Table of contents

Numerical relativity is a powerful tool for astrophysical applications where gravity is strong, as shown dramatically by numerical studies of compact binary systems. However, as a fundamental theory of space, time and gravity, general relativity has many non-astrophysical aspects, and numerical techniques can also be used to great advantage in these areas.

This is especially true now that developments in high energy physics, and particularly string theory, have led to new applications of general relativity. Numerical simulations of black holes in 4 spacetime dimensions, adapted and expanded, can be exploited to simulate higher dimensional black holes (as well as other `black objects' like black strings). Through the AdS/CFT correspondence, such simulations of higher dimensional spacetimes (in this case with AdS boundary conditions) can be used to gain information about strongly coupled field theories. High energy physics also leads us to regard general relativity as an effective low energy theory and to think of what the lowest order corrections to that theory might be. Here too, numerical relativity can be used to simulate such corrections and find their observational consequences.

A promising approach to quantizing general relativity is loop quantum gravity. In particular, for the purposes of understanding cosmology, and perhaps resolving the big bang singularity through quantum effects, it is helpful to consider loop quantum gravity truncated to a small number of degrees of freedom. Even with such a truncation, loop quantum gravity is sufficiently complicated that numerical techniques can be used to great advantage.

Though it is generally hoped that quantum gravity will resolve singularities, it is a good idea to understand exactly what it is that one hopes to resolve. Thus, one would like a thorough understanding of the singularities of classical general relativity. Numerical techniques are essential in obtaining this understanding of singularities.

This focus section covers the topics discussed above, mapping out possible strategies for using numerical techniques to make progress on these problems.

David Garfinkle and Luis Lehner Guest Editors

Physic in curved spacetime describes a multitude of phenomena, ranging from astrophysics to high-energy physics (HEP). The last few years have witnessed further progress on several fronts, including the accurate numerical evolution of the gravitational field equations, which now allows highly nonlinear phenomena to be tamed. Numerical relativity simulations, originally developed to understand strong-field astrophysical processes, could prove extremely useful to understand HEP processes such as trans-Planckian scattering and gauge–gravity dualities. We present a concise and comprehensive overview of the state-of-the-art and important open problems in the field(s), along with a roadmap for the next years.

A brief review of various numerical techniques used in loop quantum cosmology and results is presented. These include the way extensive numerical simulations provided insights into the resolution of classical singularities, resulting in the key prediction of the bounce at the Planck scale in different models, and the numerical methods used to analyze the properties of the quantum difference operator and the von Neumann stability issues. Using the quantization of a massless scalar field in an isotropic spacetime as a template, an attempt is made to highlight the complementarity of different methods to gain understanding of the new physics emerging from the quantum theory. Open directions which need to be explored with more refined numerical methods are discussed.

The formation of spacetime singularities has been studied using a variety of numerical methods. Many of these simulations are done in the case of compact Cauchy surfaces and yield singularites that are spacelike, local, and oscillatory. Other simulations are done in the case of asymptotic flatness and spherical symmetry, and yield singularities that are null and weak. This yields a picture of gravitational collapse in which both types of singularity are present, one encountered during black hole formation and the other long after the black hole has formed.

The holonomy-flux algebra $\mathfrak {A}$ of loop quantum gravity is known to admit a natural representation that is uniquely singled out by the requirement of covariance under spatial diffeomorphisms. In the cosmological context, the requirement of spatial homogeneity naturally reduces $\mathfrak {A}$ to a much smaller algebra, $\mathfrak {A}_{\rm Red}$, used in loop quantum cosmology. In Bianchi I models, it is shown that the requirement of covariance under residual diffeomorphism symmetries again uniquely selects the representation of $\mathfrak {A}_{\rm Red}$ that has been commonly used. We discuss the close parallel between the two uniqueness results and also point out a difference.

Communicated by A Corichi

Canonical methods allow the derivation of effective gravitational actions from the behavior of space-time deformations reflecting general covariance. With quantum effects, the deformations and correspondingly the effective actions change, revealing dynamical implications of quantum corrections. A new systematic way of expanding these actions is introduced showing as a first result that inverse-triad corrections of loop quantum gravity simplify the asymptotic dynamics near a spacelike collapse singularity. By generic quantum effects, the singularity is removed.

We construct an approximate static gravitational solution of the Einstein equations, with negative cosmological constant, describing a test black string stretching from the boundary of the Schwarzschild–AdS₅ black brane toward the horizon. The construction builds on a derivative expansion method, assuming that the black brane metric changes slowly along the black string direction. We provide a solution up to second order in derivatives, and it implies, in particular, that the black string must shrink to zero size at the horizon of the black brane. In the near-horizon region of the black brane, where the two horizons intersect, we provide an exact solution of a cone that describes two intersecting horizons at different temperatures. Moreover, we show that this solution equally describes a thin and long black droplet.

This paper is to introduce a new software called CBwaves which provides a fast and accurate computational tool to determine the gravitational waveforms yielded by generic spinning binaries of neutron stars and/or black holes on eccentric orbits. This is done within the post-Newtonian (PN) framework by integrating the equations of motion and the spin precession equations, while the radiation field is determined by a simultaneous evaluation of the analytic waveforms. In applying CBwaves various physically interesting scenarios have been investigated. In particular, we have studied the appropriateness of the adiabatic approximation, and justified that the energy balance relation is indeed insensitive to the specific form of the applied radiation reaction term. By studying eccentric binary systems, it is demonstrated that circular template banks are very ineffective in identifying binaries even if they possess tiny residual orbital eccentricity, thus confirming a similar result obtained by Brown and Zimmerman (2010 Phys. Rev. D 81 024007). In addition, by investigating the validity of the energy balance relation we show that, contrary to the general expectations, the PN approximation should not be applied once the PN parameter gets beyond the critical value ∼0.08 − 0.1. Finally, by studying the early phase of the gravitational waves emitted by strongly eccentric binary systems—which could be formed e.g. in various many-body interactions in the galactic halo—we have found that they possess very specific characteristics which may be used to identify these type of binary systems.

We consider high-derivative equations obtained setting to zero the divergence of the higher spin curvatures in metric-like form, showing their equivalence to the second-order equations emerging from the tensionless limit of open string field theory, which propagate reducible spectra of particles with different spins. This result can be viewed as complementary to the possibility of setting to zero a single trace of the higher spin field strengths, which yields an equation known to imply Fronsdal's equation in the compensator form. Higher traces and divergences of the curvatures produce a whole pattern of high-derivative equations whose systematics is also presented.

Observations of pulsar timing provide strong constraints on scalar–tensor theories of gravity, but these constraints are traditionally quoted as limits on the microscopic parameters (like the Brans–Dicke coupling, for example) that govern the strength of scalar–matter couplings at the particle level in particular models. For binary pulsars whose all five post-Keplerian parameters have been measured, we present fits to timing data directly in terms of the phenomenological couplings (masses, scalar charges, moment of inertia sensitivities and so on) of the stars involved, rather than to the more microscopic parameters of a specific model. For instance, for the double pulsar PSR J0737-3039A/B, we find at the 68% confidence level that the masses are bounded by 1.28 < m_A/m_⊙ < 1.34 and 1.19 < m_B/m_⊙ < 1.25, while the scalar charge-to-mass ratios satisfy |a_A| < 0.21, |a_B| < 0.21 and |a_B − a_A| < 0.002, independent of the details of the scalar–tensor model involved, and of assumptions about the stellar equations of state. Whenever it is possible to do so, we urge observers to express their results in this more model-independent way, which potentially can then be used by theorists to constrain a great variety of specific models by computing the fit quantities as functions of the microscopic parameters in any particular model. For the Brans–Dicke and quasi-Brans–Dicke models, the constraints coming from the double pulsar obtained in this manner are consistent with, and slightly weaker than, those quoted in the literature.

The equations governing null and timelike geodesics are derived within the 3+1 formalism of general relativity. In addition to the particle's position, they encompass an evolution equation for the particle's energy leading to a 3+1 expression of the redshift factor for photons. An important application is the computation of images and spectra in spacetimes arising from numerical relativity, via the ray-tracing technique. This is illustrated here by images of numerically computed stationary neutron stars and dynamical neutron stars collapsing to a black hole.

We construct explicit examples of globally regular static, spherically symmetric solutions in general relativity with scalar and electromagnetic fields which describe traversable wormholes (with flat and AdS asymptotics) and regular black holes, in particular, black universes. A black universe is a non-singular black hole where, beyond the horizon, instead of a singularity, there is an expanding, asymptotically isotropic universe. The scalar field in these solutions is phantom (i.e. its kinetic energy is negative), minimally coupled to gravity and has a nonzero self-interaction potential. The configurations obtained are quite diverse and contain different numbers of Killing horizons, from zero to four. This substantially widens the list of known structures of regular BH configurations. Such models can be of interest both as descriptions of local objects (black holes and wormholes) and as a basis for building non-singular cosmological scenarios.

Isolation from seismic motion is vital for vibration sensitive experiments. The seismic attenuation system (SAS) is a passive mechanical isolation system for optics suspensions. The low natural frequency of a SAS allows seismic isolation starting below 0.2 Hz. The desired isolation at frequencies above a few hertz is 70–80 dB in both horizontal and vertical degrees of freedom. An introduction to the SAS for the AEI 10 m Prototype, an overview of the mechanical design and a description of the major components are given.

A class of spherical collapsing exact solutions with electromagnetic charge is derived. This class of solutions—in general anisotropic—contains however as a particular case the charged dust model already known in the literature. Under some regularity assumptions that in the uncharged case give rise to naked singularities, it is shown that the process of avoidance of shell focusing singularities—already known for the dust collapse (Krasinski et al 2006 Phys. Rev. D 73 124033)—also takes place here, determining shell crossing effects or a completely regular solution.

We present an explicit example showing that the weak Gauß law of general relativity (with cosmological constant) fails in Einstein–Cartan's theory. We take this as an indication that torsion might replace dark matter.

A procedure avoiding any integration of the null geodesic equations is used to derive the direction of light propagation in a three-parameter family of static, spherically symmetric spacetimes within the post-post-Minkowskian approximation. Quasi-Cartesian isotropic coordinates adapted to the symmetries of spacetime are systematically used. It is found that the expression of the angle formed by two light rays as measured by a static observer staying at a given point is remarkably simple in these coordinates. The attention is mainly focused on the null geodesic paths that we call the quasi-Minkowskian light rays. The vector-like functions characterizing the direction of propagation of such light rays at their points of emission and reception are firstly obtained in the generic case where these points are both located at finite distances from the centre of symmetry. The direction of propagation of the quasi-Minkowskian light rays emitted at infinity is then straightforwardly deduced. An intrinsic definition of the gravitational deflection angle relative to a static observer located at a finite distance is proposed for these rays. The expression inferred from this definition extends the formula currently used in VLBI astrometry up to the second order in the gravitational constant G.

We consider a class of impulsive gravitational wave spacetimes, which generalize impulsive pp-waves. They are of the form $M=N\times \mathbb {R}^2_1$, where (N, h) is a Riemannian manifold of arbitrary dimension and M carries the line element ds² = dh² + 2 du dv + f(x)δ(u) du², with dh² being the line element of N and δ the Dirac measure. We prove a completeness result for such spacetimes M with complete Riemannian part N.

Gravity can be thought as an emergent phenomenon and it has a nice 'thermodynamic' structure. In this context, it is then possible to study the thermodynamics without knowing the details of the underlying microscopic degrees of freedom. Here, based on the ordinary thermodynamics, we investigate the phase transition of the static, spherically symmetric charged black hole solution with the arbitrary scalar curvature 2k in Hořava–Lifshitz gravity at the Lifshitz point z = 3. The analysis is done using the canonical ensemble framework, i.e. the charge is kept fixed. We find (a) for both k = 0 and k = 1, there is no phase transition, (b) while k = −1 case exhibits the second-order phase transition within the physical region of the black hole. The critical point of second-order phase transition is obtained by the divergence of the heat capacity at constant charge. Near the critical point, we find the various critical exponents. It is also observed that they satisfy the usual thermodynamic scaling laws.

In this paper, we demonstrate for the first time that it is possible to solve numerically the Cauchy problem for the linearization of the general conformal field equations near space-like infinity, which is only well defined in Friedrich's cylinder picture. We have restricted ourselves here to the 'core' of the equations—the spin-2 system—propagating on Minkowski space. We compute the numerical solutions for various classes of initial data, perform convergence tests and also compare with the exact solutions. We also choose initial data which intentionally violate the smoothness conditions and then check the analytical predictions about singularities. This paper is the first step in a long-term investigation of the use of conformal methods in numerical relativity.

Modified gravity is one of the most promising candidates for explaining the current accelerating expansion of the Universe, and even its unification with the inflationary epoch. Nevertheless, the wide range of models capable of explaining the phenomena of dark energy imposes that current research focuses on a more precise study of the possible effects of modified gravity on both cosmological and local levels. In this paper, we focus on the analysis of a type of modified gravity, the so-called f(R, G) gravity, and we perform a deep analysis on the stability of important cosmological solutions. This not only can help to constrain the form of the gravitational action, but also facilitate a better understanding of the behavior of the perturbations in this class of higher order theories of gravity, which will lead to a more precise analysis of the full spectrum of cosmological perturbations in future.

On the example of a free massless and conformally coupled scalar field, it is argued that in quantum field theory in curved spacetimes with the time-like Killing field, the corresponding KMS states (generalized Gibbs ensembles) at parameter β > 0 need not possess a definite temperature in the sense of the zeroth law. In fact, these states, although passive in the sense of the second law, are not always in local thermal equilibrium (LTE). A criterion characterizing LTE states with sharp local temperature is discussed. Moreover, a proposal is made for fixing the renormalization freedom of composite fields which serve as 'thermal observables' and a new definition of the thermal energy of LTE states is introduced. Based on these results, a general relation between the local temperature and the parameter β is established for KMS states in (anti) de Sitter spacetime.

We calculate the degree of horizon smoothness of a multi-M2 brane solution with branes along a common axis. We find that the metric is generically only thrice continuously differentiable at any of the horizons. The 4-form field strength is found to be only twice continuously differentiable. We work with Gaussian null-like co-ordinates which are obtained by solving geodesic equations for multi-M2 brane geometry. We also find different, exact co-ordinate transformations which take the metric from isotropic co-ordinates to co-ordinates in which the metric is thrice differentiable at the horizon. Both methods give the same result that the multi-M2 brane metric is only thrice differentiable at the horizon.

We study a class of $\mathbb {S}^3$-Gowdy vacuum models with a regular past Cauchy horizon which we call smooth Gowdy-symmetric generalized Taub–NUT solutions. In particular, we prove the existence of such solutions by formulating a singular initial value problem with asymptotic data on the past Cauchy horizon. We prove that also a future Cauchy horizon exists for generic asymptotic data, and derive an explicit expression for the metric on the future Cauchy horizon in terms of the asymptotic data on the past horizon. This complements earlier results about $\mathbb {S}^1 \times \mathbb {S}^2$-Gowdy models.

We analyze the periodic motion in the conformal mechanics describing particles moving near the horizon of extreme Reissner–Nordström and axion–dilaton (Clément–Gal'tsov) black holes. For this purpose, we extract the (two-dimensional) compact (angular) parts of these systems and construct their action-angle variables. In the first case, we obtain the well-known spherical Landau problem, which possesses hidden so(3) symmetry, while in the latter case the system does not have a hidden constant of motion. In both the cases, we indicate the existence of 'critical points', separating the regions of periodic motions with qualitatively different properties.

Bray and Khuri (2011 Asian J. Math.15 557–610; 2010 Discrete Continuous Dyn. Syst. A 27 741766) outlined an approach to prove the Penrose inequality for general initial data sets of the Einstein equations. In this paper we extend this approach so that it may be applied to a charged version of the Penrose inequality. Moreover, assuming that the initial data are time-symmetric, we prove the rigidity statement in the case of equality for the charged Penrose inequality, a result which seems to be absent from the literature. A new quasi-local mass, tailored to charged initial data sets is also introduced, and used in the proof.

This work makes the first ever attempt to understand the influence of the black hole background spacetime in determining the fundamental properties of the embedded relativistic acoustic geometry. To accomplish such task, we investigate the role of the spin angular momentum of the astrophysical black hole (the Kerr parameter a—a representative feature of the background black hole metric) in estimating the value of the acoustic surface gravity (the representative feature of the corresponding analogue spacetime). Since almost all astrophysical black holes are supposed to posses some degree of intrinsic rotation, the influence of the Kerr parameter on classical analogue models is very important to understand. We study the general relativistic, axially symmetric, non-self-gravitating inflow of the hydrodynamic fluid onto a rotating astrophysical black hole from the dynamical systems point of view. In this work the location of the acoustic horizon inside such fluid flow is identified and the associated acoustic surface gravity is estimated. We study the dependence of such surface gravity as a function of the Kerr parameter as well as with other dynamical and thermodynamic variables governing the fluid flow under strong gravity, and demonstrate that for retrograde flow, the surface gravity (and hence the associated analogue Hawking temperature) correlates with the black hole spin in general, whereas for the prograde flow, the surface gravity as well as the analogue temperature correlates with the black hole spin for slow to moderately rotating holes, but anti-correlates with the spin for fast to extremely rotating holes. We found that for certain values of the initial boundary conditions, more than one acoustic horizons, namely two black hole types and one white hole type, may form, and the surface gravity may become formally infinite at the acoustic white hole. We discuss the possible connection between the corresponding analogue Hawking temperature and astrophysically relevant observables associated with the spectral signature of the black hole candidates. Our result indicates that the modified dispersion relation evaluated at the close proximity of the acoustic horizon (and hence the nonuniversal feature of Hawking-like effects) is a sensitive function of the spin angular momentum of the astrophysical black hole. We propose that the black hole spin dependence of such dispersion relation may be used to distinguish a corotating flow from a counter rotating flow for axisymmetric accretion onto a Kerr black hole.

Students who are interested in quantum gravity usually face the difficulty of working through a large amount of prerequisite material before being able to deal with actual quantum gravity. A First Course in Loop Quantum Gravity by Rodolfo Gambini and Jorge Pullin, aimed at undergraduate students, marvellously succeeds in starting from the basics of special relativity and covering basic topics in Hamiltonian dynamics, Yang Mills theory, general relativity and quantum field theory, ending with a tour on current (loop) quantum gravity research. This is all done in a short 173 pages!

As such the authors cannot cover any of the subjects in depth and indeed this book should be seen more as a motivation and orientation guide so that students can go on to follow the hints for further reading. Also, as there are many subjects to cover beforehand, slightly more than half of the book is concerned with more general subjects (special and general relativity, Hamiltonian dynamics, constrained systems, quantization) before the starting point for loop quantum gravity, the Ashtekar variables, are introduced.

The approach taken by the authors is heuristic and uses simplifying examples in many places. However they take care in motivating all the main steps and succeed in presenting the material pedagogically. Problem sets are provided throughout and references for further reading are given. Despite the shortness of space, alternative viewpoints are mentioned and the reader is also referred to experimental results and bounds.

In the second half of the book the reader gets a ride through loop quantum gravity; the material covers geometric operators and their spectra, the Hamiltonian constraints, loop quantum cosmology and, more broadly, black hole thermodynamics. A glimpse of recent developments and open problems is given, for instance a discussion on experimental predictions, where the authors carefully point out the very preliminary nature of the results. The authors close with an 'open issues and controversies' section, addressing some of the criticism of loop quantum gravity and pointing to weak points of the theory. Again, readers aiming at starting research in loop quantum gravity should take this as a guide and motivation for further study, as many technicalities are naturally left out.

In summary this book fully reaches the aim set by the authors – to introduce the topic in a way that is widely accessible to undergraduates – and as such is highly recommended.

The PDF file provided contains web links to all articles in this volume.