Exact theory of the (Einstein) gravitational field in an arbitrary background space-time

The Lagrangian based theory of the gravitational field and its sources at the arbitrary background space-time is developed. The equations of motion and the energy-momentum tensor of the gravitational field are derived by applying the variational principle. The gauge symmetries of the theory and the associated conservation laws are investigated. Some properties of the energymomentum tensor of the gravitational field are described in detail and the examples of its application are given. The desire to have the total energymomentum tensor as a source for the linear part of the gravitational field leads to the universal coupling of gravity with other fields (as well as to the self-interaction) and finally to the Einstein theory.     

Modified gravity with arbitrary coupling between matter and geometry

The field equations of a generalized f(R)$f(R)$ type gravity model, in which there is an arbitrary coupling between matter and geometry, are obtained. The equations of motion for test particles are derived from a variational principle in the particular case in which the Lagrange density of the matter is an arbitrary function of the energy-density of the matter only. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the model is also considered. The perihelion precession of an elliptical planetary orbit in the presence of an extra force is obtained in a general form, and the magnitude of the extra gravitational effects is constrained in the case of a constant extra force by using Solar System observations.     

Palatini formulation of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity theory,and its cosmological implications

We consider the Palatini formulation of f(R, T) gravity theory, in which a non-minimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similar to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the f(R, T) gravity in the Palatini formulation. Cosmological models with Lagrangians of the type \(f=R-\alpha ^2/R+g(T)\) and \(f=R+\alpha ^2R^2+g(T)\) are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.     

Post-post-Newtonian equations of motion for point particles in the general theory of relativity

The post-post-Newtonian equations of motion for point particles are derived from the Einstein gravitational field equations by using the Einstein-Infeld-Hoffmann method with the help of the energy-momentum tensor proposed by Infeld and Plebanski [5, 6]. The obtained equations of motion coincide with the equations derived by Kopeikin [10] by using the Fock method.     

Gravitational Energy-Momentum and Conservation of Energy-Momentum in General Relativity

Based on a general variational principle, Einstein-Hilbert action and sound facts from geometry, it is shown that the long existing pseudotensor, non-localizability problem of gravitational energy-momentum is a result of mistaking different geometrical, physical objects as one and the same. It is also pointed out that in a curved spacetime, the sum vector of matter energy-momentum over a finite hyper-surface can not be defined. In curvilinear coordinate systems conservation of matter energy-momentum is not the continuity equations for its components. Conservation of matter energy-momentum is the vanishing of the covariant divergence of its density-flux tensor field. Introducing gravitational energy-momentum to save the law of conservation of energy-momentum is unnecessary and improper. After reasonably defining “change of a particle’s energy-momentum”, we show that gravitational field does not exchange energy-momentum with particles. And it does not exchange energy-momentum with matter fields either. Therefore, the gravitational field does not carry energy-momentum, it is not a force field and gravity is not a natural force.     

Can the Local Energy-Momentum Conservation Laws be Derived Solely from Field Equations?

The vanishing of the divergence of the matter stress-energy tensor for General Relativity is a particular case of a general identity, which follows from the covariance of the matter Lagrangian in much the same way as (generalized) Bianchi identities follow from the covariance of the purely gravitational Lagrangian. This identity, holding for any covariant theory of gravitating matter, relates the divergence of the stress tensor with a combination of the field equations and their derivatives. One could thus wonder if, according to a recent suggestion [1], the energy-momentum tensor for gravitating fields can be computed through a suitable rearrangement of the matter field equations, without relying on the variational definition. We show that this can be done only in particular cases, while in general it leads to ambiguities and possibly to wrong results. Moreover, in nontrivial cases the computations turn out to be more difficult than the standard variational technique.     

New energy-momentum tensor in relativistic gravitation theory

Expressions for a new canonical energy-momentum tensor and an internal angular momentum of the gravitational field are derived in the context of bimetric relativistic gravitation theory based on the variational principle. A system of relations for the determining parameters of the gravitational field and matter involving, in particular, the continuity condition for the energy-momentum flux density is formulated on the discontinuity surface. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 52–58, April, 2006.     

Dynamic field theory and equations of motion in cosmology

We discuss a field-theoretical approach based on general-relativistic variational principle to derive the covariant field equations and hydrodynamic equations of motion of baryonic matter governed by cosmological perturbations of dark matter and dark energy. The action depends on the gravitational and matter Lagrangian. The gravitational Lagrangian depends on the metric tensor and its first and second derivatives. The matter Lagrangian includes dark matter, dark energy and the ordinary baryonic matter which plays the role of a bare perturbation. The total Lagrangian is expanded in an asymptotic Taylor series around the background cosmological manifold defined as a solution of Einstein’s equations in the form of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric tensor. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the series expansion is gauge-invariant and all of them together form a basis for the successive post-Friedmannian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Friedmannian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background FLRW metric. The temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large scale inhomogeneities in these two components as those generated by the primordial cosmological perturbations with an effective matter density contrast δρ/ρ≤1$\delta \rho /\rho \le 1$. The small scale inhomogeneities are generated by the condensations of baryonic matter considered as the bare perturbations of the background manifold that admits δρ/ρ?1$\delta \rho /\rho ?1$. Mathematically, the large scale perturbations are given by the homogeneous solution of the linearized field equations while the small scale perturbations are described by a particular solution of these equations with the bare stress–energy tensor of the baryonic matter. We explicitly work out the covariant field equations of the successive post-Friedmannian approximations of Einstein’s equations in cosmology and derive equations of motion of large and small scale inhomogeneities of dark matter and dark energy. We apply these equations to derive the post-Friedmannian equations of motion of baryonic matter comprising stars, galaxies and their clusters.     

Field equations and conservation laws in the nonsymmetric gravitational theory

The field equations in the nonsymmetric gravitational theory are derived from a Lagrangian density using a first-order formalism. Using the general covariance of the Lagrangian density, conservation laws and tensor identities are derived. Among these are the generalized Bianchi identities and the law of energy-momentum conservation. The Lagrangian density is expanded to second-order, and treated as an Einstein plus fields theory. From this, it is deduced that the energy is positive in the radiation zone.     

On a Lagrangian formulation of gravitoelectromagnetism

We examine a Lagrangian formulation of gravity based on an approach analogous to electromagnetism, called Gravitoelectromagnetism (GEM). The gravitational analogue of the electromagnetic field tensor is a three-index tensor, \({\mathcal {F}_{\mu\nu\lambda}}\), defined in terms of a two-index gravitoelectromagnetic potential, \({\mathcal {A}_{\mu\nu}}\). The energy-momentum tensor is derived and is symmetric. We construct a Lagrangian which allows us to describe interactions between fermions, photons and gravitons. We calculate transition amplitudes of various processes involving gravitons: gravitational Møller scattering, gravitational Compton scattering, and the graviton photoproduction.     

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.     

On the phenomenological three-graviton vertex

In General Relativity, the graviton interacts in three-graviton vertex with a tensor that is not the energy-momentum tensor of the gravitational field. We consider the possibility that the graviton interacts with the definite gravitational energy-momentum tensor that we previously found in the G ² approximation. This tensor in a gauge, where nonphysical degrees of freedom do not contribute, is remarkable, because it gives positive gravitational energy density for the Newtonian center in the same manner as the electromagnetic energy-momentum tensor does for the Coulomb center. We show that the assumed three-graviton vertex does not lead to contradiction with the precession of Mercury’s perihelion. In the S-matrix approach used here, the external gravitational field has only a subsidiary role, similar to the external field in quantum electrodynamics. This approach with the assumed vertex leads to the gravitational field that cannot be obtained from a consistent gravity equation.     

Lorentz-Covariant theory of gravitation

A Lorentz-covariant theory of gravitation is proposed. It is based on a simple form of the Lagrangian for the gravitational field. The field equations have a simple mathematical structure where the energy-momentum tensor of matter and of gravitational field is the source of the field. The theory agrees with general relativity for the three well-known effects, i.e., red shift, deflection of light, and perihelion.     

Modified <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity models with curvature–matter coupling

A modified f(G) gravity model with coupling between matter and geometry is proposed, which is described by the product of the Lagrange density of the matter and an arbitrary function of the Gauss–Bonnet term. The field equations and the equations of motion corresponding to this model show the non-conservation of the energy-momentum tensor, the presence of an extra force acting on test particles and non-geodesic motion. Moreover, the energy conditions and the stability criterion at the de Sitter point in modified f(G) gravity models with curvature–matter coupling are derived, which can degenerate to the well-known energy conditions in general relativity. Furthermore, in order to get some insight in the meaning of these energy conditions, we apply them to the specific models of f(G) gravity and the corresponding constraints on the models are given. In addition, the conditions and the candidate for late-time cosmic accelerated expansion in modified f(G) gravity are studied by means of conditions of power-law expansion and the equation of state of matter ω smaller than -\frac13-\frac{1}{3}.     

Spherically symmetric solution in higher-dimensional teleparallel equivalent of general relativity

A theory of(N+1)-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity(TEGR).The fundamental gravitational field variables are the(N+1)-dimensional vector fields,defined globally on a manifold M,and the gravitational field is attributed to the torsion.The form of Lagrangian density is quadratic in torsion tensor.We then give an exact five-dimensional spherically symmetric solution(Schwarzschild(4+1)-dimensions).Finally,we calculate energy and spatial momentum using gravitational energy-momentum tensor and superpotential 2-form.     

Basic properties of the Einstein equations with a Ricci-Scalar-Dependent cosmological term

Basic properties of the Einstein equations modified by a cosmological Λ-term dependent on the Ricci scalar R are considered. We show that in addition to a nonzero divergence of the energy-momentum tensor of the matter and the consequent cold matter mass nonconservation as the Universe expands, this model suggests a significant modification of the equations for the gravitational potential and particle acceleration in the Newtonian approximation. These circumstances allow the necessary criteria for possible functional dependences Λ(R) to be formulated. Nevertheless, by introducing a variable Λ-term, we can look at the problems of dark matter and dark energy anew. In particular, we show that the model in which the cosmological term depends linearly on the Ricci scalar (this corresponds to the approximation of a more complex dependence in the case of low matter densities) makes it possible to satisfactorily describe the rotation curves of galaxies without invoking the dark matter hypothesis and to construct a cosmological model with a variable vacuum energy density, in qualitative agreement with the present views of the early Universe.     

The simplest non-minimal matter–geometry coupling in the <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) cosmology

f(R, T) gravity is an extended theory of gravity in which the gravitational action contains general terms of both the Ricci scalar R and the trace of the energy-momentum tensor T. In this way, f(R, T) models are capable of describing a non-minimal coupling between geometry (through terms in R) and matter (through terms in T). In this article we construct a cosmological model from the simplest non-minimal matter–geometry coupling within the f(R, T) gravity formalism, by means of an effective energy-momentum tensor, given by the sum of the usual matter energy-momentum tensor with a dark energy contribution, with the latter coming from the matter–geometry coupling terms. We apply the energy conditions to our solutions in order to obtain a range of values for the free parameters of the model which yield a healthy and well-behaved scenario. For some values of the free parameters which are submissive to the energy conditions application, it is possible to predict a transition from a decelerated period of the expansion of the universe to a period of acceleration (dark energy era). We also propose further applications of this particular case of the f(R, T) formalism in order to check its reliability in other fields, rather than cosmology.     

Motion of matter in metric-affine theory of gravitation

Based on the Lie derivative technique in a general space with affine connection (L₄, g), we show that in the metric-affine theory of gravitation, the law of conservation of the energy-momentum tensor for matter and consequently also the equations of motion for matter stemming from this law are (as in the general theory of relativity) a consequence of the gravitational field equations. We derive the hydrodynamic equation of motion for an ideal Weyssenhoff—Raabe spin fluid in Weyl space. We discuss the possibilities for observation of space—time nonmetricity.Moscow State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 76–82, January, 1994.     

A Non-geometric Relativistic Theory of Gravitation

A non-geometric relativistic theory of gravitation is developed by defining a semi-metric to replace the metric tensor as gravitational vector potential. The theory show that the energy-momentum tensor of the gravitational field belong to the gravitational source, gravitational radiation is contained in Einstein’s field equations that including the contribution of gravitational field, the real physical singularity in the gravitational field can be eliminated, and the dark matter in the universe is interpreted as the matter of pure gravitational field.