Table of contents

We comment on the presence of spurious observables and on a subtle violation of irreducibility in loop quantum cosmology.

The goal of this review is to present an introduction to loop quantum gravity—a background-independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird's eye view of the present status of the subject, the review should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the review is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well-established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the review to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.

We study the scalar and spinor perturbation, namely the Klein–Gordan and Dirac equations, in the Kerr–NUT spacetime. The metric is invariant under the duality transformation involving the exchange of mass and NUT parameters on the one hand and radial and angle coordinates on the other. We show that this invariance is also shared by the scalar and spinor perturbation equations. Further, by the duality transformation, one can go from the Kerr to the dual Kerr solution, and vice versa, and the same applies to the perturbation equations. In particular, it turns out that the potential barriers felt by the incoming scalar and spinor fields are higher for the dual Kerr than that for the Kerr. We also comment on the existence of horizon and singularity.

We investigate here the gravitational collapse end states for a spherically symmetric perfect fluid with an equation of state p = kρ. It is shown that given regular initial data in terms of the density and pressure profiles at the initial epoch from which the collapse develops, the black-hole or naked singularity outcomes depend on the choice of rest of the free functions available, such as the velocities of the collapsing shells and the dynamical evolutions as allowed by the Einstein equations. This clarifies the role that equation of state and initial data play in determining the final fate of gravitational collapse.

Recent observations of type Ia supernovae strongly support that the universe is accelerating now and decelerated in the recent past. This may be evidence of the breakdown of the standard Friedmann equation. We consider a general modified Friedmann equation. Three different models are analysed in detail. The current supernova data and the Wilkinson microwave anisotropy probe data are used to constrain these models. A detailed analysis of the transition from the deceleration to acceleration phase is also performed.

When using black-hole excision to numerically evolve a black-hole spacetime with no continuous symmetries, most 3 + 1 finite differencing codes use a Cartesian grid. It is difficult to do excision on such a grid because the natural r = constant excision surface must be approximated either by a very different shape such as a contained cube, or by an irregular and non-smooth 'LEGO¹ sphere' which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme which uses multiple {r × (angular coordinates)} grid patches, each patch using a different (nonsingular) choice of angular coordinates. This allows excision on a smooth r = constant 2-sphere. I discuss the key design choices in such a multiple-patch scheme, including the choice of ghost-zone versus internal-boundary treatment of the interpatch boundaries (I use a ghost-zone scheme), the number and shape of the patches (I use a 6-patch 'inflated-cube' scheme), the details of how the ghost zones are 'synchronized' by interpolation from neighbouring patches, the tensor basis for the Einstein equations in each patch, and the handling of non-tensor field variables such as the BSSN (I use a scheme which requires ghost zones which are twice as wide for the BSSN conformal factor ϕ as for and the other BSSN field variables). I present sample numerical results from a prototype implementation of this scheme. This code simulates the time evolution of the (asymptotically flat) spacetime around a single (excised) black hole, using fourth-order finite differencing in space and time. Using Kerr initial data with J/m² = 0.6, I present evolutions to t ⪆ 1500m. The lifetime of these evolutions appears to be limited only by outer boundary instabilities, not by any excision instabilities or by any problems inherent to the multiple-patch scheme.

We attempt to match the most general cylindrically symmetric vacuum spacetime with a Robertson–Walker interior. The matching conditions show that the interior must be dust filled and that the boundary must be comoving. Further, we show that the vacuum region must be polarized. Imposing the condition that there are no trapped cylinders on an initial time slice, we can apply a result of Thorne's and show that trapped cylinders never evolve. This results in a simplified line element which we prove to be incompatible with the dust interior. This result demonstrates the impossibility of the existence of an isotropic cylindrically symmetric star (or even a star which has a cylindrically symmetric portion). We investigate the problem from a different perspective by looking at the expansion scalars of invariant null geodesic congruences and, applying to the cylindrical case, the result that the product of the signs of the expansion scalars must be continuous across the boundary. The result may also be understood in relation to recent results about the impossibility of the static axially symmetric analogue of the Einstein–Straus model.

A set of conditions are given which must be satisfied if two spacetimes are to be properly matched at a corner. Also given are the physical implications of such a matching and hence the requirements for modelling sharp-edged objects in space.

Following Sanchez's approach we investigate the effect of scalar mass in the absorption and emission problems of a four-dimensional (4D) Schwarzschild black hole. The absorption cross sections for arbitrary angular momentum of the scalar field are computed numerically in the full range of energy by making use of the analytic near-horizon and asymptotic solutions and their analytic continuations. The scalar mass has an interesting effect in the low-energy absorption cross section for the S-wave. Unlike the massless case, the cross section decreases with increasing energy in the extremely low-energy regime. As a result the universality, i.e., low-energy cross section for the S-wave is equal to the horizon area, is broken in the presence of mass. If the scalar mass is larger than the critical mass, the absorption cross section becomes a monotonically decreasing function in the entire range of energy. The Hawking emission is also calculated numerically. It turns out that the Planck factor generally suppresses the contribution of higher partial waves except the S-wave. The scalar mass in general tends to reduce the emission rate.

The kinematical setting of spherically symmetric quantum geometry, derived from the full theory of loop quantum gravity, is developed. This extends previous studies of homogeneous models to inhomogeneous ones where interesting field theory aspects arise. A comparison between a reduced quantization and a derivation of the model from the full theory is presented in detail, with an emphasis on the resulting quantum representation. Similar concepts for Einstein–Rosen waves are discussed briefly.

We exhibit three families of complete curvature homogeneous pseudo-Riemannian manifolds which are modelled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov–Petrova.

Using the multipolar post-Minkowskian and matching formalism, we compute the gravitational wave form of inspiralling compact binaries moving in quasi-circular orbits at the second and a half post-Newtonian (2.5PN) approximation to general relativity. The inputs we use include notably the mass-type quadrupole at the 2.5PN order, the mass octupole and current quadrupole at the 2PN order, the mass 2⁵-pole and current 2⁴-pole at 1PN. The nonlinear hereditary terms come from the monopole–quadrupole multipole interactions or tails, present at the 1.5PN, 2PN and 2.5PN orders, and the quadrupole–quadrupole interaction arising at the 2.5PN level. In particular, the specific effect of nonlinear memory is computed using a simplified model of binary evolution in the past. The 'plus' and 'cross' wave polarizations at the 2.5PN order are obtained in ready-to-use form, extending the 2PN results calculated earlier by Blanchet, Iyer, Will and Wiseman.

Fomalont and Kopeikin have recently succeeded in measuring the velocity-dependent component of the Shapiro time delay of light from a quasar passing behind Jupiter. While there is general agreement that this observation tests the gravitomagnetic properties of the gravitational field, a controversy has emerged over the question of whether the results depend on the speed of light, c, or the speed of gravity, c_g. By analysing the Shapiro time delay in a set of 'preferred frame' models, I demonstrate that this question is ill-posed: the distinction can only be made in the context of a class of theories in which c ≠ c_g, and the answer then depends on the specific class of theories one chooses. It remains true, however, that for a large class of theories 'close enough' to general relativity, the leading contribution to the time delay depends on c and not c_g; within this class, observations are thus not yet accurate enough to measure the speed of gravity.

Recently, Flanagan (2004 Phys. Rev. Lett.92 071101) has argued that the Palatini form of 1/R gravity is ruled out by experiments such as electron–electron scattering. His argument involves adding minimally coupled fermions in the Jordan frame and transforming to the Einstein frame. This produces additional terms that are ruled out experimentally. Here I argue that this conclusion is false. It is well known that conformally related theories are mathematically equivalent but not physically equivalent. As discussed by Magnano and Sokolowski (1999 Phys. Rev. D 50 5039) one must decide, in the vacuum theory, which frame is the physical frame and add the minimally coupled Lagrangian in this frame. If this procedure is followed the resulting theory is not ruled out experimentally. The discussions in this paper also show that the equivalence between the generalized gravitational theories and scalar tensor theories discussed by Flanagan (2003 Class. Quantum Grav.21 417) is only mathematical, not physical.

It has frequently been claimed in the literature that the classical physical predictions of scalar–tensor theories of gravity depend on the conformal frame in which the theory is formulated. We argue that this claim is false, and that all classical physical predictions are conformal-frame invariants. We also respond to criticisms by Vollick (2003 Preprint gr-qc/0312041), in which this issue arises, of our recent analysis of the Palatini form of 1/R gravity.