The confrontation between general relativity and experiment: An update

The status of experimental tests of general relativity to the end of 1983 is reviewed. The experimental support for the Einstein equivalence principle is summarized. If this principle is valid, gravitation must be described by a curved space-time, “metric” theory of gravity. General properties of metric theories are described and the parametrized post-Newtonian (PPN) formalism for treating the weak-field, slow-motion limit of such theories is set up. A zoo of selected metric theories of gravity is presented. Experimental tests of metric theories are then described, including the “classical” tests, tests of the strong equivalence principle, and others. The possibility of using gravitational-wave observations to test metric theories is discussed. A review is presented of the binary pulsar, in which the first evidence for gravitational radiation has been found. Finally cosmological tests of alternative theories are briefly described.     

Modified dispersion relations in Hořava–Lifshitz gravity and Finsler brane models

We study possible links between quantum gravity phenomenology encoding Lorentz violations as nonlinear dispersions, the Einstein–Finsler gravity models, EFG, and nonholonomic (non-integrable) deformations to Hořava–Lifshitz, HL, and/or Einstein’s general relativity, GR, theories. EFG and its scaling anisotropic versions formulated as Hořava–Finsler models, HF, are constructed as covariant metric compatible theories on (co) tangent bundle to Lorentz manifolds and respective anisotropic deformations. Such theories are integrable in general form and can be quantized following standard methods of deformation quantization, A-brane formalism and/or (perturbatively) as a nonholonomic gauge like model with bi-connection structure. There are natural warping/trapping mechanisms, defined by the maximal velocity of light and locally anisotropic gravitational interactions in a (pseudo) Finsler bulk spacetime, to four dimensional (pseudo) Riemannian spacetimes. In this approach, the HL theory and scenarios of recovering GR at large distances are generated by imposing nonholonomic constraints on the dynamics of HF, or EFG, fields.     

Loop quantum modified gravity and its cosmological application

A general nonperturvative loop quantization procedure for metric modified gravity is reviewed. As an example, this procedure is applied to scalar-tensor theories of gravity. The quantum kinematical framework of these theories is rigorously constructed. Both the Hamiltonian and master constraint operators are well defined and proposed to represent quantum dynamics of scalar-tensor theories. As an application to models, we set up the basic structure of loop quantum Brans-Dicke cosmology. The effective dynamical equations of loop quantum Brans-Dicke cosmology are also obtained, which lay a foundation for the phenomenological investigation to possible quantum gravity effects in cosmology.     

Violation of the equivalence principle in modified theories of gravity

We show that the metric in f(R) theories of gravity in Palatini formalism can be solved as the product of a rank-two tensor times a scalar function which is very sensitive to the local energy-momentum densities. This local dependence of the metric generates new gravitationally-induced microscopic interactions, which eventually would lead to self-accelerated test body trajectories. These facts make very unlikely the viability of Palatini f(R) models designed to change the late-time cosmic evolution.     

Combinatorial Space from Loop Quantum Gravity

The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectively invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.     

Kinematics and dynamics off(R) theories of gravity

We generalise the equations governing relativistic fluid dynamics given by Ehlers and Ellis for general relativity, and by Maartens and Taylor for quadratic theories, to generalisedf(R) theories of gravity. In view of the usefulness of this alternative framework to general relativity, its generalisation can be of potential importance for deriving analogous results to those obtained in general relativity. We generalise, as an example, the results of Maartens and Taylor to show that within the framework of generalf(R) theories, a perfect fluid spacetime with vanishing vorticity, shear and acceleration is Friedmann-Lemaître-Robertson-Walker only if the fluid has in addition a barotropic equation of state. It then follows that the Ehlers-Geren-Sachs theorem and its almost extension also hold forf(R) theories of gravity.     

Energy Distribution for a Power Model of the Plane Symmetric Spacetime in f(R) Gravity

The study of the energy localization in f(R) theories of gravity has attracted much interest in recent years. In this paper, the vacuum solutions of the modified field equations for a power model of plane symmetric metric are studied in metric f(R) gravity with the assumption of constant Ricci scalar. Next, we determine the energy-momentum complexes in f(R) theories of gravity for this spacetime for some important models. We also show that these models satisfy the stability and constant curvature conditions.     

Energy Distribution for a Power Model of the Plane Symmetric Spacetime in f (R) Gravity

The study of the energy localization in f(R)theories of gravity has attracted much interest in recent years.In this paper,the vacuum solutions of the modified field equations for a power model of plane symmetric metric are studied in metric f(R)gravity with the assumption of constant Ricci scalar.Next,we determine the energy-momentum complexes in f(R)theories of gravity for this spacetime for some important models.We also show that these models satisfy the stability and constant curvature conditions.     

First order formalism off(R) gravity

The first order formalism is applied to study the field equations of a general Lagrangian density for gravity of the form . These field equations correspond to theories which are a subclass of conformally metric theories in which the derivative of the metric is proportional to the metric by a Weyl vector field. The resulting geometrical structure is unique, except whenf(R)=aR ², in the sense that the Weyl field is identifiable in terms of the trace of the energy-momentum tensor and its derivatives. In the casef(R)=aR ² the metric is only defined up to a conformai factor. We discuss the matter conservation equations which are implied by the invariance of the theories under diffeomorphisms. We apply the results to the case of dust and obtain that in general the dust particles will not follow geodesic Unes. We consider the linearized field equations and apply them to obtain the weak field slow motion limit. It is found that the gravitational potential acquires a new term which depends linearly on the mass density. The importance of these new equations is briefly discussed.     

Black holes in scalar-tensor gravity

Hawking has proven that black holes which are stationary as the end point of gravitational collapse in Brans-Dicke theory (without a potential) are no different than in general relativity. We extend this proof to the much more general class of scalar-tensor and f(R) gravity theories, without assuming any symmetries apart from stationarity.     

Loop-quantum-gravity vertex amplitude

Spin foam models are hoped to provide the dynamics of loop-quantum gravity. However, the most popular of these, the Barrett-Crane model, does not have the good boundary state space and there are indications that it fails to yield good low-energy n-point functions. We present an alternative dynamics that can be derived as a quantization of a Regge discretization of Euclidean general relativity, where second class constraints are imposed weakly. Its state space matches the SO(3) loop gravity one and it yields an SO(4)-covariant vertex amplitude for Euclidean loop gravity.     

Kerr black holes are not unique to general relativity

Considerable attention has recently focused on gravity theories obtained by extending general relativity with additional scalar, vector, or tensor degrees of freedom. In this Letter, we show that the black-hole solutions of these theories are essentially indistinguishable from those of general relativity. Thus, we conclude that a potential observational verification of the Kerr metric around an astrophysical black hole cannot, in and of itself, be used to distinguish between these theories. On the other hand, it remains true that detection of deviations from the Kerr metric will signify the need for a major change in our understanding of gravitational physics.     

Probing the formalism through gravitational wave polarizations

The direct observation of gravitational waves (GW) in the near future, and the corresponding determination of the number of independent polarizations, is a powerful tool to test general relativity and alternative theories of gravity. In the present work we use the Newman–Penrose formalism to characterize GWs in quadratic gravity and in a particular class of f(R) Lagrangians. We find that both quadratic gravity and the f(R) theory belong to the most general invariant class of GWs, i.e., they can present up to six independent polarizations of GWs. For a particular combination of the parameters, we find that quadratic gravity can present up to five polarizations states. On the other hand, if we use the Palatini approach for f(R) theories, GWs present only the usual two transverse-traceless polarizations such as in general relativity. Thus, we conclude that the observation of GWs can strongly constrain the suitable formalism for these theories.     

Scalar–tensor representation of f(R) gravity and Birkhoff's theorem

Birkhoff's theorem is discussed in the frame of f(R) gravity by using its scalar–tensor representation. Modified gravity has become very popular in recent times as it is able to reproduce the unification of inflation and late‐time acceleration with no need of a dark energy component or an inflation field. Here, another aspect of modified f(R) gravity is studied, specifically the range of validity of Birkhoff's theorem, compared with another alternative to general relativity, the well‐known Brans–Dicke theory. As a novelty, here both theories are studied using a conformal transformation and writing the actions in the Einstein frame, where spherically symmetric solutions are studied using perturbation techniques. The differences between both theories are analyzed as well as the validity of the theorem within the Jordan and Einstein frames, where interesting results are obtained.     

Einstein Gravity in Almost Kähler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics

We argue that the Einstein gravity theory can be reformulated in almost Kähler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation on manifolds enabled with nonholonomic distributions is considered. We prove that, for certain types of nonholonomic constraints, there are modelled effective Lagrangians which do not develop instabilities. It is also elaborated a linearization formalism for anholonomic noncommutative gravity theories models and analyzed the stability of stationary ellipsoidal solutions defining some nonholonomic and/or nonsymmetric deformations of the Schwarzschild metric. We show how to construct nonholonomic distributions which remove instabilities in nonsymmetric gravity theories. It is concluded that instabilities do not consist a general feature of theories of gravity with nonsymmetric metrics but a particular property of some models and/or unconstrained solutions.     

Deformations of the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity

We show that the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity can be nontrivially deformed by allowing the cosmological constant to become an arbitrary function of the (Weyl) curvature. Our result implies that there is not one but infinitely many (parametrized by an arbitrary function) four-dimensional generally covariant local gravity theories propagating 2 degrees of freedom.     

Discreteness of the volume of space from Bohr-Sommerfeld quantization

A major challenge for any theory of quantum gravity is to quantize general relativity while retaining some part of its geometrical character. We present new evidence for the idea that this can be achieved by directly quantizing space itself. We compute the Bohr-Sommerfeld volume spectrum of a tetrahedron and show that it reproduces the quantization of a grain of space found in loop gravity.     

北京光速各向异性实验的广义相对论(及度规理论)预言值

基于广义相对论及度规引力理论的基本原理,计算了北京“光速各向异性检验”实验的预期值。结论是该实验将不可能在广义相对论传统观点和文献[1]提出的观点之间作出判断。     

Cosmological tests of gravity

Modifications of general relativity provide an alternative explanation to dark energy for the observed acceleration of the universe. We review recent developments in modified gravity theories, focusing on higher-dimensional approaches and chameleon/f(R) theories. We classify these models in terms of the screening mechanisms that enable such theories to approach general relativity on small scales (and thus satisfy solar system constraints). We describe general features of the modified Friedman equation in such theories.The second half of this review describes experimental tests of gravity in light of the new theoretical approaches. We summarize the high precision tests of gravity on laboratory and solar system scales. We describe in some detail tests on astrophysical scales ranging from ∼ kpc (galaxy scales) to ∼ Gpc (large-scale structure). These tests rely on the growth and inter-relationship of perturbations in the metric potentials, density and velocity fields which can be measured using gravitational lensing, galaxy cluster abundances, galaxy clustering and the integrated Sachs-Wolfe effect. A robust way to interpret observations is by constraining effective parameters, such as the ratio of the two metric potentials. Currently tests of gravity on astrophysical scales are in the early stages — we summarize these tests and discuss the interesting prospects for new tests in the coming decade.     

Canonical quantization of general relativity in discrete space-times

It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. We analyze discrete lattice general relativity and develop a canonical formalism that allows one to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a "dynamical gauge" fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. The problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories. A simple cosmological application exhibits quantum elimination of the singularity at the big bang.