Table of contents

We present a brief, and to a great extent, pedagogical, review on renormalization in curved spacetime and some recent results on the derivation and better understanding of quantum corrections to the action of gravity. The paper is mainly devoted to the semiclassical approach, but we also discuss its importance for quantum gravity and string theory.

As one candidate of the higher dimensional black holes, the 5D Ricci-flat black string is considered in this paper. By means of a non-trivial potential V_n, the quasi-normal modes of a massless scalar field around this black string space are studied. By using the classical third-order WKB approximation, we carefully analyze the evolution of frequencies in two aspects, one is the induced cosmological constant Λ and the other is the quantum number n. The massless scalar field decays more slowly because of the existence of the fifth dimension and the induced cosmological constant. If an extra dimension has in fact existed near the black hole, the quasi-normal frequencies may have some indication of it.

We describe a method to obtain an astrophysical result from the output of a search for gravitational waves from coalescing binaries. Specifically, we introduce a method based on the loudest event statistic to calculate an upper limit or interval on the astrophysical rate of binary coalescence. The calculation depends upon the sensitivity and noise background of the detectors, and a model for the astrophysical distribution of coalescing binaries. There are significant uncertainties in the calculation of the rate due to both astrophysical and instrumental uncertainties as well as errors introduced by using the post-Newtonian waveform to approximate the full signal. We catalog these uncertainties in detail and describe a method for marginalizing over them. Throughout, we provide an example based on the initial LIGO detectors.

The generalized uncertainty principle arises from the Heisenberg uncertainty principle when gravity is taken into account, so the leading order correction to the standard formula is expected to be proportional to the gravitational constant G_N = L²_Pl. On the other hand, the emerging picture suggests a set of departures from the standard theory which demand a revision of all the arguments used to deduce heuristically the new rule. In particular, one can now argue that the leading order correction to the Heisenberg uncertainty principle is proportional to the first power of the Planck length L_Pl. If so, the departures from ordinary quantum mechanics would be much less suppressed than what is commonly thought.

We compare the Brown–York (BY) and the standard Misner–Sharp (MS) quasilocal energies for round spheres in spherically symmetric spacetimes from the point of view of radial geodesics. In particular, we show that the relation between the BY and MS energies is precisely analogous to that between the (relativistic) energy E of a geodesic and the effective (Newtonian) energy E_eff appearing in the geodesic equation, thus shedding some light on the relation between the two. Moreover, for Schwarzschild-like metrics we establish a general relationship between the BY energy and the geodesic effective potential which explains and generalizes the recently observed connection between negative BY energy and the repulsive behaviour of geodesics in the Reissner–Nordström metric. We also comment on the extension of this connection between geodesics and the quasilocal BY energy to regions inside a horizon.

After a suitable gauge fixing, the local gravitational degrees of freedom of the Gowdy S¹ × S² and S³ cosmologies are encoded in an axisymmetric field on the sphere S². Recently, it has been shown that a standard field parametrization of these reduced models admits no Fock quantization with a unitary dynamics. This lack of unitarity is surpassed by a convenient redefinition of the field and the choice of an adequate complex structure. The result is a Fock quantization where both the dynamics and the SO(3)-symmetries of the field equations are unitarily implemented. The present work proves that this Fock representation is in fact unique inasmuch as, up to equivalence, there exists no other possible choice of SO(3)-invariant complex structure leading to a unitary implementation of the time evolution.

We describe a modification of a fourth-order accurate 'moving-puncture' evolution code, where by replacing spatial fourth-order accurate differencing operators in the bulk of the grid by a specific choice of sixth-order accurate stencils we gain significant improvements in accuracy. We illustrate the performance of the modified algorithm with an equal-mass simulation covering nine orbits.

Recent detailed analysis within the loop quantum gravity calculation of black hole entropy shows a stair-like structure in the behavior of entropy as a function of horizon area. The non-trivial distribution of the degeneracy of the black hole horizon area eigenstates is at the origin of this behavior. This degeneracy distribution is analyzed and a phenomenological model is put forward to study the implications of this distribution in the black hole radiation spectrum. Some qualitative quantum effects are obtained within the isolated horizon framework. This result provides us with a possible observational test of this model for quantum black holes.

In a previous paper we showed that static spherically symmetric objects which, in the vicinity of their surface, are well described by a polytropic equation of state with 3/2 < Γ < 2 exhibit a curvature singularity in Palatini f(R) gravity. We argued that this casts serious doubt on the validity of Palatini f(R) gravity as a viable alternative to general relativity. In the present paper, we further investigate this characteristic of Palatini f(R) gravity in order to clarify its physical interpretation and consequences.

We consider the application of stable marginally outer trapped surfaces to problems concerning the size of material bodies and the area of black holes. The results presented extend to general initial data sets (V, g, K) previous results assuming either maximal (tr_gK = 0) or time-symmetric (K = 0) initial data.

Recently, two new spin-foam models have appeared in the literature, both motivated by a desire to modify the Barrett–Crane model in such a way that the imposition of certain second class constraints, called cross-simplicity constraints, are weakened. We refer to these two models as the FKLS (Freidel–Krasnov–Livine–Speziale) model and the flipped model. Both of these models are based on a reformulation of the cross-simplicity constraints. This paper has two main parts. First, we clarify the structure of the reformulated cross-simplicity constraints and the nature of their quantum imposition in the new models. In particular, we show that in the FKLS model quantum cross-simplicity implies no restriction on states. The deeper reason for this seems to be that, with the symplectic structure relevant for FKLS, the reformulated simplicity constraints, among themselves, now form a first class system, and this seems to cause the coherent state method of imposing the constraints, key in the FKLS model, to fail to give any restriction on states. Nevertheless, the cross-simplicity can still be seen as implemented via suppression of intertwiner degrees of freedom in the dynamical propagation. In the second part of the paper, we investigate area spectra in the models. The results of these two investigations will highlight how, in the flipped model, the Hilbert space of states, as well as the spectra of area operators, exactly match those of loop quantum gravity, whereas in the FKLS (and Barrett–Crane) models the boundary Hilbert spaces and area spectra are different.

Recently, uniqueness theorems were constructed for the representation used in loop quantum gravity. We explore the existence of alternate representations by weakening the assumptions of the Lewandowski–Okolow–Sahlmann–Thiemann (LOST) uniqueness theorem. The weakened assumptions seem physically reasonable and retain the key requirement of explicit background independence. For simplicity, we restrict attention to the case of gauge group U(1).

We describe spacetime fluctuations by means of small fluctuations of the metric on a given background metric. From a minimally coupled Klein–Gordon equation we obtain a modified Schrödinger equation within a weak-field approximation up to second order and an averaging procedure over a finite spacetime scale given by the quantum particle in the nonrelativistic limit. The dominant modification consists of an anomalous inertial mass tensor which depends on the type of particle and the fluctuation scenario. The scenario considered in this paper is a most simple picture of spacetime fluctuations and gives an existence proof for an apparent violation of the weak equivalence principle and, in general, for a violation of Lorentz invariance.

The behavior of a charged massive Dirac field on a Reissner–Nordström–AdS black hole background is investigated. We first analyze the problem of the essential self-adjointness of the Dirac Hamiltonian, which is made difficult by the boundary-like behavior of spatial infinity, and we find that the Hamiltonian is essentially self-adjoint iff ; moreover, we determine the essential spectrum of the Hamiltonian. Then we focus on the analysis of the discharge problem for the case . We follow the Ruffini–Damour–Deruelle approach and, as in the standard Reissner–Nordström black hole case, we find that the existence of level-crossing between the positive and negative energy solutions of the Dirac equation is at the root of the pair-creation process associated with the discharge of the black hole.

The vacuum Gowdy models provide much studied, non-trivial midi-superspace examples. Various technical issues within loop quantum gravity can be studied in these models and one can hope to understand singularities and their resolution in the loop quantization. The first step in this program is to reformulate the model in real connection variables in a manner that is amenable to loop quantization. We begin with the unpolarized model and carry out a consistent reduction to the polarized case. Carrying out complete gauge fixing, the known solutions are recovered.

We consider the properties of stress–energy tensors compatible with a null big bang, i.e., cosmological evolution starting from a Killing horizon rather than a singularity. For Kantowski–Sachs cosmologies, it is shown that if matter satisfies the null energy condition, then (i) regular cosmological evolution can only start from a Killing horizon, (ii) matter is absent at the horizon and (iii) matter can only appear in the cosmological region due to interaction with vacuum. The latter is understood phenomenologically as a fluid whose stress tensor is insensitive to boosts in a particular direction. We also argue that matter is absent in a static region beyond the horizon. All this generalizes the observations recently obtained for a mixture of dust and a vacuum fluid. If, however, we admit the existence of phantom matter, its certain special kinds (with the parameter w ⩽ −3) are consistent with a null big bang without interaction with vacuum (or without vacuum fluid at all). Then in the static region there is matter with w ⩾ −1/3. Alternatively, the evolution can begin from a horizon in an infinitely remote past, leading to a scenario combining the features of a null big bang and an emergent universe.

We solve the Killing–Yano equation on manifolds with a G structure for G = SO(n), U(n), SU(n), Sp(n) ⋅ Sp(1), Sp(n), G₂ and Spin(7). Solutions include nearly-Kähler, weak holonomy G₂, balanced SU(n) and holonomy G manifolds. As an application, we find that particle probes on AdS₄ × X compactifications of type IIA and 11-dimensional supergravity admit a type of symmetry generated by the fundamental forms. We also explore the symmetries of string and particle actions in heterotic and common sector supersymmetric backgrounds. In the heterotic case, the generators of the symmetries completely characterize the solutions of the gravitino Killing spinor equation, and the structure constants of the symmetry algebra depend on the solution of the dilatino Killing spinor equation.

We describe a scheme for the exploration of quantum gravity phenomenology focusing on effects that could be thought of as arising from a fundamental granularity of spacetime. In contrast with the simplest assumptions, such granularity is assumed to respect Lorentz invariance but is otherwise left unspecified. The proposal is fully observer covariant, it involves non-trivial couplings of curvature to matter fields and leads to a well-defined phenomenology. We present the effective Hamiltonian which could be used to analyze concrete experimental situations, some of which are briefly described, and we shortly discuss the degree to which the present proposal is in line with the fundamental ideas behind the equivalence principle.

Quantization of systems with constraints can be carried out with several methods. In the Dirac formulation the classical generators of gauge transformations are required to annihilate physical quantum states to ensure their gauge invariance. Carrying on BRST symmetry it is possible to get a condition on physical states which, different from the Dirac method, requires them to be invariant under the BRST transformation. Employing this method for the action of general relativity expressed in terms of the spin connection and tetrad fields with path integral methods, we construct the generator of the BRST transformation associated with the underlying local Lorentz symmetry of the theory and write a physical state condition following from BRST invariance. This derivation is based on the general results on the dependence of the effective action used in path integrals and consequently of Green's functions on the gauge-fixing functionals used in the DeWitt–Faddeev–Popov method. The condition we gain differs from the one obtained within Ashtekar's canonical formulation, showing how we recover the latter only by a suitable choice of the gauge-fixing functionals. Finally we discuss how it should be possible to obtain all of the requested physical state conditions associated with all the underlying gauge symmetries of the classical theory using our approach.

We use a recent result by Cabezas et al (2007 Gen. Rel. Grav.39 707) to build up an approximate solution to the gravitational field created by a rigidly rotating polytrope. We solve the linearized Einstein equations inside and outside the surface of zero pressure including second-order corrections due to rotational motion to get an asymptotically flat metric in a global harmonic coordinate system. We prove that if the metric and their first derivatives are continuous on the matching surface up to this order of approximation, the multipole moments of this metric cannot be fitted to those of the Kerr metric.

We construct a covariant phase space for rotating weakly isolated horizons in Einstein–Maxwell–Chern–Simons theory in all (odd) D ⩾ 5 dimensions. In particular, we show that horizons on the corresponding phase space satisfy the zeroth and first laws of black-hole mechanics. We show that the existence of a Killing spinor on an isolated horizon in four dimensions (when the Chern–Simons term is dropped) and in five dimensions requires that the induced (normal) connection on the horizon has to vanish, and this in turn implies that the surface gravity and rotation 1-form are zero. This means that the gravitational component of the horizon angular momentum is zero, while the electromagnetic component (which is attributed to the bulk radiation field) is unconstrained. It follows that an isolated horizon is supersymmetric only if it is extremal and nonrotating. A remarkable property of these horizons is that the Killing spinor only has to exist on the horizon itself. It does not have to exist off the horizon. In addition, we find that the limit when the surface gravity of the horizon goes to zero provides a topological constraint. Specifically, the integral of the scalar curvature of the cross sections of the horizon has to be positive when the dominant energy condition is satisfied and the cosmological constant Λ is zero or positive, and in particular rules out the torus topology for supersymmetric isolated horizons (unless Λ < 0) if and only if the stress–energy tensor T_ab is of the form such that T_abℓ^an^b = 0 for any two null vectors ℓ and n with normalization ℓ_an^a = −1 on the horizon.

An extended local Lorentz symmetry in four-dimensional (4D) theory is considered. A source of this symmetry is a group of general linear transformations of four-component Majorana spinors GL(4, M) which is isomorphic to GL(4, R) and is the covering of an extended Lorentz group in a 6D Minkowski space M(3, 3) including superluminal and scaling transformations. Physical spacetime is assumed to be a 4D pseudo-Riemannian manifold. To connect the extended Lorentz symmetry in the M(3, 3) space with the physical spacetime, a fiber bundle over the 4D manifold is introduced with M(3, 3) as a typical fiber. The action is constructed which is invariant with respect to both general 4D coordinate and local GL(4, M) spinor transformations. The components of the metric on the 6D fiber are expressed in terms of the 4D pseudo-Riemannian effective metric and two extra complex fields: 4D vector and scalar ones. These extra fields describe in the general case massive particles interacting with an extra U(1) gauge field and weakly interacting with ordinary particles, i.e. possessing properties of invisible (dark) matter.

We investigate the late-time behavior of a scalar field on a fixed Kerr background using a 2 + 1 dimensional pseudo-spectral evolution code. We compare evolutions of pure axisymmetric multipoles in both Kerr–Schild and Boyer–Lindquist coordinates. We find that the late-time power-law decay rate depends upon the slicing of the background, confirming previous theoretical predictions for those decay rates. The accuracy of the numerical evolutions is sufficient to decide unambiguously between competing claims in the literature.

The Plebański–Demiański solution is a very general axially symmetric analytical solution of the Einstein field equations generalizing the Kerr solution. This solution depends on seven parameters which under certain circumstances are related to mass, rotation, cosmological constant, NUT parameter, electric and magnetic charges and acceleration. In this paper we present a general description of matter wave interferometry in the general Plebański–Demiański black hole spacetime. Particular emphasis is placed on a gauge invariant description of the symmetries of the gauge field. We show that it is possible to have access to all parameters separately except the acceleration. For neutral particles there is only access to a combination of electric and magnetic charge.

In searches for gravitational-wave bursts, a standard technique used to reject noise is to discard burst event candidates that are not seen in coincidence in multiple detectors. A coincidence test in which Bayesian inference is used to measure how noise-like a tuple of events appears is presented here. This technique is shown to yield higher detection efficiencies for a given false alarm rate than do techniques based on per-parameter thresholds when applied to a toy model covering a broad class of event candidate populations. Also presented is the real-world example of a use of the technique for noise rejection in a time–frequency burst search conducted on simulated gravitational-wave detector data. Besides achieving a higher detection efficiency, the technique is significantly less challenging to implement well than is a per-parameter threshold method.

Non-perturbative theories of quantum gravity inevitably include configurations that fail to resemble physically reasonable spacetimes at large scales. Often, these configurations are entropically dominant and pose an obstacle to obtaining the desired classical limit. We examine this 'entropy problem' in a model of causal set quantum gravity corresponding to a discretization of 2D spacetimes. Using results from the theory of partial orders we show that, in the large volume or continuum limit, its partition function is dominated by causal sets which approximate to a region of 2D Minkowski space. This model of causal set quantum gravity thus overcomes the entropy problem and predicts the emergence of a physically reasonable geometry.

A Penrose diagram is constructed for an example black hole that evaporates at a steady rate as measured by a distant observer, until the mass vanishes, yielding a final state Minkowski spacetime. Coordinate dependences of significant features, such as the horizon and coordinate anomalies, are clearly demonstrated on the diagram. The large-scale causal structure of the spacetime is briefly discussed.