Showing papers in "Classical and Quantum Gravity in 2002"

Lecture notes on holographic renormalization

Abstract: We review the formalism of holographic renormalization. We start by discussing mathematical results on asymptotically anti-de Sitter (AdS) spacetimes. We then outline the general method of holographic renormalization. The method is illustrated by working all details in a simple example: a massive scalar field on anti-de Sitter spacetime. The discussion includes the derivation of the on-shell renormalized action, holographic Ward identities, anomalies and renormalization group (RG) equations, and the computation of renormalized one-, two- and four-point functions. We then discuss the application of the method to holographic RG flows. We also show that the results of the near-boundary analysis of asymptotically AdS spacetimes can be analytically continued to apply to asymptotically de Sitter spacetimes. In particular, it is shown that the Brown–York stress energy tensor of de Sitter spacetime is equal, up to a dimension-dependent sign, to the Brown–York stress energy tensor of an associated AdS spacetime.

Penrose limits and maximal supersymmetry

Abstract: We show that the maximally supersymmetric pp-wave of IIB superstring and M-theories can be obtained as a Penrose limit of the supersymmetric AdS × S solutions. In addition, we find that in a certain large tension limit, the geometry seen by a brane probe in an AdS × S background is either Minkowski space or a maximally supersymmetric pp-wave.

Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes

Abstract: A general formalism for understanding the thermodynamics of horizons in spherically symmetric spacetimes is developed. The formalism reproduces known results in the case of black-hole spacetimes and can handle more general situations such as: (i) spacetimes which are not asymptotically flat (such as the de Sitter spacetime) and (ii) spacetimes with multiple horizons having different temperatures (such as the Schwarzschild–de Sitter spacetime) and provide a consistent interpretation for temperature, entropy and energy. I show that it is possible to write Einstein's equations for a spherically symmetric spacetime in the form T dS − dE = P dV near any horizon of radius a with S equal; 1/4(4πa2), |E| = (a/2) and the temperature T determined from the surface gravity at the horizon. The pressure P is provided by the source of Einstein's equations and dV is the change in the volume when the horizon is displaced infinitesimally. The same results can be obtained by evaluating the quantum mechanical partition function without using Einstein's equations or the WKB approximation for the action. Both the classical and quantum analyses provide a simple and consistent interpretation of entropy and energy for de Sitter spacetime as well as for (1 + 2) dimensional gravity. For the Rindler spacetime the entropy per unit transverse area turns out to be 1/4 while the energy is zero. The approach also shows that the de Sitter horizon—like the Schwarzschild horizon—is effectively one dimensional as far as the flow of information is concerned, while the Schwarzschild–de Sitter, Reissner–Nordstrom horizons are not. The implications for spacetimes with multiple horizons are discussed.

General logarithmic corrections to black-hole entropy

Abstract: We compute leading-order corrections to the entropy of any thermodynamic system due to small statistical fluctuations around equilibrium. When applied to black holes, these corrections are shown to be of the form −k ln(Area). For BTZ black holes, k = 3/2, as found earlier. We extend the result to anti-de Sitter Schwarzschild and Reissner–Nordstrom black holes in arbitrary dimensions. Finally we examine the role of conformal field theory in black-hole entropy and its corrections.

Penrose limits, supergravity and brane dynamics

Abstract: We investigate the Penrose limits of classical string and M-theory backgrounds. We prove that the number of (super)symmetries of a supergravity background never decreases in the limit. We classify all the possible Penrose limits of AdS ? S spacetimes and of supergravity brane solutions. We also present the Penrose limits of various other solutions: intersecting branes, supersymmetric black holes and strings in diverse dimensions, and cosmological models. We explore the Penrose limit of an isometrically embedded spacetime and find a generalization to spaces with more than one time. Finally, we show that the Penrose limit is a large tension limit for all branes including those with fields of Born?Infeld type.

Super-Acceleration from Massless, Minimally Coupled $\phi^4$

Abstract: We derive a simple form for the propagator of a massless, minimally coupled scalar in a locally de Sitter geometry of arbitrary spacetime dimension. We then employ it to compute the fully renormalized stress tensor at one- and two-loop orders for a massless, minimally coupled 4 theory which is released in Bunch?Davies vacuum at t = 0 in co-moving coordinates. In this system, the uncertainty principle elevates the scalar above the minimum of its potential, resulting in a phase of super-acceleration. With the non-derivative self-interaction the scalar's breaking of de Sitter invariance becomes observable. It is also worth noting that the weak-energy condition is violated on cosmological scales. An interesting subsidiary result is that cancelling overlapping divergences in the stress tensor requires a conformal counterterm which has no effect on purely scalar diagrams.

Hawking radiation in different coordinate settings: Complex paths approach

Abstract: We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild spacetime. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semiclassical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation as far as these coordinates are concerned.

The cosmological constant problem and quintessence

Abstract: I briefly review the cosmological constant problem and the issue of dark energy (or quintessence). Within the framework of quantum field theory, the vacuum expectation value of the energy momentum tensor formally diverges as k4. A cutoff at the Planck or electroweak scale leads to a cosmological constant which is, respectively, 10123 or 1055 times larger than the observed value, Λ/8πG 10−47 GeV4. The absence of a fundamental symmetry which could set the value of Λ to either zero or a very small value leads to the cosmological constant problem. Most cosmological scenarios favour a large time-dependent Λ-term in the past (in order to generate inflation at z 1010), and a small Λ-term today, to account for the current acceleration of the universe at z 1. Constraints arising from cosmological nucleosynthesis, CMB and structure formation constrain Λ to be sub-dominant during most of the intermediate epoch 1010 < z < 1. This leads to the cosmic coincidence conundrum which suggests that the acceleration of the universe is a recent phenomenon and that we live during a special epoch when the density in Λ and in matter are almost equal. Time varying models of dark energy can, to a certain extent, ameliorate the fine-tuning problem (faced by Λ), but do not resolve the puzzle of cosmic coincidence. I briefly review tracker models of dark energy, as well as more recent brane inspired ideas and the issue of horizons in an accelerating universe. Model independent methods which reconstruct the cosmic equation of state from supernova observations are also assessed. Finally, a new diagnostic of dark energy—statefinder— is discussed.

Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings

Abstract: We report on thermal noise from the internal friction of dielectric coatings made from alternating layers of Ta2O5 and SiO2 deposited on fused silica substrates. We present calculations of the thermal noise in gravitational wave interferometers due to optical coatings, when the material properties of the coating are different from those of the substrate and the mechanical loss angle in the coating is anisotropic. The loss angle in the coatings for strains parallel to the substrate surface was determined from ringdown experiments. We measured the mechanical quality factor of three fused silica samples with coatings deposited on them. The loss angle, ||(f), of the coating material for strains parallel to the coated surface was found to be 4.2 ± 0.3 × 10−4 for coatings deposited on commercially polished slides, and 1.0 ± 0.3 × 10−4 for a coating deposited on a superpolished disc. Using these numbers, we estimate the effect of coatings on thermal noise in the initial LIGO and Advanced LIGO interferometers. We also find that the corresponding prediction for thermal noise in the 40 m LIGO prototype at Caltech is consistent with the noise data. These results are complemented by results for a different type of coating, presented in a companion paper.

Lectures on string/brane cosmology

Abstract: An overview of some cosmological aspects of string theory is presented. Recent developments are emphasized, especially the attempts to derive inflation or alternatives to inflation from the dynamics of branes in string theory. Time-dependent backgrounds with potential cosmological implications, such as those provided by negative tension branes and S-branes and the rolling string tachyon are also discussed.

Scalar perturbations during multiple-field slow-roll inflation

Abstract: We calculate the scalar gravitational and matter perturbations in the context of slow-roll inflation with multiple scalar fields, that take values on a (curved) manifold, to first order in slow roll. For this purpose a basis for these perturbations determined by the background dynamics is introduced and multiple-field slow-roll functions are defined. To obtain analytical solutions to first order, the scalar perturbation modes have to be treated in three different regimes. Matching is performed by identifying leading order asymptotic expansions analytically in different regions. The possible sources for multiple-field effects in the gravitational potential are the particular solution caused by the coupling to the field perturbation perpendicular to the field velocity, and the rotation of the basis. The former can contribute even to leading order if the corresponding multiple-field slow-roll function is sizable during the last 60 e-folds. Making some simplifying assumptions, the evolution of adiabatic and isocurvature perturbations after inflation is discussed. The analytical results are illustrated and checked numerically with the example of a quadratic potential.

The GEO 600 gravitational wave detector

Abstract: The GEO 600 laser interferometer with 600 m armlength is part of a worldwide network of gravitational wave detectors. Due to the use of advanced technologies like multiple pendulum suspensions with a monolithic last stage and signal recycling, the anticipated sensitivity of GEO 600 is close to the initial sensitivity of detectors with several kilometres armlength. This paper describes the subsystems of GEO 600, the status of the detector by September 2001 and the plans towards the first science run.

Isotropic loop quantum cosmology

Abstract: Isotropic models in loop quantum cosmology allow explicit calculations, thanks largely to a completely known volume spectrum, which is exploited in order to write down the evolution equation in a discrete internal time. Because of genuinely quantum geometrical effects, the classical singularity is absent in those models in the sense that the evolution does not break down there, contrary to the classical situation where spacetime is inextendible. This effect is generic and does not depend on matter violating energy conditions, but it does depend on the factor ordering of the Hamiltonian constraint. Furthermore, it is shown that loop quantum cosmology reproduces standard quantum cosmology and hence (e.g., via WKB approximation) classical behaviour in the large volume regime where the discreteness of space is insignificant. Finally, an explicit solution to the Euclidean vacuum constraint is discussed which is the unique solution with semiclassical behaviour representing quantum Euclidean space.

General Gauss-Bonnet brane cosmology

Abstract: We consider 5-dimensional spacetimes of constant 3-dimensional spatial curvature in the presence of a bulk cosmological constant. We find the general solution of such a configuration in the presence of a Gauss-Bonnet term. Two classes of non-trivial bulk solutions are found. The first class is valid only under a fine tuning relation between the Gauss-Bonnet coupling constant and the cosmological constant of the bulk spacetime. The second class of solutions are static and are the extensions of the AdS-Schwarzchild black holes. Hence in the absence of a cosmological constant or if the fine tuning relation is not true, the generalised Birkhoff's staticity theorem holds even in the presence of Gauss-Bonnet curvature terms. We examine the consequences in brane world cosmology obtaining the generalised Friedmann equations for a perfect fluid 3-brane and discuss how this modifies the usual scenario.

On non-uniform black branes

Abstract: Certain black branes are unstable towards fluctuations that lead to non-uniform mass distributions. We study static, non-uniform solutions that differ only perturbatively from uniform ones. For uncharged black strings in five dimensions, we find evidence of a first-order transition from uniform to non-uniform solutions.

The cosmological term as a source of mass

Abstract: In the spherically symmetric case, the dominant energy condition, together with the requirement of regularity at the centre, asymptotic flatness and finiteness of the ADM mass, defines the family of asymptotically flat globally regular solutions to the Einstein equations which includes the class of metrics asymptotically de Sitter as r ? 0. The source term corresponds to an r-dependent cosmological term ??? invariant under boosts in the radial direction and evolving from the de Sitter vacuum ?g?? in the origin to the Minkowski vacuum at infinity. The ADM mass is related to a cosmological term by m = (2G)?1?0??ttr2 dr, with the de Sitter vacuum replacing a central singularity at the scale of symmetry restoration. Spacetime symmetry changes smoothly from the de Sitter group near the centre to the Lorentz group at infinity through radial boosts in between. In the range of masses m ? mcrit, de Sitter?Schwarzschild geometry describes a vacuum nonsingular black hole (?BH), and for m < mcrit, it describes a G-lump?a vacuum self-gravitating particle-like structure without horizons. The quantum energy spectrum of the G-lump is shifted down by the binding energy and zero-point vacuum mode is fixed at the value corresponding (up to the coefficient) to the Hawking temperature from the de Sitter horizon.

Geometry of Generic Isolated Horizons

Abstract: Geometrical structures intrinsic to non-expanding, weakly-isolated and isolated horizons are analysed and compared with structures which arise in other contexts within general relativity, e.g. at null infinity. In particular, we address in detail the issue of singling out the preferred normals to these horizons required in various applications. This study provides powerful tools to extract invariant, physical information from numerical simulations of the near-horizon, strong-field geometry. While it complements the previous analysis of laws governing the mechanics of weakly-isolated horizons, prior knowledge of those results is not assumed.

Is quantum Einstein gravity nonperturbatively renormalizable

Abstract: We find considerable evidence supporting the conjecture that four-dimensional quantum Einstein gravity is 'asymptotically safe' in Weinberg's sense. This would mean that the theory is likely to be nonperturbatively renormalizable and thus could be considered a fundamental (rather than merely effective) theory which is mathematically consistent and predictive down to arbitrarily small length scales. For a truncated version of the exact flow equation of the effective average action, we establish the existence of a non-Gaussian renormalization group fixed point which is suitable for the construction of a nonperturbative infinite cut-off limit. The truncation ansatz includes the Einstein–Hilbert action and a higher derivative term.

Celestial mechanics in Kerr spacetime

Abstract: The dynamical parameters conventionally used to specify the orbit of a test particle in Kerr spacetime are the energy E, the axial component of the angular momentum, Lz, and Carter's constant Q. These parameters are obtained by solving the Hamilton–Jacobi equation for the dynamical problem of geodesic motion. Employing the action-angle variable formalism, on the other hand, yields a different set of constants of motion, namely, the fundamental frequencies ωr, ωθ and ω associated with the radial, polar and azimuthal components of orbital motion, respectively. These frequencies, naturally, determine the time scales of orbital motion and, furthermore, the instantaneous gravitational wave spectrum in the adiabatic approximation. In this paper, it is shown that the fundamental frequencies are geometric invariants and explicit formulae in terms of quadratures are derived. The numerical evaluation of these formulae in the case of a rapidly rotating black hole illustrates the behaviour of the fundamental frequencies as orbital parameters, such as the semi-latus rectum p, the eccentricity e or the inclination parameter θ− are varied. The limiting cases of circular, equatorial and Keplerian motion are investigated as well and it is shown that known results are recovered from the general formulae.

All spacetimes with vanishing curvature invariants

Abstract: All Lorentzian spacetimes with vanishing invariants constructed from the Riemann tensor and its covariant derivatives are determined. A subclass of the Kundt spacetimes results and we display the corresponding metrics in local coordinates. Some potential applications of these spacetimes are discussed.

Relativity without relativity

Abstract: We give a derivation of general relativity (GR) and the gauge principle that is novel in presupposing neither spacetime nor the relativity principle. We consider a class of actions defined on superspace (the space of Riemannian 3-geometries on a given bare manifold). It has two key properties. The first is symmetry under 3-diffeomorphisms. This is the only postulated symmetry, and it leads to a constraint linear in the canonical momenta. The second property is that the Lagrangian is constructed from a 'local' square root of an expression quadratic in the velocities. The square root is 'local' because it is taken before integration over 3-space. It gives rise to quadratic constraints that do not correspond to any symmetry and are not, in general, propagated by the Euler–Lagrange equations. Therefore these actions are internally inconsistent. However, one action of this form is well behaved: the Baierlein–Sharp–Wheeler (Baierlein R F, Sharp D and Wheeler J A 1962 Phys. Rev. 126 1864) reparametrization-invariant action for GR. From this viewpoint, spacetime symmetry is emergent. It appears as a 'hidden' symmetry in the (underdetermined) solutions of the Euler–Lagrange equations, without being manifestly coded into the action itself. In addition, propagation of the linear diffeomorphism constraint together with the quadratic square-root constraint acts as a striking selection mechanism beyond pure gravity. If a scalar field is included in the configuration space, it must have the same characteristic speed as gravity. Thus Einstein causality emerges. Finally, self-consistency requires that any 3-vector field must satisfy Einstein causality, the equivalence principle and, in addition, the Gauss constraint. Therefore we recover the standard (massless) Maxwell equations.

New cosmological singularities in braneworld models

Abstract: Higher-dimensional braneworld models which contain both bulk and brane curvature terms in the action admit cosmological singularities of rather unusual form and nature. These 'quiescent' singularities, which can occur both during the contracting as well as the expanding phase, are characterized by the fact that while the matter density and Hubble parameter remain finite, all higher derivatives of the scale factor ( etc) diverge as the cosmological singularity is approached. The singularities are the result of the embedding of the (3 + 1)-dimensional brane in the bulk and can exist even in an empty homogeneous and isotropic (FRW) universe. The possibility that the present universe may expand into a singular state is discussed.

Gravitational quasinormal modes for anti-de Sitter black holes

Abstract: The quasinormal mode spectra for gravitational perturbations of black holes in four-dimensional de Sitter and anti-de Sitter space are investigated. The de Sitter case lends itself to approximation by a Poshcl–Teller potential. The anti-de Sitter case is relevant to the AdS-CFT correspondence in superstring theory. The ADS-CFT correspondence suggests a preferred set of boundary conditions.

Cosmology with positive and negative exponential potentials

Abstract: We present a phase-plane analysis of cosmologies containing a scalar field with an exponential potential V ∝ exp(−λ κ ) where κ2 = 8πG and V may be positive or negative. We show that power-law kinetic–potential scaling solutions only exist for sufficiently flat (λ2 6) negative potentials. The latter correspond to a class of ever-expanding cosmologies with negative potential. However, we show that these expanding solutions with a negative potential are unstable in the presence of ordinary matter, spatial curvature or anisotropic shear, and generic solutions always recollapse to a singularity. Power-law kinetic–potential scaling solutions are the late-time attractor in a collapsing universe for steep negative potentials (the ekpyrotic scenario) and stable against matter, curvature or shear perturbations. Otherwise kinetic-dominated solutions are the attractor during collapse (the pre-big-bang scenario) and are only marginally stable with respect to anisotropic shear.

Existence of non-trivial, vacuum, asymptotically simple spacetimes

Abstract: We construct non-trivial vacuum spacetimes with a global +. The construction proceeds by proving extension results for initial data sets across compact boundaries, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is the existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon. Finally, the construction allows one to add Einstein?Rosen bridges to time-symmetric initial data sets at points satisfying a local parity condition, with the perturbation of the metric localized in an arbitrarily small neighbourhood of the bridge.

Quantization ambiguities in isotropic quantum geometry

Abstract: Some typical quantization ambiguities of quantum geometry are studied within isotropic models. Since this allows explicit computations of operators and their spectra, one can investigate the effects of ambiguities in a quantitative manner. It is shown that these ambiguities do not affect the fate of the classical singularity, demonstrating that the absence of a singularity in loop quantum cosmology is a robust implication of the general quantization scheme. The calculations also allow conclusions about modified operators in the full theory. In particular, using holonomies in a non-fundamental representation of SU(2) to quantize connection components turns out to lead to significant corrections to classical behaviour at macroscopic volume for large values of the spin of the chosen representation.

Current status of TAMA

Abstract: TAMA300, an interferometric gravitational-wave detector with 300 m baseline length, has been developed and operated with sufficient sensitivity to detect gravitational-wave events within our galaxy and sufficient stability for observations. The interferometer was operated for over 24 h stably and continuously. With a strain-equivalent noise level of h ~ 5 × 10−21 Hz−1/2, a signal-to-noise ratio (SNR) of 30 is expected for gravitational waves generated by a coalescence of 1.4 M⊙−1.4 M⊙ binary neutron stars at 10 kpc distance. In the summer of 2000, we carried out a two-week data-taking run, called data taking 4 (DT4), collecting 160 h of data to be analysed in the search for gravitational waves. In this paper, we review the design of the TAMA300 interferometer and the results of DT4. In addition, improvements after DT4 and recent results are also reported.

On N = 1, 2, 4 higher spin gauge theories in four dimensions

Abstract: We study = 1, 2, 4 higher spin superalgebras in four dimensions and higher spin gauge theories based on them. We extend the existing minimal = 2, 4 theories and find a minimal = 1 theory. Utilizing the basic structure of the minimal = 8 theory, we express the full field equations for the = 1, 2, 4 theories in a universal form without introducing Kleinian operators. We also use a non-minimal = 4 higher spin algebra tensored with U(3) to describe a higher spin extension of = 4 supergravity coupled to the massless vector multiplets arising in the KK spectrum of 11D supergravity on the = 3 supersymmetric AdS4 × N 010 background. The higher spin theory also contains a triplet of vector multiplets which may play a role in the super-Higgs effect in which = 4 is broken down to = 3.

Hamiltonian Quantization of Chern-Simons theory with SL(2, C) Group

Abstract: We analyse the Hamiltonian quantization of Chern–Simons theory associated with the real group SL(2, ), universal covering group of the Lorentz group SO(3, 1). The algebra of observables is generated by finite-dimensional spin networks drawn on a punctured topological surface. Our main result is a construction of a unitary representation of this algebra. For this purpose, we use the formalism of combinatorial quantization of Chern–Simons theory, i.e., we quantize the algebra of polynomial functions on the space of flat SL(2, ) connections on a topological surface Σ with punctures. This algebra, the so-called moduli algebra, is constructed along the lines of Fock–Rosly, Alekseev–Grosse–Schomerus, Buffenoir–Roche using only finite-dimensional representations of Uq(sl(2, )). It is shown that this algebra admits a unitary representation acting on a Hilbert space which consists of wave packets of spin networks associated with principal unitary representations of Uq(sl(2, )). The representation of the moduli algebra is constructed using only Clebsch–Gordan decomposition of a tensor product of a finite-dimensional representation with a principal unitary representation of Uq(sl(2, )). The proof of unitarity of this representation is nontrivial and is a consequence of the properties of Uq(sl(2, )) intertwiners which are studied in depth. We analyse the relationship between the insertion of a puncture coloured with a principal representation and the presence of a worldline of a massive spinning particle in de Sitter space.