International Journal of Modern Physics A

This article provides a brief overview of recent research into the gravitational-wave driven instability of the r-modes in rotating neutron stars.

Contrary to everyday experience it is possible to construct models with more than three spatial dimensions which are compatible with linearized 4D Einstein gravity and homogeneous isotropic cosmologies. The simplest of these models have co-dimension one and are subjected to reflection symmetry. We discuss various extensions, in particular the inclusion of a scalar field and the relaxation of the assumption of reflection symmetry.

We briefly review recent work which established the existence of chaos in the generic cosmological solutions of the tree-level low-energy effective actions coming out of string theory, and linked this chaos to the Weyl groups of some hyperbolic Kac-Moody algebras.

I review briefly, primarily for relativists, a series of recent results, obtained with A. Waldron, on the novel behavior of massive higher (s > 1) spin systems in constant curvature backgrounds. We find that the cosmological constant Λ, together with the mass parameter, define a "phase plane" in which partially massless gauge invariant lines separate allowed regions from forbidden, non-unitary, ones. These lines represent short multiplet systems, with missing lower helicities, removed by novel local gauge invariances, and (despite having m ≠ 0) propagating on the light cone. In the limit of an infinite tower of these higher spin bosons and fermions, unitarity requires Λ to vanish.

This article considers the two-way relationship between local physics and the large scale structure of the universe - in particular considering Olber's paradox, Mach's principle, and the various arrows of time. Thus the focus is various ways in which local physics is influenced by the universe itself.

We discuss the mechanism of energy mining from black holes by cosmic strings and its efficiency.

It is now widely recognized that strong and weak gravitational lensing both play important roles in the observational cosmology. We shall here report our proposal and recent observation on the determination of the dark energy using strong gravitational lensing, and our recent analysis of weak shear field around Cl1604+4304 where we found mass concentration with no optical counterparts. Our recent weak shear observation by Subaru/S-Cam is also briefly mentioned.

Binary black hole systems in the pre-coalescence stage are numerically constructed by demanding that the associated spacetime admits a helical Killing vector. Comparison with third order post-Newtonian calculations indicates a rather good agreement until the innermost stable circular orbit.

Within the framework of generally covariant (pre-metric) electrodynamics, we specify a local vacuum spacetime relation between the excitation and the field strength F = (E,B). We study the propagation of electromagnetic waves in such a spacetime by Hadamard's method and arrive, with the constitutive tensor density κ ~ ∂H/∂F, at a Fresnel equation which is algebraic of 4th order in the wave covector. We determine how the different pieces of κ, in particular the axion and the skewon pieces, affect the propagation of light.

The aim of this contribution is to provide a short introduction to recently investigated models in which our accessible universe is a four-dimensional submanifold, or brane, embedded in a higher dimensional spacetime and ordinary matter is trapped in the brane. I focus here on the gravitational and cosmological aspects of such models with a single extra-dimension.

This paper gives a short review of the status and applications of computer algebra systems for calculations in relativity and gravitation.

Location-dependence of physical parameters such as the electromagnetic fine structure constant and Newton's G produce body accelerations which violate universality of free fall rates testable with laboratory and space experiments. Theoretically related cosmological time variation of these same parameters are also constrained by experiments such as lunar laser ranging, and these time variations produce accelerations of bodies relative to a preferred cosmological inertial frame.

The configuration space of general relativity is superspace, the space of Riemannian three-geometries and the Hamiltonian is just a sum of constraints, with Lagrange multipliers. One can go from this Hamiltonian, via a Legandre transformation, back and forth to the Lagrangian. The Lagrange multiplier (the lapse function) can be eliminated from the Lagrangian and one is left with an action which is a product of square roots. This is the Baierlein-Sharp-Wheeler action for general relativity. This action is unique in that all other square root actions are not self-consistent. This paper shows how to express this result in phase space. The only selfconsistent constrained Hamiltonian on superspace which is ultralocal and quadratic in the momenta in the ADM Hamiltonian for general relativity.

The study of acoustic black holes has been undertaken to provide new insights about the role of high frequencies in black hole evaporation. Because of the infinite gravitational redshift from the event horizon, Hawking quanta emerge from configurations which possessed ultra high (trans-Planckian) frequencies. Therefore Hawking radiation cannot be derived within the framework of a low energy effective theory; and in all derivations there are some assumptions concerning Planck scale physics. The analogy with condensed matter physics was thus introduced to see if the asymptotic properties of the Hawking phonons emitted by an acoustic black hole, namely stationarity and thermality, are sensitive to the high frequency physics which stems from the granular character of matter and which is governed by a non-linear dispersion relation. In 1995 Unruh showed that they are not sensitive in this respect, in spite of the fact that phonon propagation near the (acoustic) horizon drastically differs from that of photons. In 2000 the same analogy was used to establish the robustness of the spectrum of primordial density fluctuations in inflationary models. This analogy is currently stimulating research for experimenting Hawking radiation. Finally it could also be a useful guide for going beyond the semi-classical description of black hole evaporation.

Gamma-Ray Bursts (GRBs) are the most relativistic objects discovered so far. I describe here two aspects of the relativistic nature of GRBs. Their likely association with the formation of black holes and their possible role as sources of gravitational radiation.

A class of solutions to the Bianchi identities in linearized gravitational theory is given which includes fields which vanish asymptotically and are nowhere singular.

This paper is a short summary of the talk on Robinson manifolds and Cauchy–Riemann spaces given at Journées Relativistes 2001 in Dublin. It recalls two generalizations of the definition of shear-free congruences of null geodesics. These generalizations are equivalent in dimension 4, but not in higher dimensions, as illustrated on an example described by Ivor Robinson during that conference.

I review the main aspects of the simulated gravity that arises in brane world models in which the reflection symmetry is broken.

The use of lower-dimensional models is exploited in many areas of physics as a way to simplify the mathematical treatment of very complicated phenomena while, at the same time, retaining the main physical ingredients. For the case of black holes, the Hawking evaporation process¹ can be understood using a simple model where the gravity action is coupled to quantized matter in the form of free two-dimensional minimal scalar fields. In particular, the effective action one derives by straightforward integration of the trace anomaly² gives a stress energy tensor which in the Schwarzschild spacetime perfectly agrees with the results one gets by standard canonical quantization. Trying to improve the model, i.e. by employing a spherically reduced 4d minimal scalar field³, one faces a number of difficulties. Unphysical results obtained for the evaporation of black holes (such as antievaporation^3,4) using the anomaly induced effective action led us to perform a rigorous calculation of the quantum stress tensor in Schwarzschild by using the point-splitting regularization technique⁵: exact asymptotic results close to the horizon and at infinity have been derived in the three quantum states of interest (namely Boulware, Hartle-Hawking and Unruh) and two analytic approximations proposed , one valid for large r (based on the WKB) and the other being physically meaningful in the region close to the horizon. Finally, based on these results a "phenomenological" modification of the (unsatisfactory) anomaly induced effective action has been carried out⁶.

A general characterisation of an impulsive light–like signal was given^1,2. The signal may consist of a shell of null matter and/or an impulsive gravitational wave. Both parts of the signal can be unambiguously identified^3,4. The signals can be used to model bursts of gravitational radiation and light– like matter accompanying cataclysmic events such as supernovae and neutron star collisions. Also in high speed collisions of compact objects such as black–holes or neutron stars the gravitational fields of these objects resemble those of impulsive light–like signals when the objects are boosted to the speed of light. Several examples of impulsive light–like signals were presented, in particular those produced by recoil effects⁵ and by the Aichelburg–Sexl boost of an isolated gravitating multipole source⁶. The detection of these signals was also discussed⁷.

The singularity theorems of general relativity predict that gravitational collapse finally ends up in a spacetime singularity¹. The cosmic censorship hypothesis (CCH) states that such a singularity is covered by an event horizon². Despite much effort, there is no rigorous formulation or proof of the CCH. In view of this, examples that appear to violate the CCH and lead to naked singularities, in which non-spacelike curves can emerge, rather than black holes, are important to shed more light on the issue. We have studied several collapse scenarios which can lead to both situations³. In the case of the Vaidya-de Sitter spacetime⁴, we have shown that the naked singularities that arise are of the strong curvature type. Both types of singularities can also arise in higher dimensional Vaidya and Tolman-Bondi spacetimes, but black holes are favoured in some sense by the higher dimensions. The charged Vaidya-de Sitter spacetime also exhibits both types of singularities⁵.

A rank-r tensor on a Lorentzian manifold of dimension N is causal if the contraction with r arbitrary causal future-directed vectors is non-negative. General superenergy tensors¹, such as the Bel and Bel-Robinson tensors, are examples of even ranked causal tensors^1,2, and may therefore be useful when defining norms for geometric evolution equations³. We here show that any symmetric rank-2 causal tensor (energy-momentum tensors satisfying the dominant energy condition) can be written as a sum of at most N superenergy tensors of simple forms. If N=4 this can be expressed in an elegant way as the sum of four spinors squared. Since, for arbitrary N, the superenergy of any simple form is a self-map of the cone (its square is proportional to the metric) this leads to new representations and classifications of all conformal Lorentz transformations and to generalisations of the classical four dimensional Rainich-Misner-Wheeler (RMW) theory of determining the space-time physics from its geometry⁴. For non-simple forms more complicated equations are satisfied by the superenergy tensors, but also in this case we are able to generalise the RMW theory, and as an example the complete algebraic RMW theory in five dimensions is obtained⁵.

The quantization of gravity coupled to a perfect fluid model leads to a Schrödinger-like equation, where the matter variable plays the role of time. The wave function can be determined, in the flat case, for an arbitrary barotropic equation of state p = α ρ; solutions can also be found for the radiative non-flat case. The wave packets are constructed, from which the expectation value for the scale factor is determined. The quantum scenarios reveal a bouncing Universe, free from singularity. Such quantum cosmological perfect fluid models admit a universal classical analogue, represented by the addition, to the ordinary classical model, of a repulsive stiff matter fluid^1,2. The existence of this universal classical analogue may imply that this perfect fluid coupled to gravity model is not a real quantum system. The quantum cosmological perfect fluid model is, for a flat spatial section, formally equivalent to a free particle in ordinary quantum mechanics, for any value of α, while the radiative non-flat case is equivalent to the harmonic oscillator. The repulsive fluid needed to reproduce the quantum results is the same in both cases.

The exact quasinormal frequencies for scalar perturbations of the three-dimensional BTZ black hole were presented¹. The timescale for the decay of the scalar perturbation is given by the imaginary part of the quasinormal frequencies. Via the AdS/CFT correspondence, one then obtains a prediction of the timescale for return to equilibrium of the dual conformal field theory. By studying a two-particle collision process, an exact analytic expression for the Choptuik scaling parameter of the BTZ black hole was obtained². A possible connection between the Choptuik scaling parameter and the imaginary part of the quasinormal frequencies was explored¹.

The motion of the relativistic vortices has in general been studied in the context of a model based on non-linear wave equations of a complex scalar field. These studies have enlightened the theoretical similarities between superfluid vortices and global cosmic strings^1-5. In this work we take a purely hydrodynamical point of view⁶. By a weak deformation of the cylindrical symmetry of a potential vortex in a relativistic perfect isentropic fluid, we study the possible dynamics of the central line of this vortex. In "stiff" material the Nambu-Goto equations are obtained⁷.

We studied the quasi-normal modes (QNM) of electromagnetic and gravitational perturbations of a Schwarzschild black hole in an asymptotically anti-de Sitter (AdS) spacetime, extending previous works^1,2 on the subject. Some of the electromagnetic modes do not oscillate, they only decay, since they have pure imaginary frequencies. The gravitational modes show peculiar features: the odd and even gravitational perturbations no longer have the same characteristic quasinormal frequencies. There is a special mode for odd perturbations whose behavior differs completely from the usual one in scalar¹ and electromagnetic perturbation in an AdS spacetime, but has a similar behavior to the Schwarzschild black hole³ in an asymptotically flat spacetime: the imaginary part of the frequency goes as , where r₊ is the horizon radius. We also investigated the small black hole limit showing that the imaginary part of the frequency goes as . These results are important to the AdS/CFT⁴ conjecture since according to it the QNMs describe the approach to equilibrium in the conformal field theory. For other geometries see^5,6.

A possible origin of gamma-ray bursts is the gravitational collapse of compact objects. A semiclassical analysis, which employs the Born-Oppenheimer approximation in order to treat gravity as classical while matter is quantized¹, was first applied to homogeneous spheres of dust^2,3 and (thick) non-radiating shells ⁴. We later derived an effective Lagrangian for (thin) radiating shells⁵ which allowed us to study the evolution of a self-gravitating shell of bosonic matter coupled to a scalar radiation field. We showed that the matter can form a condensate and the matter degrees of freedom are excited, because of the non-adiabaticity of the collapse, at the expense of gravitational energy^4,6. Upon decaying back to the ground state, the excited matter energy is subsequently transferred to the radiation field. In fact, the system can convert large fractions of its gravitational energy into radiation and the time evolution of the radius of the shell is then significantly slowed down with respect to the non-radiating case. This provides a new mechanism to extract energy from collapsing matter which might play a role as a source of gamma-ray bursts.

A new family of euclidean Einstein metrics with self-dual Weyl tensor have been obtained using ideas from extended supersymmetries^1,2. The basic supersymmetric formalism³, known as harmonic superspace, was adapted to the computation of self-dual Einstein metrics in ⁴. The resulting metric depends on 4 parameters besides the Einstein constant and has for isometry group U(1) × U(1), with hypersurface generating Killing vectors. In the limit of vanishing Einstein constant we recover a family of hyperkähler metrics within the Multicentre family ⁵ (in fact the most general one with two centres). Our results include the metrics of Plebanski and Demianski⁶ when these ones are restricted to be self-dual Weyl. From Flaherty's equivalence ⁷ these metrics can also be interpreted as a solution of the coupled Einstein-Maxwell field equations, for which we have given the Maxwell field strength forms².

The dynamical properties of spatially homogeneous and isotropic cosmological models containing a barotropic perfect fluid and multiple scalar fields with independent exponential potentials is investigated. It is shown that the assisted inflationary scaling solution is the global late-time attractor for the parameter values for which the model is inflationary, even when curvature and barotropic matter are included. For all other parameter values the multi-field curvature scaling solution is the global late-time attractor (in these solutions the curvature is not dynamically negligible asymptotically). Consequently, in general all of the scalar fields in multi-field models with exponential potentials are non-negligible in late-time behaviour, contrary to what is commonly believed.

Starting with an exact and simple geodesic, we generate approximate geodesics^1,2 by summing up higher-order geodesic deviations in a fully relativistic scheme. We apply this method to the problem of orbit motion of test particles in Schwarzschild³ and Kerr metrics; from a simple circular orbit as the initial geodesic we obtain finite eccentricity orbits as a Taylor series with respect to the eccentricity. The explicit expressions of these higher-order geodesic deviations are derived using successive systems of linear equations with constant coefficients, whose solutions are of harmonic oscillator type. This scheme is best adapted for small eccentricities, but arbitrary values of M/R. We also analyse the possible application to the calculation of the emission of gravitational radiation from non-circular orbits around a very massive body³.

The invariant classification (Karlhede classification)¹ of a particular spacetime fixes (up to isotropy) an intrinsic tetrad for the spacetime, in terms of the Riemann tensor and its derivatives. Exploiting this tetrad, it has been shown how to determine the number of Killing vectors which exist ², and also whether a proper homothetic Killing vector exists ³. The first part of the talk outlined a procedure for finding explicitly homothetic and Killing vectors by exploiting the symmetry properties of the intrinsic tetrad ^4,5.

In general, it is not always possible to use this same tetrad to investigate proper conformal Killing vectors by the same methods. In this talk it was suggested to modify the invariant classification procedure so that only conformally invariant conditions are used for fixing the tetrad in terms of the Riemann tensor and its derivatives. This would mean that the existence of proper conformal Killing vectors could be deduced directly from the modified invariant classification scheme, and then found explicitly ⁶.

We compute the spectral index for scalar perturbations generated in a primordial inflationary model². In this model, the transition of the inflationary phase to the radiative era is achieved through the decay of the cosmological term leading a second order phase transition and the characteristics of the model allow to implement a set of initial conditions where the perturbations display a thermal spectrum when they emerge from the horizon.

The obtained value for the spectral index is equal to 2, a result that depends very weakly on the various parameters of the model and on the initial conditions used. This "negative" result pleads for the standard inflationary scenario, but of course cannot prove it and the problem of the interpretation of the inflaton remains open².

Long before the general theory of relativity was finally formulated in 1916, it was claimed that the following argument, based on (the strong) Einstein's equivalence principle (EP for short), predicted the well known and experimentally observed phenomenon of the gravitational red shift; precisely the same argument is being used to this day in almost all books on general relativity to derive the same phenomenon (see, for example, Refs. 1, 2):

Consider "Einstein's elevator" at rest on the Earth's surface with an emitter (E) fixed on the floor of the elevator, and a receiver (R) fixed on the ceiling directly above E and distance h from it. Let E send light signals, at frequency ν_E, to R and let the frequency at which they are received by R be ν_R. To find the relationship between ν_E and ν_R we invoke the EP and consider, instead, E and R as fixed in an elevator which is accelerating relative to an inertial frame S in gravitation-free space with constant acceleration g in the direction , g being the acceleration due to gravity on the Earth's surface. At time t = 0, when E is assumed to be at rest in S, E emits a light wave towards R. The time it takes the wave to reach R is roughly t = h/c, where c is the speed of light in S. But in this time R has acquired the velocity and, therefore, there is a consequent Doppler shift given by . By the EP the same result must hold when E and R are fixed near the Earth's surface. In this case gh = △Φ = Φ(R) - Φ(E), so that in the Earth's gravitational field we have , which is the standard formula for the gravitational red shift.

Simple and straight forward as the above argument may seem, we shall show in this lecture that it is, in fact, fundamentally flawed in two important respects. We shall present a new argument, entirely within the framework of classical mechanics (just as the above argument), which is free from these two flaws. Alas!, it leads to zero gravitational red shift for the case dealt with in the above argument. It is argued that this result not only does it not invalidate the general theory of relativity but it strengthens it; for, the full theory of general relativity alone, irrespective of its historical development, can correctly and unambiguously predict the observed gravitational red shift.

We consider the computation of the energy difference between spaces having the same asymptotic behavior. Following the example of the Schwarzschild case which asymptotically tends to flat space^1,2, we consider the case of a cosmological constant. The analysis is realized by means of variational methods in a Hamiltonian formulation and it is restricted to transverse-traceless (TT) tensors to one loop approximation. In particular, we consider the Schwarzschild-Anti-de Sitter (S-AdS) case which asymptotically tends to the Anti-de Sitter space and the Schwarzschild-de Sitter (SdS) case which asymptotically tends to the de Sitter space^3,4. In both cases (S-AdS, SdS), we discover the existence of an unstable mode at zero temperature. This result leads to consider a different vacuum space which differs in a case to case. The configuration is stabilized by allowing N copies of the same initial system to contribute to the final energy⁵. A selection rule for different scenarios of a foam-like space can be given in terms of transition frequencies of the emitted radiation of a black hole⁶.

The introduction of the warp drive metric by Alcubierre¹ has aroused great interest over the past few years. Using an uncertainty-type principle, Ford and Pfenning² proved that the warp drive transport of a spaceship in a regular bubble having a radius of 100 m is unrealistic. However, Van Den Broeck³ has shown that the situation largely improves when one uses a warp drive bubble with a small surface area and large spatial volume.

Putting aside many physics problems related to the realization of the warp drive concept, we show in this paper⁴ how to modify Van Den Broeck's idea to improve his results. We find new lower bounds for the warp drive energy by working on parameters whose latitude has never been considered before. We also consider micro warp drive bubbles which can be treated as physical entities of their own and could possibly be used to transmit information faster than the speed of light. The conditions prevailing just after the Big Bang allow the spontaneous formation of such micro bubbles which could still be present in our period of time.

It has been shown^1,2 that the usual 3+1 form of Einstein's equations may be ill-posed. This result has been previously observed in numerical simulations^3,4. We present a 3+1 type formalism inspired by these works to decompose Einstein's equations. This decomposition is motivated by the aim of stable numerical implementation and resolution of the equations. We introduce the conformal 3-"metric" (scaled by the determinant of the usual 3-metric) which is a tensor density of weight -2/3. The Einstein equations are then derived in terms of this "metric", of the conformal extrinsic curvature and in terms of the associated derivative. We also introduce a flat 3-metric (the asymptotic metric for isolated systems) and the associated derivative. Finally, the generalized Dirac gauge (introduced by Smarr and York⁵) is used in this formalism and some examples of formulation of Einstein's equations are shown.

The nature of an ordinary cosmic string¹ in the framework of a scalar-tensor extension of gravity is investigated. It is shown that in this case, the dilaton field can act as a timelike or a spacelike current traveling along the string, and localized into it². Apart from the fact that the current can not be formed after the string forming phase transition but exactly at the same time, since the dilaton is not an ordinary scalar field but a component of the gravitational interaction configuration would resemble very much the usual superconducting strings first proposed by Witten³. This means that a network of strings here produced would suffer from the vorton excess problem⁴, leading to an actual cosmological catastrophe from which one can derive strong constraints on the relevant theories.

In general relativity, only relative acceleration has an observer-independent meaning: curvature and non-gravitational forces determine the rate at which world lines of test bodies diverge or converge. We derive the equations governing this effect not only in the well-known case of point masses, but also for test bodies with charge and/or spin. The application of our method to the motion of particles in a central field results in an elegant, fully relativistic version of the Ptolemaean epicycle scheme.

The obstruction for the existence of an energy-momentum tensor for the gravitational field in GR is connected with vanishing of first order invariants in (pseudo) Riemannian geometry. This specific geometric property is not valid in alternative geometrical structures^1,2.

A parallelizable differentiable 4D-manifold endowed with a class of smooth coframe fields ϑ^a is considered. A general 3-parameter class of global Lorentz invariant teleparallel models is considered. It includes a 1-parameter subclass of models with the Schwarzschild coframe solution (generalized teleparallel equivalent of gravity) ³. By introducing the notion of a 3-parameter conjugate field strength F linear in the strength C^a = dϑ^a the coframe Lagrangian is rewritten in the Maxwell-Yang-Mills form L = 1/2F^a ∧ C_a. The field equation turns out to have a form d * F^a = T^a completely similar to the Maxwell field equation.

By applying the Noether procedure, the source 3-form T^a is shown to be connected with the diffeomorphism invariance of the Lagrangian. Thus the source T^a of the coframe field is interpreted as the total conserved energy-momentum current of the coframe field and matter⁴. The energy-momentum tensor is defined as a map of the module of current 3-forms into the module of vector fields ⁵. Thus an energy-momentum tensor for the coframe is defined in a diffeomorphism invariant and a translational covariant way. The total energy-momentum current of a system is conserved. Thus a redistribution of the energy-momentum current between material and coframe (gravity) field is possible in principle, unlike as in the standard GR.

The result is: The standard GR has a neighborhood of viable models with the same Schwarzschild solutions. These models however have a better Lagrangian behavior and produce an invariant energy-momentum tensor.

General Relativity is unique, among the class of field theories, in the treatment of the equations of motion. The equations of motion of massive particles are completely determined by the field equation ^1-5. Einstein's field equations, as well as most field equations in gravity theory, have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order, with coefficients that depend on the variables.

We propose for the N-body problem of equations that are Lorentz invariant a novel algorithm for the derivation of the equations of motion from the field equations. It is:

(1) Compute a static, spherically symmetric solution of the field equation. It will be singular at the origin. This will be taken to be the field generated by a single particle.

(2) Move the solution on a trajectory P and apply the instantaneous Lorentz transformation based on instantaneous velocity.

(3) Take, as first approximation, the field generated by N particles to be the superposition of the fields generated by the single particles.

(4) Compute the leading part of the equation. Hopefully, only terms that involve the acceleration will be dominant. This is the "inertial" part.

(5) Compute the quadratic part of the equation. This is the agent of the "force".

(6) Equate for each singularity, the highest order terms of the singularities that came from the linear part and the quadratic parts, respectively. This is an equation between the inertial part and the force.

The algorithm was applied to Einstein's equations. The approximate evolution of the scalar curvature leads, in turn, to an invariant scalar equation. The algorithm for it did produce Newton's law of gravitation. This is, also, the starting point for the embedding of the trajectories in a common field.

In the context of the AdS/CFT conjecture¹, a Schwarzschild-anti-de Sitter (SAdS) black hole may be looked at as a thermal state in the CFT. Perturbing the black hole corresponds in the CFT to perturb the thermal state. We considered an important specific perturbation - the radial infall of a small test particle coupled to a scalar field into a SAdS black hole. We computed the spectra, waveforms and total scalar energy radiated during this process. For small black holes, the spectra is dominated by a resonance, and the waveform by quasinormal ringing^2,3,4. For large black holes we find that the waveform quickly settles down to its final zero value, always in a quasinormal way. The approach to thermal equilibrium in the CFT is therefore dictated by the lowest quasinormal frequency. We also commented on the interpretation of the bulk process when viewed from the brane: to the black hole corresponds a thermal bath, to the infalling probe corresponds an expanding bubble, and to the scalar field waves correspond particles decaying into bosons of the associate operator of the gauge theory. For more details see^5,6.

Black holes in string theories can be studied by means of effective gravity-matter actions. These contain, among others, non-minimally coupled scalar fields, as the dilaton and moduli arising from the compactification of the extra dimensions¹. The presence of non-minimal couplings prevents the application of the well known no-hair theorems², so that it is possible to find regular black hole solutions with scalar hair. Solutions with non-trivial dilaton hair are known for Maxwell³ and Gauss-Bonnet⁴ black holes. In these cases the dilaton charge is a function of the other parameters of the solutions (secondary hair). However, it has recently been shown that when moduli fields are taken into account as well, one can obtain solutions where the scalar charges of the moduli are independent parameters (primary scalar hair)^5,6. The thermodynamics of these black holes has been investigated. In particular, in the Maxwell case a condition for extremality has been obtained, which sets a lower bound for the mass in terms of the magnetic and scalar charges⁷. Similarly in the Gauss-Bonnet case, a minimal allowed value for the mass has been found, which depends only on the scalar charges and can be interpreted as a ground state for the Hawking evaporation process of the black hole⁶.

We revisit the problem of the development of singularities in the gravitational collapse of an inhomogeneous dust sphere. As shown by Yodzis et al¹, naked singularities may occur at finite radius where shells of dust cross one another. These singularities are gravitationally weak ², and it has been claimed that at these singularities, the metric may be written in continuous form ², with locally L^∞ connection coefficients ³. We correct these claims, and show how the field equations may be reformulated as a first order, quasi-linear, non-conservative, non-strictly hyperbolic system. We discuss existence and uniqueness of generalized solutions of this system using bounded functions of bounded variation (BV) ⁴, where the product of a BV function and the derivative of another BV function may be interpreted as a locally finite measure. The solutions obtained provide a dynamical extension to the future of the singularity.

The equivalence problem for the third order ODEs solved by E. Cartan¹ and S. S. Chern² is reconsidered. We consider third order ODEs of the form y′′′ = F(x,y,y′,y′′) for which the Wunshman invariant I vanishes. All such ODEs split into equivalence classes with respect to the contact transformations of the variables. As shown by E. T. Newman³ and collaborators such equations are also in one-to-one correspondence with conformal classes of Lorentian three-metrics. We supplement Cartan-Chern-Newman results by providing explicit expressions for all the contact invariants of an ODE with I = 0. The invariants are explicitly written in terms of the function F and its partial derivatives. Explicit expression for the associated Cartan's O(2,3) connection is also given. The curvature of this conformal connection is reinterpreted in terms of the Cotton-York tensor of the Lorentzian three-metric associated with the equation.

Boost-rotation symmetric spacetimes are the only axially symmetric vacuum spacetimes with an additional symmetry that are radiative and admit global null infinity^1,2. They are the only radiative asymptotically flat exact solutions representing the gravitational field of moving objects – uniformly accelerated "particles" of various kinds³. The general theory of boost-rotation symmetric spacetimes was first developed for nonspinning sources⁴ (i.e. for hypersurface orthogonal Killing vectors) and later partially generalized for spinning sources⁵.

The C-metric³ and the spinning C-metric^6,7,8 belonging to this class were analyzed. With appropriately chosen parameters they represent the gravitational field of uniformly accelerated Schwarzschild and Kerr black or white holes. Stability of geodesics describing co-accelerated particles was also studied using an effective potential⁹ and a Lyapunov function⁸.

A non-standard cosmological model in Weyl-Cartan spacetime was presented¹. Within this model the non-Riemannian quantities, i.e. torsion T^α and nonmetricity Q_αβ, are proportional to the Weyl 1-form. The hypermomentum Δ_αβ depends on our ansatz for the nonmetricity and vice versa. In contrast to earlier investigations² this model incorporates an additional quadratic nonmetricity term in the gauge Lagrangian, the Lagrangian under consideration reads

A Yukawa coupling of fermions to the string forming Higgs field leads to the existence of massless¹ and massive^2,3 fermionic bound states in the vortex that can drastically modify the string dynamics^3,4. An exhaustive study of these states is performed. In particular, it is shown that there exists transitions between subsonic and supersonic regimes (i.e. regimes in which the transverse perturbation propagation velocity is less or greater than the longitudinal, sound-like one, respectively), which appear as soon as the massless and massive modes are respectively filled⁵. This results in modifying the classical stability of cosmic string loops. On the other hand, this confinement mechanism can also be transposed to the the brane context as a viable way for localizing fermions on the brane⁶.

In the last few years, brane cosmological models with extra dimensions have been actively studied. In these scenarios, ordinary matter is confined in a (3+1)-dimensional surface (the world-sheet of a 3-brane) while gravity lives in the bulk space-time^2,3. In this regard, it has been shown that the cosmology of such models is characterized by a non standard Friedmann equation^4,5. Some analytic solutions have been found when the scalar field couples to the brane⁶, but they correspond to static solutions where the cosmological evolution is due to the motion of the brane into the bulk. In this talk I have presented two kinds of exact solutions obtained by solving Einstein's equations in the light-cone coordinate system and making two different assumptions : a. proportionality between the scalar field and the logarithm of the bulk scale factor⁷ and b. separable solutions. It has been shown that the first kind of solutions correspond to moving branes into static backgrounds : somehow the hypothesis of proportionality froze the evolution of the metric. On the contrary, in the second kind of solution the bulk is really dynamic. The induced cosmology in the brane has been explored for both types of solutions, confirming, in particular, the non standard cosmological evolution of the scale factor.

The mathematical properties of causal tensors,¹ which are the tensors satisfying the dominant property (generalization of the dominant energy condition), were presented. The whole class of the so-called superenergy tensors², including in particular the Bel-Robinson tensor, are causal tensors. Conversely, the superenergy tensors are the basic building blocks of all causal tensors¹. Actually, the superenergy tensors of simple forms are intimately related to conformally Lorentz transformations as they leave the null cone invariant.¹ This provides, in particular, a wide generalization of the algebraic Rainich conditions¹ in arbitrary dimension and for many different fields. See ^1,2 and references therein for details.

The study of singular spacetimes by distributional methods faces the fundamental obstacle of the inherent nonlinearity of the field equations. Staying strictly within the distributional (in particular: linear) regime, as determined by Geroch and Traschen² excludes a number of physically interesting examples (e.g., cosmic strings). In recent years, several authors have therefore employed nonlinear theories of generalized functions (Colombeau algebras, in particular) to tackle general relativistic problems^1,5,8.

Under the influence of these applications in general relativity the nonlinear theory of generalized functions itself has undergone a rapid development lately, resulting in a diffeomorphism invariant global theory of nonlinear generalized functions on manifolds^3,4,6. In particular, a generalized pseudo-Riemannian geometry allowing for a rigorous treatment of generalized (distributional) spacetime metrics has been developed⁷. It is the purpose of this talk to present these new mathematical methods themselves as well as a number of applications in mathematical relativity.

Presently, there exists some renewed interest in time varying speed of light theories as possible solutions of the major cosmological problems¹. It is often believed that the local Lorentzian invariance is broken if the speed of light in a vacuum is not a constant. We point out that this belief is not necessarily founded and that a variable speed of light is perfectly consistent with general relativity under the assumption of non-minimal coupling between electromagnetism and curvature. Two kinds of arguments may be invoked in favour of such an assumption. First, a theorem due to Horndeski² shows that in a four-dimensional space-time the Einstein-Maxwell field equations are not the only second-order vector potential field equations which stem from a Lagrangian scalar density, are consistent with the charge conservation and reduce to Maxwell's equations in a flat space-time (see also³). Second, according to QED^4,5, vacuum polarization induces tidal gravitational effects which imply that photons propagating in a curved space-time have velocities exceeding the value of the "Lorentzian structural constant" c. The modified electromagnetic field equations given by Horndeski² are studied here in the geometrical optics limit. Considering the case of Friedmann-Robertson-Walker cosmological models, we find the value of the speed of light as a function of the energetic content of the universe. We deduce from this result a new equation of state for a photon gas and we discuss the consequences of this equation on the evolution of the scale factor during the radiation-dominated era.

Inspired by the central equation of canonical quantum gravity, the timeless Wheeler-DeWitt equation, we investigate timeless properties for quantum cosmological models. In particular we are interested in the following question: What is the probability of the system entering certain regions of its configuration space without reference to time at all? For the classical case this is a property of the entire system trajectories and can be formulated using a phase space probability distribution ¹. In the quantum case we propose and investigate a probability which we obtain by adapting the decoherent histories approach to quantum theory ² to the timeless case and by using the induced inner product on the physical state space of quantum cosmological models ³. For the example of the Klein-Gordon equation we have investigated the case of space–like surfaces and our results agree with the standard operator methods ⁴. For a more general model we consider a semiclassical approximation as well as the effect of an environment ¹. When the histories are decoherent, the probabilities approximately coincide with the classical case, with the phase space probability distribution replaced by the Wigner function of the quantum state. The influence functional of the environment respects the reparametrization invariance of the quantum cosmological Wheeler-DeWitt equation.

According to the Cosmic Censorship hypothesis realistic singularities should be hidden by an event horizon. However there are many examples of physically realistic space–times which are geodesically incomplete, and hence possess singularities according to the usual definition, which are not inside an event horizon. Many of these counterexamples to the cosmic censorship conjecture have a curvature tensor which is reasonably behaved (for example bounded or integrable) as one approaches the singularity. We give a class of weak singularities which may be described as having distributional curvature¹. Because of the non–linear nature of Einstein's equations such distributional geometries are described using a diffeomorphism invariant theory of non–linear generalised functions². We also investigate the propagation of test fields on space–times with weak singularities. We give a class of singularities^3,4 which do not disrupt the Cauchy development of test fields and result in space–times which satisfy Clarke's criterion of 'generalised hyperbolicity'. We consider that points which are well behaved in this way, and where Einstein's equations make sense distributionally, should be regarded as interior points of the space–time rather than counterexamples to cosmic censorship.

A numerical and linear study of modes driven toward instability by PN radiation-reaction force (current quadrupole) in slowly rotating neutron stars was presented. It consists in a time evolution using spectral methods in spherical coordinates for spatial operators. Whatever the noisy initial data, there exists a hydrodynamic instability. Yet, depending on the background, the symmetric properties of the mode may change. Thus, in a rigid rotating Newtonian star, the expected and purely axial r-mode¹ is growing. But, when the Newtonian background is assumed to be differentially rotating or when the evolution is done in the framework of general relativity with Cowling approximation (frozen spacetime), a coupling between axial and polar modes appears. Then, the mode driven to instability no longer belongs to one of these subclasses. Preliminary results were presented, showing the common features and main differences between Newtonian and relativistic cases².

We compute the density of scalar and Dirac particles created by a cosmological anisotropic universe^1,2 in the presence of a time dependent homogeneous electric field. In order to compute the rate of particles created we apply a quasiclassical approach that has been used successfully in different scenarios^3,4. The idea behind the method is the following: First, we solve the relativistic Hamilton-Jacobi equation and, looking at its solutions, we identify positive and negative frequency modes. Second, after separating variables^5,6, we solve the Klein-Gordon and Dirac equations and, after comparing with the results obtained for the quasiclassical limit, we identify the positive and negative frequency states. We show that the particle distribution becomes thermal when one neglects the electric interaction.

Since the formulation of the AdS/CFT correspondence ¹, there has been great interest in space-times which are asymptotically anti-de Sitter, and the properties of the Kerr-Newman-anti-de Sitter (KN-AdS) space-time in various dimensions have been extensively studied ². However, the properties of classical or quantum fields propagating on this background have not been widely studied, and, in particular, the question of whether super-radiance occurs has not been addressed. This is an important issue since a detailed understanding of classical super-radiance is necessary before tackling quantum field theory on rotating black hole geometries ³.

We considered a classical scalar field on the KN-AdS background ⁴, and examined the form of the separated field modes. Given the structure of infinity in asymptotically anti-de Sitter space-times, we paid particular attention to the boundary conditions at infinity.

Unlike the situation for asymptotically flat Kerr-Newman black holes ⁵, super-radiance is not inevitable. It depends partly on our choice of boundary condition at infinity. For reflective boundary conditions at infinity, there is no super-radiance. On the other hand, if we consider transparent boundary conditions at infinity, then the presence of super-radiance depends on our choice of positive frequency. For those KN-AdS black holes possessing a globally time-like Killing vector, then the natural definition of positive frequency implies that there are no super-radiant modes. For other KN-AdS black holes, then this same definition of positive frequency again leads to no super-radiance.