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In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a specific volume. Here, we generalize the Einstein gravitational energy–momentum pseudotensor to non-local theories of gravity where analytic functions of the non-local integral operator □ − 1 are taken into account. We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations. The invariance of non-local gravitational action under global translations leads to a locally conserved Noether current, and thus, to the definition of a gravitational energy–momentum pseudotensor, which is an affine object transforming like a tensor under affine transformations. Furthermore, the energy–momentum complex remains locally conserved, thanks to the non-local contracted Bianchi identities. The continuity equations for the gravitational pseudotensor and the energy–momentum complex, taking into account both gravitational and matter components, can be derived. Finally, the weak field limit of pseudotensor is performed to lowest order in metric perturbation in view of astrophysical applications. [ABSTRACT FROM AUTHOR]

An unambiguous definition of gravitational energy remains one of the unresolved issues of physics today. This problem is related to the non-localization of gravitational energy density. In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed by Einstein, Tolman, Landau and Lifshitz, Papapetrou, Møller, and Weinberg. In this review, we firstly explored the energy–momentum complex in an n t h order gravitational Lagrangian L = L g μ ν , g μ ν , i 1 , g μ ν , i 1 i 2 , g μ ν , i 1 i 2 i 3 , ⋯ , g μ ν , i 1 i 2 i 3 ⋯ i n and then in a gravitational Lagrangian as L g = (R ¯ + a 0 R 2 + ∑ k = 1 p a k R □ k R) − g . Its gravitational part was obtained by invariance of gravitational action under infinitesimal rigid translations using Noether's theorem. We also showed that this tensor, in general, is not a covariant object but only an affine object, that is, a pseudo-tensor. Therefore, the pseudo-tensor τ α η becomes the one introduced by Einstein if we limit ourselves to General Relativity and its extended corrections have been explicitly indicated. The same method was used to derive the energy–momentum complex in f R gravity both in Palatini and metric approaches. Moreover, in the weak field approximation the pseudo-tensor τ α η to lowest order in the metric perturbation h was calculated. As a practical application, the power per unit solid angle Ω emitted by a localized source carried by a gravitational wave in a direction x ^ for a fixed wave number k under a suitable gauge was obtained, through the average value of the pseudo-tensor over a suitable spacetime domain and the local conservation of the pseudo-tensor. As a cosmological application, in a flat Friedmann–Lemaître–Robertson–Walker spacetime, the gravitational and matter energy density in f (R) gravity both in Palatini and metric formalism was proposed. The gravitational energy–momentum pseudo-tensor could be a useful tool to investigate further modes of gravitational radiation beyond two standard modes required by General Relativity and to deal with non-local theories of gravity involving □ − k terms. [ABSTRACT FROM AUTHOR]

We derive gravitational waves in a theory with non-local curvature corrections to the Hilbert–Einstein Lagrangian. In addition to the standard two massless tensor modes, with plus and cross polarizations, helicity 2 and angular frequency ω1, we obtain a further scalar massive mode with helicity 0 and angular frequency ω2, whose polarization is transverse. It is a breathing mode, which, at the lowest order of an effective parameter γ, presents a speed difference between nearly null and null plane waves. Finally, the quasi-Lorentz E(2)-invariant class for the non-local gravity is type N3, according to the Petrov classification. This means that the presence (or absence) of gravitational wave modes is observer-independent. [ABSTRACT FROM AUTHOR]

We investigate gravitational waves in the f (Q) gravity, i.e., a geometric theory of gravity described by a non-metric compatible connection, free from torsion and curvature, known as symmetric-teleparallel gravity. We show that f (Q) gravity exhibits only two massless and tensor modes. Their polarizations are transverse with helicity equal to two, exactly reproducing the plus and cross tensor modes typical of General Relativity. In order to analyze these gravitational waves, we first obtain the deviation equation of two trajectories followed by nearby freely falling point-like particles and we find it to coincide with the geodesic deviation of General Relativity. This is because the energy-momentum tensor of matter and field equations are Levi-Civita covariantly conserved and, therefore, free structure-less particles follow, also in f (Q) gravity, the General Relativity geodesics. Equivalently, it is possible to show that the curves are solutions of a force equation, where an extra force term of geometric origin, due to non-metricity, modifies the autoparallel curves with respect to the non-metric connection. In summary, gravitational waves produced in non-metricity-based f (Q) gravity behave as those in torsion-based f (T) gravity and it is not possible to distinguish them from those of General Relativity only by wave polarization measurements. This shows that the situation is different with respect to the curvature-based f (R) gravity where an additional scalar mode is always present for f (R) ≠ R. [ABSTRACT FROM AUTHOR]

The teleparallel equivalent of higher order Lagrangians like L□R = −R + a0R2 + a1R□R can be obtained by means of the boundary term B = 2∇μTμ. In this perspective, we derive the field equations in presence of matter for higher-order teleparallel gravity considering, in particular, sixth-order theories where the □ operator is linearly included. In the weak field approximation, gravitational wave solutions for these theories are derived. Three states of polarization are found: the two standard + and × polarizations, namely 2-helicity massless transverse tensor polarizations, and a 0-helicity massive, with partly transverse and partly longitudinal scalar polarization. Moreover, these gravitational waves (GWs) exhibit four oscillation modes related to four degrees of freedom: the two classical + and × tensor modes of frequency ω1, related to the standard Einstein waves with ; two mixed longitudinal-transverse scalar modes for each frequencies ω2 and ω3, related to two different 4-wave vectors, and. The four degrees of freedom are the amplitudes of each individual mode, i.e. , , , and. To describe a general teleparallel gravity model of order (2p + 2), we used the teleparallel Lagrangian which demonstrates to be not equivalent to. By varying its action, the related field equations in the presence of matter are derived. Hence we obtain the GWs for these teleparallel gravity models, i.e. , that are exactly the Einstein GWs as in f(T) teleparallel gravity. In conclusion, the boundary term B generates extra polarizations and additional modes beyond the standard ones. In particular B2 and B□B terms generate both the additional scalar polarization and the extra scalar modes. [ABSTRACT FROM AUTHOR]

We derive the gravitational waves for f T , B gravity which is an extension of teleparallel gravity and demonstrate that it is equivalent to f(R) gravity by linearized the field equations in the weak field limit approximation. f(T, B) gravity shows three polarizations: the two standard of general relativity, plus and cross, which are purely transverse with two-helicity, massless tensor polarization modes, and an additional massive scalar mode with zero-helicity. The last one is a mix of longitudinal and transverse breathing scalar polarization modes. The boundary term B excites the extra scalar polarization and the mass of scalar field breaks the symmetry of the TT gauge by adding a new degree of freedom, namely a single mixed scalar polarization. [ABSTRACT FROM AUTHOR]

We derive the gravitational energy–momentum pseudotensor τ λ σ in metric f (R) gravity and in teleparallel f (T) gravity. In the first case, R is the Ricci curvature scalar for a torsionless Levi-Civita connection; in the second case, T is the curvature-free torsion scalar derived by tetrads and Weitzenböck connection. For both classes of theories the continuity equations are obtained in presence of matter. f (R) and f (T) are non-equivalent, but differ for a quantity ω (T , B) containing the torsion scalar T and a boundary term B. It is possible to obtain the field equations for ω (T , B) and the related gravitational energy–momentum pseudotensor τ λ σ | ω. Finally we show that, thanks to this further pseudotensor, it is possible to pass from f (R) – f (T) and vice versa through a simple relation between gravitational pseudotensors. [ABSTRACT FROM AUTHOR]

We derive the gravitational energy–momentum pseudo-tensor τ μ ν μ in both Palatini and metric approaches to f (R) gravity. We then obtain the related cosmological gravitational energy density. Considering a flat Friedmann–Lemaître–Robertson–Walker spacetime, the energy density complex of matter and gravitation vanishes in the metric approach, but results non-vanishing in the Palatini formalism. This feature could be relevant in order to physically discriminate between the two approaches. • Gravitational energy–momentum pseudotensor. • Palatini f(R) gravity. • Metric f (R) gravity. • Cosmological background. [ABSTRACT FROM AUTHOR]