theory. Extensions of our method to other bosonic terms in the type-II effective actions are also discussed.

Static, spherically symmetric solutions with regular origin are investigated of the Einstein–Yang–Mills theory with a negative cosmological constant Λ. A combination of numerical and analytical methods leads to a clear picture of the 'moduli space' of the solutions. Some issues discussed in the existing literature on the subject are reconsidered and clarified. In particular the stability of the asymptotically AdS solutions is studied. Like for the Bartnik–McKinnon (BK) solutions obtained for Λ = 0 there are two different types of instabilities—'topological' and 'gravitational'. Regions with any number of these instabilities are identified in the moduli space. While for BK solutions there is always a non-vanishing equal number of instabilities of both types, this degeneracy is lifted and there exist stable solutions, genuine sphalerons with exactly one unstable mode and so on. The boundaries of these regions are determined.

In this paper we present a solution for the Kaluza–Klein magnetic monopole in a five-dimensional global monopole spacetime. This new solution is a generalization of the previous ones, obtained by Gross and Perry (1983 Nucl. Phys. B 226 29), containing a magnetic monopole in a Ricci-flat formalism, and by Banerjee et al (1996 Class. Quantum Grav.13 3141) for a global monopole in a five-dimensional spacetime, setting the specific integration constant equal to zero. Also we analyse the classical motion of a massive charged test particle on this manifold and present the equation for the classical trajectory obeyed by this particle.

We discuss the topology of the symmetry groups appearing in compactified (super)gravity, and discuss two applications. First, we demonstrate that for three-dimensional sigma models on a symmetric space G/H with G non-compact and H the maximal compact subgroup of G, the possibility of oxidation to a higher dimensional theory can immediately be deduced from the topology of H. Second, by comparing the actual symmetry groups appearing in maximal supergravities with the subgroups of and Spin(32), we argue that these groups cannot serve as a local symmetry group for M-theory in a formulation of de Wit–Nicolai type.

We introduce a novel type of phase diagram for black holes and black strings on cylinders. The phase diagram involves a new asymptotic quantity called the relative binding energy. We plot the uniform string and the non-uniform string solutions in this new phase diagram using Wiseman's data. Intersection rules for branches of solutions in the phase diagram are deduced from a new Smarr formula that we derive.