Energy-momentum complex in higher order curvature-based local gravity

Citation: Capozziello, S.; Capriolo, M.; Lambiase, G. Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity. Particles 2022,5, 298–330. https://doi.org/10.3390/ particles5030026 Academic Editor: Armen Sedrakian Received: 15 July 2022 Accepted: 1 August 2022 Published: 10 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Review Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity Salvatore Capozziello 1,2,3,4,*,† , Maurizio Capriolo 2,5,† and Gaetano Lambiase 5,6,† 1 Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II”, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 Napoli, Italy 2Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 Napoli, Italy 3Scuola Superiore Meridionale, Largo S. Marcellino 10, I-80138 Napoli, Italy 4Department of Mathematics, Faculty of Civil Engineering, VSB-Technical University of Ostrava, Ludvika Podeste 1875/17, 708 00 Ostrava, Czech Republic 5Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano, Italy 6 Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Gruppo Collegato di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano, Italy *Correspondence: [email protected] † These authors contributed equally to this work. Abstract: 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 nth order gravitational Lagrangian L=Lgµν,gµν,i1,gµν,i1i2,gµν,i1i2i3,··· ,gµν,i1i2i3···in and then in a gravitational Lagrangian as Lg= (R+a0R2+∑p k=1akRkR)√−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 τη α 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 τη α 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 ˆ 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 −kterms. Keywords: energy–momentum complex; pseudo-tensor; gravitational energy 1. Introduction A widely accepted definition of gravitational energy density and its localization in curved spacetime are serious problems that afflict the general relativity. Several prescriptions for gravitational contribution to energy–momentum density and more generally for energy–momentum complex have been suggested by Einstein, Tolman, Landau and Particles 2022,5, 298–330. https://doi.org/10.3390/particles5030026 https://www.mdpi.com/journal/particles Particles 2022,5299 Lifshitz, Papapetrou, Møller, and Weinberg [ 1 – 10 ]. These attempts are based on the introduction of a super-potential or through the expansion of the Ricci tensor in the metric perturbation h . Thus, the gravitational part of the energy–momentum density transforms as an affine tensor not as a covariant tensor, and for this reason, it is not really a tensor but a pseudo-tensor. This affine property of the gravitational stress–energy tensor makes the gravitational energy–momentum density not localizable. However, integrating the density over a suitable spatial region at a certain time such as over an asymptotically flat spacetime, viable for isolated systems, the gravitational energy–momentum becomes a four-vector, as meaning that changes in right way under asymptotically flat coordinate transformations. Over all space it becomes quasi independent of the coordinate system, that is, the gravitational energy–momentum of the spacetime exists, but it cannot be localized. In this review a generalization of Einstein’s pseudo-tensor to Extended Theories of Gravity [ 11 , 12 ] is proposed by imposing the invariance of the higher order gravitational Lagrangian under an infinitesimal rigid translation and by using Noether’s theorem. Then, thanks to a continuity equation, a Noether current and a Noether charge were derived that correspond to a gravitational energy–momentum pseudo-tensor and gravitational energy–momentum, respectively, both locally conserved. By weakly perturbing the metric tensor around the Minkowskian metric, a weak-field limit, in a suitable gauge, the gravitational energy–momentum pseudo-tensor for a Lagrangian of nth order appears an object easier to handle. Then, by averaging of the pseudo-tensor over a suitable spacetime domain, it is possible to calculate the power emitted by some localized astrophysical source carried away by the gravitational waves. This approach could be relevant for searching for polarization states of gravitational waves in addition to the two standards of general relativity [ 13 , 14 ]. Finally, after deriving the gravitational energy–momentum pseudo-tensor in F(R) gravity formulated in Palatini and metric formalism, some cosmological applications were discussed, wherein a flat FLRW metric the total energy density was obtained in both approaches [15,16]. For more details on the issue of energy–momentum localization in modified theories of gravity such as f(R) , f(R,R, . . . , kR) [ 16 , 17 ], teleparallel gravity and its extended version f(T) , see Ref. [ 18 ]. Meanwhile, for a study of wavelike solutions of modified teleparallel gravity necessary for future applications of the pseudo-tensor, see references [19,20]. The review is organized as follows. Firstly in Section 2some definitions of gravitational pseudo-tensors in general relativity are listed. In Section 3.1 we derived the gravitational energy–momentum pseudo-tensor for a general Lagrangian of nth order through two procedures: the first method uses a variational principle under rigid transformations via Noether’s theorem and the second adopts the Landau–Lifshitz procedure [ 1 ] without the introduction of the super-potential. Hence, in Section 3.2, we proved that a stress– energy object is transformed in the correct manner under linear transformations but not under diffeomorphisms and, therefore, is a pseudo-tensor and not a covariant tensor. In Section 3.3, we calculated the Euler–Lagrange equations and the gravitational energy– momentum pseudo-tensor for f(R) gravity, always using Noether’s theorem applied to a particular one-parameter group of diffeomorphisms given by rigid translations. Therefore, in all models of gravity we obtained the continuity equation for an energy–momentum complex. In Section 3.4, we derived the gravitational energy–momentum pseudo-tensor of a gravitation field for a particular Lagrangian Lg= (R+a0R2+∑p k=1akRkR)√−g . Section 3.5, is devoted to the weak-field limit of the gravitational stress–energy pseudotensor expanded to lowest order in a small perturbation h , i.e., up to h2 order, and we have shown two simple cases where the index p is equal to zero and one. Hence, in Section 4.1, we averaged the pseudo-tensor over an suitable region containing the isolated sources and then we found the emitted power carried by the gravitational radiation. Afterward, in Section 5.1, in Palatini f(R) gravity, related field equations and related gravitational energy–momentum pseudo-tensor were obtained. Therefore in Section 6, by adopting a flat FLRW spacetime, an explicit calculus of an energy density complex for Particles 2022,5300 power law cosmological solutions was performed, also in the metric formalism of f(R) . Conclusions are summarized in Section 7. Finally in Appendix A.1, we proved that the additive terms related to the symmetries of gµν and its derivatives yield a mean of zero, i.e., hApη αi=hBpη αi= 0. While in Appendix A.2, we explicitly showed the six polarization tensors associated with the gravitational waves present in higher-order theories. 2. Several Definitions of Gravitational Energy–Momentum Pseudo-Tensor in General Relativity Here are some of the most important definitions of gravitational energy–momentum pseudo-tensor in general relativity in the scientific literature, for details see [21]. 2.1. Einstein Energy–Momentum Complex In special relativity the law of conservation of energy and momentum is given by ∂Tµν ∂xµ=0 , (1) with Tµν the energy–momentum tensor of matter and non-gravitational fields. In general relativity this principle becomes for general covariance ∇µTµν =0 , (2) which does not correspond to any law of conservation of physical quantities. Einstein therefore formulated the conservation law in the following way ∂θ ν µ ∂xν=∂ ∂xνp−gTν µ+tν µ=0 , (3) where tν µ is an pseudo-tensor. So what is conserved is not only the tensor of nongravitational fields and matter Tν µ but a pseudo-tensor tν µ must be added to it. This pseudo-tensor added can be interpreted as associated with the gravitational field and the energy due to the sum of the contributions of the gravitational fields plus those due to the matter is conserved. However, the pseudo-tensoriality behaviour of tν µ makes it dependent on coordinates and the gravitational energy becomes non localizable. In order to write the Equation (2) in the form of an ordinary divergence equation Equation (3) , Einstein starting from the following Lagrangian density which is a non-covariant scalar density L=p−ggµνΓσ µνΓρ σρ −Γσ µρΓρ νσ, (4) introduced a pseudo-tensor defined by the relation p−gt ν µ=1 16π δν µL−∂L ∂gρσ,ν gρσ,µ!. (5) 2.2. Landau–Lifshitz Energy–Momentum Pseudo-Tensor The gravitational energy–momentum pseudo-tensor defined by Landau–Lifshitz has the great advantage of being symmetric unlike Einstein’s, which in general is not.This allows defining the angular momentum and therefore the related conservation law. We adopt a system of geodetic coordinates where the first derivatives of the metric tensor gµν vanish. Then, the Equation (2) is reduced to (1) which can be written in terms of the following antisymmetric quantity in the last two indices ηµνσ =−ηµσν Tµν =∂ηµνσ ∂xσ. (6) Particles 2022,5301 Since the Levi–Civita connection Γ vanishes at one point, in such coordinate system it is possible using Einstein’s equations in the presence of matter written in such coordinates, to express the stress–energy tensor of matter Tµν as Tµν =1 (−g) ∂ ∂xσ1 16π ∂ ∂xρ[(−g)(gµνgσρ −gµσgνρ)], (7) where indicating the term in braces with the antisymmetric quantity in the last two indices hµνσ =−hµσν, we get ∂hµνσ ∂xσ−(−g)Tµν =0 . (8) Returning to an arbitrary coordinate system the previous difference does not cancel anymore so we can indicate it with (−g)tµν or (−g)(Tµν +tµν)=∂hµνσ ∂xσ. (9) Quantities tµν are symmetric but are not the components of a covariant tensor but affine. Using Einstein’s field equations again it is possible from Equation (9) to get an explicit expression of tµν , defined as the energy–momentum pseudo-tensor of the gravitational field, by means of the derivatives of the components of the metric tensor, that is 16π(−g)tµν =gµν,ρgρσ,σ−gµρ,ρgνσ,σ+1 2gµνgρσgρα ,βgβσ ,α −gµρgσαgνα,βgσβ ,ρ+gνρgσαgµα ,βgσβ ,ρ+gρσgαβgµρ,αgνσ ,β +1 8(2gµρgνσ −gµνgρσ)2gαβgγλ −gβγgαλgαλ,ρgβγ ,σ, (10) where gµν =√−ggµν. 2.3. Møller Energy–Momentum Complex The energy–momentum pseudo-tensors tµν of both Einstein and Landau–Lifshitz besides having the flaw of being tensors only affine and not covariant also depend on the choice of coordinates. Then, Møller looked for an expression for energy and gravitational momentum independent of the particular coordinate system. To do this Møller exploited the fact that the pseudo-tensor including matter plus gravity θµν =Tµν +tµν can be defined at less than a magnitude Sµν at zero divergence ∂µSµν = 0. In 1958 Møller proposed the following complex tensor of energy–momentum complex Tν µ=θν µ+Sν µ looking for the Sν µsuch that Tν µtransformed as a tensor for only spatial transformations Tν µ=1 8π∂ρhp−ggµσ,λ−gµλ,σgλνgσρi, (11) where the expression in square brackets is the antisymmetric super-potential Uνρ µ=−Uρν µ such that ∂νTν µ=0 . (12) 2.4. Papapetrou Energy–Momentum Pseudo-Tensor Papapetrou in 1948 used the generalized Belifante method to derive his pseudo-tensor due to the complex of energy–momentum contributions using Tolman’s expression of Einstein’s total pseudo-tensor θν µ(3), i.e., θν µ=1 8π ∂ ∂xρ −gνσ ∂L ∂gµσ,ρ +1 2δν µgαβ ∂L ∂gαβ ,ρ , (13) Particles 2022,5302 where L is Einstein Lagrangian give by (4) and gνσ have been defined in Equation (10) . Belifante’s method consists in finding a symmetric quantity Ωµν =Ωνµ divergence free which differs by ηµρθν ρ only for an antisymmetric quantity divergence in the first two indices Bµνρ =−Bµνρ i.e. Ωµν =ηµρθν ρ+∂ ∂xρBµνρ , (14) such that ∂ ∂xνΩµν =0 , (15) with ηµν =diag(1, −1, −1, −1). Expressing Bµνρ in terms of of the field spin density Sµνρ Bµνρ =−1 2(Sµνρ +Sρµν +Sρνµ), (16) you get after a few counts the expression for the total pseudo-tensor Ωµν Ωµν =1 16π ∂2 ∂xρxσp−g(gµνηρσ −gµρηνσ −gρσηµν −gνσηµρ). (17) This geometric object is symmetric with respect to the first two indicesµeν. 2.5. Weinberg Gravitational Energy–Momentum Pseudo-Tensor Weinberg [22] derived the gravitational energy–momentum pseudo-tensor by adopting a quasi-minkowskian coordinate system. In this system the metric tensor gµν tends to that of Minkowski ηµν at great distances from a localized material system. We write the metric gµν as the sum of the metric of Minkowski ηµν plus hµν that goes to zero to infinity gµν =ηµν +hµν . (18) We linearize Einstein equations Gµν =− 8 πGTµν , expanding Ricci tensor Rµν in terms of powers of hµν as R(1) µν −1 2ηµνR(1)=−8πGTµν +tµν, (19) where tµν =1 8πGRµν −1 2gµνR−R(1) µν +1 2ηµνR(1), (20) is the gravitational energy–momentum pseudo-tensor. So in the Equation (19) you see that reading the equation from right to left, tµν assumes the meaning of the source of the linearized curvature together with the tensor of the non-gravitational fields and of the matter Tµν . From the linearized Bianchi law to which quantity R(1) µν must satisfy, we get the following local conservation law ∂ ∂xν(Tµν +tµν)=0 . (21) The pseudo-tensor tµν to the second order in his tµν =1 8πG−1 2hµνR(1)+1 2ηµνηρσR(1) ρσ +R(2) µν −1 2ηµνηρσR(2) ρσ+Oh3, (22) where Rµν to first order in his R(1) µν =1 2 ∂2hλλ ∂xµ∂xν−∂2hλµ ∂xλ∂xν−∂2hλν ∂xλ∂xµ+∂2hµν ∂xλ∂xλ!, (23) Particles 2022,5303 while to second order hbecomes R(2) µν =−1 2hλρ ∂2hλρ ∂xν∂xµ−∂2hµρ ∂xν∂xλ−∂2hλν ∂xρ∂xµ+∂2hµν ∂xρ∂xλ! +1 4 2∂hρσ ∂xρ−∂hρρ ∂xσ! ∂hσµ ∂xν+∂hσν ∂xµ−∂hµν ∂xσ! −1 4∂hσν ∂xλ+∂hσλ ∂xν−∂hλν ∂xσ ∂hσµ ∂xλ +∂hσλ ∂xµ−∂hλµ ∂xσ! . (24) 3. Energy–Momentum Complex in Curvature Based Gravity 3.1. The gravitational Energy–Momentum “Tensor”’ of nth Order Lagrangian Let us examine the energy–momentum complex for a fourth order gravitational Lagrangian, that is, which depends up to fourth derivatives of the metric tensor gµν as L=L(gµν , gµν,ρ , gµν,ρλ , gµν,ρλξ , gµν,ρλξσ) , whose field equations, in general, are of eighth order in metric formalism (see also [ 23 , 24 ]). In this manner we include all possible curvature invariants, not only  operators, into the gravitational action. Then, we will generalize the approach to a gravitational Lagrangian of n -th order, i.e., which depends up to nth derivatives of metric tensor. We will derive the energy–momentum tensor using the Noether’s theorem, imposing that gravitational action is invariant under global translations [ 1 ]. In this review the metric signature of gµν adopted is (+ , − , − , −) , while Ricci tensor is defined as Rµν =Rρµρν and Riemann tensor as Rαβµν =Γα βν,µ+. . .. Let us vary the gravitational action with respect to metric gµν and coordinates xµ[11,25,26] I=ZΩd4xL →˜ δI=ZΩ0d4x0L0−ZΩd4xL =ZΩd4xδL+∂µ(Lδxµ), (25) where ˜ δ stands for the local variation while δ means the total variation, that is, keeping the value of coordinate xfixed. By infinitesimal transformations as x0µ=xµ+eµ(x), (26) the total variation of the metric tensor reads δgµν =g0 µν(x)−gµν(x)=−eα∂αgµν −gµα∂νeα−gνα∂µeα. (27) Under global transformation, ∂λeµ= 0, the functional variation of the metric becomes δgµν =−eα∂αgµν . If we also require that the action to be invariant under this transformation, that is, ˜ δI=0, from arbitrariness of domain of integration Ω, we have 0=δL+∂µ(Lδxµ)=∂L ∂gµν −∂ρ∂L ∂gµν,ρ +∂ρ∂λ ∂L ∂gµν,ρλ −∂ρ∂λ∂ξ ∂L ∂gµν,ρλξ +∂ρ∂λ∂ξ∂σ∂L ∂gµν,ρλξσ δgµν +∂η2χp−gτη αeα, (28) where the explicit expression of gravitational energy–momentum tensor, that we will see being a pseudo-tensor or affine tensor, is Particles 2022,5304 τη α=1 2χ√−g ∂L ∂gµν,η−∂λ ∂L ∂gµν,ηλ +∂λ∂ξ ∂L ∂gµν,ηλξ −∂λ∂ξ∂σ∂L ∂gµν,ηλξσ !gµν,α + ∂L ∂gµν,ρη −∂ξ ∂L ∂gµν,ρηξ +∂ξ∂σ∂L ∂gµν,ρηξσ !gµν,αρ + ∂L ∂gµν,ρλη −∂σ∂L ∂gµν,ρλησ !gµν,ρλα +∂L ∂gµν,ρλησ gµν,ρλξα −δη αL.(29) If the metric tensor gµν satisfies the Euler–Lagrange equations for our gravitational Lagrangian δL δgµν =∂L ∂gµν −∂ρ∂L ∂gµν,ρ +∂ρ∂λ ∂L ∂gµν,ρλ −∂ρ∂λ∂ξ ∂L ∂gµν,ρλξ +∂ρ∂λ∂ξ∂σ∂L ∂gµν,ρλξσ =0 , (30) for an arbitrary eα, we get a local continuity equation for our Noether current ∂ηp−gτη α=0 . (31) In a more compact form, the gravitational energy–momentum tensor takes the following form τη α=1 2χ√−g3 ∑ m=0 (−1)m ∂L ∂gµν,ηi0···im!,i0···im gµν,α + 2 ∑ j=0 3 ∑ m=j+1 (−1)j ∂L ∂gµν,ηi0···im!,i0···ij gµν,ij+1···imα−δη αL, (32) where we used the following notation (),i0=I;(),i0···im=           (),i1if m=1 (),i1i2if m=2 (),i1i2i3if m=3 and so on ;(),ikik=(),ik Let us now generalize our approach considering a general Lagrangian density depending up to nth derivative of gµν , that is, L=Lgµν,gµν,i1,gµν,i1i2,gµν,i1i2i3,··· ,gµν,i1i2i3···in . Total variation of Lagrangian Land its Euler–Lagrange equations yield δL= n ∑ m=0 ∂L ∂gµν,i0···im δgµν,i0···im= n ∑ m=0 ∂L ∂gµν,i0···im ∂i0···imδgµν , (33) δL δgµν = n ∑ m=0 (−1)m∂i0···im ∂L ∂gµν,i0···im =0 , (34) where δ/δgµν is the functional derivative, while it is possible to exchange the variation δ with the derivatives δgµν,i0···im=∂i0···imδgµν , because we are varying keeping x fixed. So, we can find a most general local continuity equation which allows us to define the energy–momentum pseudo-tensor (which is an affine tensor as it will be proved later) for the gravitational field of 2nth order gravity Particles 2022,5305 τη α=1 2χ√−gn−1 ∑ m=0 (−1)m ∂L ∂gµν,ηi0···im!,i0···im gµν,α +Θ[2,+∞[(n) n−2 ∑ j=0 n−1 ∑ m=j+1 (−1)j ∂L ∂gµν,ηi0···im!,i0···ij gµν,ij+1···imα−δη αL, (35) where Θis the Heaviside function Θ[a,+∞[(n)=(1 if n∈[a,+∞[ 0 otherwise . (36) If fields and its derivatives vanish on boundary of our spatial region or rapidly decreasing to the spatial infinite on an infinity spacelike hypersurface, the gravitational energy– momentum tensor is totally conserved and satisfies a more general conservation law. An alternative way to obtain the tensor (35) is the procedure developed by Landau [ 1 ]. For example, we start by deriving the tensor (32) , because its generalization to higher order Lagrangians is the same. First of all, let us impose the stationary condition and vary the action with respect to the metric to find the field equations under the hypothesis that both δgµν and the variation of derivative δ∂ng vanish on the boundary of integration domain, canceling the surface integrals. Hence, the following occurs: δI=δZΩd4xLgµν,gµν,ρ,gµν,ρλ,gµν,ρλξ,gµν,ρλξσ=0 , (37) l ∂L ∂gµν −∂ρ∂L ∂gµν,ρ +∂ρ∂λ ∂L ∂gµν,ρλ −∂ρ∂λ∂ξ ∂L ∂gµν,ρλξ +∂ρ∂λ∂ξ∂σ∂L ∂gµν,ρλξσ =0 . (38) Now, we perform the derivative of Lagrangian respect to metric tensor and then we put it into the field equations (37). We obtain ∂L ∂xα=∂L ∂gµν ∂gµν ∂xα+∂L ∂gµν,ρ ∂gµν,ρ ∂xα+∂L ∂gµν,ρλ ∂gµν,ρλ ∂xα +∂L ∂gµν,ρλξ ∂gµν,ρλξ ∂xα+∂L ∂gµν,ρλξσ ∂gµν,ρλξσ ∂xα =∂ρ∂L ∂gµν,ρgµν,α−∂ρ∂λ ∂L ∂gµν,ρλ gµν,α+∂ρ∂λ∂ξ ∂L ∂gµν,ρλξ gµν,α−∂ρ∂λ∂ξ∂σ∂L ∂gµν,ρλξσ gµν,α +∂L ∂gµν,ρgµν,ρα +∂L ∂gµν,ρλ gµν,ρλα +∂L ∂gµν,ρλξ gµν,ρλξα +∂L ∂gµν,ρλξσ gµν,ρλξσα =∂ρ∂L ∂gµν,ρgµν,α−∂ρ ∂λ ∂L ∂gµν,ρλ gµν,α!+∂λ ∂L ∂gµν,ρλ gµν,ρα! +∂ρ ∂λ∂ξ ∂L ∂gµν,ρλξ gµν,α!+∂λ ∂L ∂gµν,ρλξ gµν,ρξα! −∂ξ ∂λ ∂L ∂gµν,ρλξ gµν,αρ!−∂ρ ∂λ∂ξ∂σ∂L ∂gµν,ρλξσ gµν,α! +∂λ ∂L ∂gµν,ρλξσ gµν,ρξσα!−∂ξ ∂λ ∂L ∂gµν,ρλξσ gµν,ρσα! +∂σ ∂ξ∂λ ∂L ∂gµν,ρλξσ gµν,ρα!.(39) Particles 2022,5306 Grouping together terms and renaming dumb indices, we obtain ∂ηp−gτη α=0 , (40) that is, the pseudo-tensor is locally conserved, where τη αis the tensor defined in (32). The energy–momentum complex, instead, can be derived considering the material Lagrangian Lm=2χ√−gLmwith stress–energy tensor given by Tηα =2 √−g δ(√−gLm) δgηα . (41) Thus, we use the field equations in presence of matter, namely Pηα =χTηα , (42) where Pηα =−1 √−g δLg δgηα with the coupling χ=8πG c4. (43) By field Equation (42), we obtain 2χp−gτη α,η=−p−gPρσgρσ,α=−χp−gTρσgρσ,α =2χp−gTη α;η−2χp−gTη α,η, (44) ∂ηhp−gτη α+Tη αi=p−gTη α;η, (45) being δL+∂µ(Lδxµ)=−Pµνp−gδgµν +∂η2χp−gτη αeα =hp−gPµνgµν,α+∂η2χp−gτη αieα=0 , (46) and because from symmetry of tensor Tη α, one gets p−gTη α;η=p−gTη α,η−1 2gρσ,αTρσp−g. (47) The relation (45) tells us that the conservation law of the energy–momentum complex, i.e., the sum of two stress–energy tensors due to matter plus gravitational fields, is related to the covariant derivative of the only matter part. From contracted Bianchi identities we get the total conservation law and conversely Gηα ;η=0↔Pηα ;η=0↔Tηα ;η=0↔∂ηhp−gτη α+Tη αi=0 , (48) where Gηα =Rηα −1 2gηαR is the Einstein tensor and the locally conserved energy–momentum complex is given by Tη α=p−gτη α+Tη α. (49) In a nutshell, the contracted Bianchi identities lead to the local conservation of energy– momentum complex or, viceversa, the local conservation of matter and gravitational fields involves the contracted Bianchi identities (see also [ 27 ] for a detailed discussion in modified gravity). From the local continuity Equation (48) , it is possible to derive some conserved quantities, Noether charges, such as the total 4-momentum of matter plus gravitational field. If we Particles 2022,5313 An important extension of local Lagrangian (81) to non-local Lagrangian is possible allowing p→∞. Let Dpbe a linear differential operator defined by Dp= p ∑ k=0 akk. (89) If the weak or strong convergence is guaranteed under suitable assumptions for the coefficients ak (e.g. ∑∞ k=0|ak|<∞ ) and for the domain of the operator Dp , we obtain the following non-local operator F() lim p→∞ p ∑ k=0 akk=F()(90) and also our local action becomes non local, i.e. I=ZΩd4xR+RF()Rp−g. (91) Accordingly integral operator acts as Φ(x)=ZΩd4yF(x−y)R(x)=F()R(x). (92) Let us carry out now the limit n→∞ for the energy–momentum pseudo-tensor of n -order Lagrangian (35), we may obtain the non-local pseudo-tensor, that is lim n→∞τη α(x)=τη α(x). (93) Whereas τη α(x) transforms as an affine tensor, we could show that also its limit for n→∞ , i.e., τη α(x), is an affine tensor. For an linear transformation x0µ=Λµ νxν|Λ| 6=0 (94) the following affine pseudo-tensor changes as τη α(x)=Λ−1η σΛτ ατ0σ τx0. (95) Substituting (95) in (93), we have τη α(x)=lim n→∞Λ−1η σΛτ ατ0σ τx0=Λ−1η σΛτ αlim n→∞τ0σ τx0=Λ−1η σΛτ ατ0σ τx0(96) which implies that τσ τ(x)transforms as an affine object also in the limit n→∞. 3.5. The Weak-Field Limit of Energy–Momentum Pseudo-Tensor The gravitational energy–momentum pseudo-tensor (87) related to Lagrangian (81) in weak field approximation can be performed perturbing weakly spacetime metric around the Minkowski metric ηµν as gµν =ηµν +hµν being |hµν|  1 , (97) where h=ηµνhµν is the trace of perturbation. Thus, we expand the energy–momentum pseudo-tensor to lower order in h , namely, retaining terms up to h2 . Let’s see what becomes the weakly perturbed pseudo-tensor (88) in harmonic coordinates where gµνΓσ µν = 0. The quadratic part of the Ricci scalar Ryields R=−gµνΓρ µσΓσ νρ, (98) Particles 2022,5314 that is R=−1 4gµνgσλgρegeµ,σ+geσ,µ−gµσ,egλν,ρ+gλρ,ν−gνρ,λ. (99) Keeping terms up to second order in h2, we get ∂R ∂gαβ,γ!(1) gαβ,δ(1)h2 =1 2hαβ γ ,hαβ,δ−hγα β ,hαβ,δ, (100) according to ∂R ∂gαβ,γ gαβ,δ=−1 4gµβgσαgeγ +gµγgσαgβe −gµαgσγgβegeµ,σ+geσ,µ−gσµ,e +gβνgγλgρα +gγνgβλgρα −gαλgβνgργgλν,ρ+gλρ,ν−gνρ,λgαβ,δ, (101) and also R(2)=−1 4hσλ ,ρhρ λσ,−2hσλ ,ρhρ λ,σ. (102) Hence, when we put these terms into (88) , the stress–energy pseudo-tensor in general relativity up to order h2takes the form τη α|GR =1 2χ1 2hµν,ηhµν,α−hηµ,νhµν,α−1 4δη αhσλ ,ρh,ρ λσ −2hσλ ,ρhρ λ,σ. (103) Now, we have to expand to second order in h the corrections of the pseudo-tensor (87) due to extended gravity terms. To lower order in hwe consider the following expansions ∂R ∂gµν,ηλ !(0) =1 2gµη gνλ +gµλgνη −2gµνgηλ(0) =1 2ηµηηνλ +ηµληνη −2ηµνηηλ, (104) ∂R ∂gµν,ηλ !(0) gµν,λα(1)=hλη ,λα −h,ηα=hλη −ηηλh,λα h.g. =−1 2h,ηα, (105) ∂R ∂gµν,ηλ !(0) gµν,α(1)=hλη −ηηλh,α, (106) ∂hR ∂gµν,ηi0···im!(0) = ∂hR ∂gµν,ηi0···iq!(0) = ∂hR ∂gµν,ηi0···i2h+1!(0) =ηi2i3···ηi2hi2h+1ηµi1ηνη −ηµνηηi1+··· . (107) Then, we take into account only the terms up to h2in harmonic gauge, as 2a0R+ p ∑ k=1 akkR!∂R gµν,ηλ gµν,λα h2 h.g. =1 4 p ∑ k=0 akk+1h!h,ηα+1 4a0h,ηαh, (108) −∂λ"p−g 2a0R+ p ∑ k=1 akkR!∂R ∂gµν,ηλ #gµν,α h2 h.g. =a0h,λhλη −ηηλh,α Particles 2022,5315 +1 2 p ∑ k=1 akk+1h,λhλη −ηλη h,α, (109) p ∑ h=1 2h+1 ∑ q=0 (−1)q∂i0···iq"p−gahR∂hR ∂gµν,ηi0···iq#gµν,α h2 h.g. =1 2 p ∑ h=1 ahh+1h,λhηλ −ηηλh,α+Apη α, (110) p ∑ h=1 2h ∑ j=0 2h+1 ∑ m=j+1 (−1)j∂i0···ij"p−gahR∂hR ∂gµν,ηi0···im#gµν,ij+1···imα h2 h.g. =1 4 p ∑ h=1 ahhhh,ηα +1 2 1 ∑ h=0 p−1+h ∑ j=h p ∑ m=j+1−h (−1)hamm−jhηλ −ηηλh,ihα j+1−hhih ,λ+Bpη α. (111) In Equations (110) , (111) and (107) , we have disregarded the index permutations ( µν ) and (ηi1···i2h+1) because Apη α and Bpη α terms, averaged on a suitable spacetime region, vanish, according to Appendix (A.1) . Hence we calculated only the term deriving from (A1) without considering the index permutations ( µν ) and (ηi1···i2h+1) . This because, taking into account terms obtained from permutations in Apη α and Bpη α , averaged on a suitable spacetime region, we obtain that are equal to zero as we will see below in Appendix A.1. This mathematical trick is essential to calculated the averaged gravitational energy–momentum pseudo-tensor and the power emitted by a source. So, by inserting equalities (108) , (109) , (110) and (111) into (87) , we find the extra term of pseudo-tensor τη α to second order owing to extension of general relativity , that we call ˜ τη α, that is ˜ τη αh2 =1 2χ(1 4 p ∑ k=0 akk+1h!h,ηα+1 2 p ∑ t=0 att+1h,λhηλ −ηηλh,α +1 2 1 ∑ h=0 p ∑ j=h p ∑ m=j (−1)hamm−jhηλ −ηηλh,αih j+1−hhih ,λ +1 4 p ∑ l=0 allh,ηα−hδη αh+Θ[1,+∞[(p)hApη α+Bpη αi), (112) where conventions used are (),αi0=(),αhi0 ,λ=h,λ. In summary, we can split the gravitational energy–momentum pseudo-tensor in the general relativity part and in the Extended Gravity part, that is τη αh2 =τη α|GR +˜ τη α. (113) Now in the particular case when p is equal to 0 and 1, extended corrections of the pseudotensor ˜ τη α was derived. Then, for p= 0, that is, Lg=R+a0R2√−g as in the case discussed in [25], we obtain τη αh2 =τη α|GR +˜ τη α, Particles 2022,5316 with ˜ τη αh2 =a0 2χ1 2h,ηαh+hη λ,αh,λ−h,αh,η−1 4(h)2δη α. (114) While for p=1, that is Lg=R+a0R2+a1RR√−g, one has τη αh2 =τη α|GR +˜ τη α, where extended corrections to pseudo-tensor are ˜ τη αh2 =1 2χ(1 42a0h+a12hh,ηα+1 22a0h,λ+a12h,λhηλ −ηηλh,α +1 2a1hηλ −ηηλh,α h,λ+1 2a1hηλ −ηηλh,α 2h,λ−1 2a1hηλ −ηηλh,σα hσ ,λ +1 4a1h,ηαh−1 4δη αha0(h)+a12hih+(A1)η α+(B1)η α).(115) The iteration can be performed to every pintroducing new contributions into dynamics. 4. Power Emitted Carried by a Gravitational Wave We wish to calculate the power emitted in the form of gravitational waves by an isolated massive system considering the local conservation of the energy–momentum pseudo-tensor (40). 4.1. The Average of the Energy–Momentum Pseudo-Tensor Let us now regard the wavelike solutions of the linearized field equations in vacuum associated with Lagrangian (81) , for details see Ref. [ 42 ]. Gravitational waves solutions can be expressed as hµν(x)= p+2 ∑ m=1ZΩ d3k (2π)3(Bm)µν(k)ei(km)αxα+c.c. , (116) where (Bm)µν(k)=   Cµν(k)for m=1 1 3ηµν 2+(km)µ(km)ν k2 (m)Am(k)for m≥2, (117) with Cµν(k) related to transverse-traceless polarization tensor typical of general relativity and Am(k) the amplitude of wave at k fixed. Here “c.c.” stands for the complex conjugate. The trace of tensor (117) is (Bm)λ λ(k)=(Cλ λ(k)for m=1 Am(k)for m≥2, (118) and the kµ m=(ωm,k) is the wave vector with k2 m=ω2 m−|k|2=M2 where k2 1= 0 and k2 m6=0 for m≥2. Keeping kfixed, we derive the following relations hη ,α=2Re(p+2 ∑ j=1 (−1)kjαkjηAjeikjx), (119) mh,λ=2Re((−1)mi p+2 ∑ j=1kjλk2 jmAjeikjx), (120) qhηλ −ηηλh,α=2Re((−1)qi p+2 ∑ l=1 (kl)αk2 lqh(Bl)ηλ −ηηλ(Bl)ρ ρieiklx), (121) Particles 2022,5317 mhσ ,λ=2Re((−1)m+1p+2 ∑ j=1kjλkjσk2 jmAjeikjx), (122) qhηλ −ηηλh,σα =2Re((−1)q+1p+2 ∑ l=1 (kl)σ(kl)αk2 lqh(Bl)ηλ −ηηλ(Bl)ρ ρieiklx), (123) nh=2Re((−1)np+2 ∑ r=2k2 rnAreikrx). (124) Now, we choose a domain of the spacetime Ω such that |Ω|  1 |k| [ 22 ]. Then, we can perform the average of the gravitational energy–momentum pseudo-tensor τη α over our region and all integrals, including terms such as ei(ki−kj)αxα , tend to zero, by means of following identities Re{f}Re{g}=1 2Re{f g}+1 2Re{f¯ g}, (125) (kl)λh(Bl)ηλ −ηηλ(Bl)ρ ρi=−(kl)η 2Al. (126) In the harmonic gauge, after averaging and some algebraic manipulations, we find (see Appendix A.1) mh,λqhηλ −ηηλh,α=(−1)m+q+1p+2 ∑ l=2 (kl)α(kl)ηk2 l(m+q)|Al|2, mhσ ,λqhηλ −ηηλh,σα=(−1)m+q+1p+2 ∑ l=2 (kl)α(kl)ηk2 l(m+q)+1|Al|2, Dqh,η αmhE=2(−1)m+q+1p+2 ∑ r=2 (kr)α(kr)ηk2 r(m+q)|Ar|2, hmhhi=2(−1)m+1p+2 ∑ j=2k2 jm+1|Aj|2, hApη αi=hBpη αi=0 . (127) A set of polarization tensors forming a basis for the linearized solutions hµν is given in Appendix A.2. According to equalities (127) , we can calculate the average value of the energy–momentum pseudo-tensor as Dτη αE=1 2χ(k1)η(k1)αCµνC∗ µν −1 2|Cλ λ|2 +1 2χ"−1 6p+2 ∑ j=2kjηkjα−1 2k2 jδη α|Aj|2# +1 2χ(" p ∑ l=0 (l+2)(−1)lal p+2 ∑ j=2kjηkjαk2 jl+1|Aj|2# −1 2 p ∑ l=0 (−1)lal p+2 ∑ j=2k2 jl+2|Aj|2δη α), (128) with gravitational coupling χ=8πG c4 . In TT gauge for the first mode associated with k1 and only in harmonic gauge for residual modes km, in the momentum space, it gets ((k1)µCµν =0∧Cλ λ=0 if m=1 (km)µ(Bm)µν =1 2(Bm)λ λkνif m≥2. (129) Particles 2022,5318 We now explore a gravitational wave propagating in the +z -direction at k fixed, with 4-wave vector given by kµ=(ω, 0, 0, kz) where ω2 1=k2 z if k2 1= 0 and k2 m=m2=ω2 m−k2 z otherwise with kz> 0. Accordingly the averaged time-space tensorial component which can be seen as flux of gravitational energy along the z axis through the surface that delimits our domain Ω, reads Dτ3 0E=c4 8πGω2 1C2 11 +C2 12+c4 16πG"−1 6p+2 ∑ j=2 ωjkz|Aj|2 + p ∑ l=0 (l+2)(−1)lal p+2 ∑ j=2 ωjkzm2(l+1) j|Aj|2#. (130) Finally, we can calculate the emitted power per unit solid angle Ω , radiated by the localized sources, in a direction ˆ x at k fixed. By choosing of the suitable gauge, for the local conservation of the energy–momentum pseudo-tensor (40), the power is given by dP dΩ=r2ˆ xiDτi 0E. (131) By ranging the index p of the pseudo-tensor (130) over { 0, 1, 2 } , we obtain the following three cases for p= 0 Dτ3 0E=c4ω2 1 8πGhC2 11 +C2 12i+c4 16πG−1 6ω2|A2|2kz+2a0ω2m2 2|A2|2kz, (132) for p= 1 Dτ3 0E=c4ω2 1 8πGhC2 11 +C2 12i+c4 16πG−1 6ω2|A2|2+ω3|A3|3kz +2a0hω2m2 2|A2|2+ω3m2 3|A3|2|2kzi−3a1hω2m4 2|A2|2+ω3m4 3|A3|2kzi, (133) and for p= 2 Dτ3 0E=c4ω2 1 8πGhC2 11 +C2 12i+c4 16πG−1 6ω2|A2|2+ω3|A3|3+ω4|A4|2kz +2a0hω2m2 2|A2|2+ω3m2 3|A3|2+ω4m2 4|A4|2kzi −3a1hω2m4 2|A2|2+ω3m4 3|A3|2+ω4m4 4|A4|2kzi +4a2hω2m6 2|A2|2+ω3m6 3|A3|2+ω4m6 4|A4|2i, (134) where the gravitational coupling χ has been explicitly indicated. By Formulas (132) – (134) it is obvious that the first term comes out of general relativity and the corrections strongly depends on p . In any context where corrections to general relativity can be investigated, this approach could constitute a paradigm to search for higher order effects. Particles 2022,5319 5. Energy–Momentum Complex of f(R)Gravity in Palatini Approach 5.1. The Gravitational Pseudo-Tensor of f (R)Gravity in Palatini Formulation In Palatini approach the metric tensor gµν and the connection Γα µν are independent, that means that we do not assume any relation between the metric and the connection, and Riemann and Ricci tensors are, in general, defined as Rµν(Γ) =∂αΓα µν −∂νΓα µα +Γα µν Γσ ασ −Γα νλ Γλ µα, (135) R(g,Γ) =Rµν(Γ)gµν. (136) So, the Palatini gravitational action of f(R)appears as [43] S=1 2κ2Zd4xp−g f (R), (137) with the coupling κ2= 8 πG/c4 and g the determinant of metric tensor gµν . By varying the metric gµν and the connection Γα µν , for a general infinitesimal transformation coordinate xµ it gets x0µ=xµ+δxµ, (138) g0µν(x0) =gµν(x) + ˜ δgµν,g0µν(x) =gµν(x) + δgµν, (139) Γ0α µν(x0) =Γα µν(x) + ˜ δΓα µν,Γ0α µν(x) =Γα µν(x) + δΓα µν, (140) where ˜ δ is the local variation and δ is the variation that keeps the coordinates x fixed. The variation of the gravitational action with respect to the metric gµν and the connection Γα βγ yield ˜ δS=1 2κ2Zd4x(p−gfRRµν −1 2gµν fδgµν +fRgµν δRµν+∂µp−g f δxµ), (141) where fR:=df(R)/dR. According to the following Palatini identity δRµν =∇αδΓα µν−∇νδΓα αµ. (142) the action (141) takes the form ˜ δS=1 2κ2Zd4x(p−gfRRµν −1 2gµν fδgµν +δΓλνµh−∇λp−ggνµ fR+∇αp−ggµαδν λfRi +∂λhp−g fRgµνδλ α−gµλδν αδΓαµν +p−g f δxλi). (143) By the principle of least action or stationary action (137) , by imposing that the variation of metric and its derivatives vanish at the boundary, we obtain field equations for the metric tensor and the connection in vacuum, i.e., fRR(µν)−1 2gµν f=0, (144) ∇λp−ggνµ fR=0. (145) Particles 2022,5320 Given that we adopting an arbitrary non-compatible connection, the symmetric part of the Ricci tensor, R(µν) , enter in the Equation (144) and then the Ricci tensor is non symmetric, that is Rµν =Rνµ +Rλλµν , (146) being Riemann tensor Rσλµν no longer antisymmetric on its first two indices, i.e., the term Rλλµν does not vanishes. For a generic infinitesimal transformation, the metric tensor and the connection change as x0µ=xµ+ξµ, (147) g0µν(xλ)≃gµν(xλ)−ξλ∂gµν ∂xλ+gµα ∂ξν ∂xα+gνα ∂ξµ ∂xα, (148) Γ0α µν(xλ)≃Γαµν(xλ)−ξλ∂Γαµν ∂xλ+Γρµν ∂ξα ∂xρ−Γασν ∂ξσ ∂xµ−Γαµσ ∂ξσ ∂xν−∂2ξα ∂xµ∂xν, (149) where we have neglected terms of higher order in ξµ in the series expansion. Under a rigid infinitesimal translation, that is, ∂µξν=0, we obtain g0µν(xλ)≃gµν(xλ)−ξλ∂gµν ∂xλ, (150) Γ0α µν(xλ)≃Γαµν(xλ)−ξλ∂Γαµν ∂xλ. (151) Therefore, the Palatini action (143) becomes ˜ δSg=1 2κ2Zd4x(−p−gfRRµν −1 2gµν fξλgµν ,λ −ξλΓβ νµ,λh−∇βp−ggνµ fR+∇αp−ggµαδν βfRi +∂λh−p−g fRgµνδλ α−gµλδν αξβΓαµν,β+p−g f ξλi). (152) If the metric gµν and the Palatini connection Γα βγ are solution of Equations (144) and (145) , the stationary of the local variation of the action (152) , gives the local conservation of gravitational energy–momentum pseudo-tensor τλβof Palatini f(R)gravity, namely ∂λp−gτλβ=0, (153) where τλβis defined as τλβ=1 2κ2hf(R)δλ β−fR(R)gµνδλ α−gµλδν αΓαµν,βi. (154) It is worth noting that the pseudo-tensor defined in Equation (154) has the opposite sign of the one defined above. In order to derive the energy–momentum complex, let us analyze the action containing the matter part, that is Sm=Zd4xp−gLm. (155) Generally, the matter Lagrangian Lm depends on the connection as, for example, occurs in presence of fermion fields. Here, we consider only material Lagrangian which does not depend on the affine connection Γ. Then, the matter energy–momentum tensor is defined Particles 2022,5321 as in (75) . Hence, field equations for metric and connection, i.e., Equations (144) and (145) , in presence of matter yield fRR(µν)−1 2gµν f=κ2Tµν, (156) ∇λp−ggνµ fR=0. (157) As already pointed out above the connection can be non compatible with the metric gµν , i.e., ∇λgµν 6= 0. In compact form, we can define a new metric, conformally related to the metric gµν, as hµν :=fRgµν. (158) so that Equation (157) becomes ∇λ√hhµν=0. (159) Thus the Palatini connection Γαµν appears as the Christoffel connection for the new metric hµν, i.e., Γαµν =1 2fR(R)gαβ∂µfR(R)gνβ+∂νfR(R)gµβ−∂βfR(R)gµν. (160) The Palatini connection Γαµν and Levi–Civita connection ◦ Γαµν are related as Γαµν =◦ Γαµν +δα µAν+δα νAµ−gµν Aα, (161) where the four-vector Aµis defined as Aµ:=1 2fR∇µfR. (162) For f(R) = R, we recover the Christoffel symbols constructed by the metric gµν, that is Γαµν =◦ Γαµν=1 2gαβgβµ,ν+gβν,µ−gµν,β(163) this means that in general relativity no difference results in metric and Palatini formalism. The Ricci tensor Rµν in Palatini formalism and that in metric formalism Rµν , are related as follows Rµν =Rµν +3 2 1 (fR(R))2◦ ∇µfR(R)◦ ∇µfR(R) −1 fR(R)◦ ∇µ◦ ∇ν−1 2gµν ◦ fR(R), (164) where ◦ :=◦ ∇µ◦ ∇µ and ◦ ∇ denotes the covariant derivative associated with the Levi–Civita connection. Contracting tensorial equality (164) with gµν , we obtain the relation between Rand R, that is, the Ricci scalar in both approach R=R+3 2(fR(R))2◦ ∇µfR(R)◦ ∇µfR(R)+3 fR(R)◦ fR(R). (165) Particles 2022,5322 Adopting the Palatini connection Γαµν (160) , the symmetry of Ricci tensor is restored on account of the relation Γλ=∂λf2 R√−g f2 R√−g, (166) which implies R[µν]=∂[µΓν]=0. (167) Furthermore the connection is non compatible with metric gµν being ∇λgµν =−gµν fR∇λfR. (168) Despite this, the covariant derivatives associated with Palatini connection commute each other, as displayed below ∇ρ,∇λgµν =0 . (169) Thus, we restore the antisymmetry on the first two indices of Riemann tensor, namely Rµνλρ =−Rνµλρ. (170) by the definition of Riemann tensor for an arbitrary tensor Jµν ∇ρ,∇λJµν =−Rαµρλ Jαν −Rανρλ Jµα. (171) In addition, the contracted Bianchi identities are fulfilled, that is ∇µRµν −1 2gµνR=0. (172) According to the Palatini connection Equation (160) and from the symmetry of energy– momentum tensor Tµν, taking into account that for the new metric hµν we have Γλ=∂λ√−h √−h, (173) and Γµνλ +Γνµλ =1 fR∂λhµν, (174) so we derive the following useful expression √−h∇σTσν=∂σ√−hTσν−1 2fRTλρ∂νhλρ√−h. (175) Field equations in matter (156) lead to 0=√−h 2f2 R Tµνgµν,βξβ+∂λp−g1 2κ2hf(R)δλ β−fRgµνδλ α−gµλδν αΓαµν,βiξβ, (176) and from Equation (175) , after some algebraic manipulations, we get the following 4divergence of energy–momentum complex not vanishing ∂σhp−gTσβ+tσβi=√−h f2 R∇λTλβ+2√−h f3 R Tλβ∇λfR−√−h 2f3 R T∇βfR. (177) From contracted Bianchi identities and the field equations, the following relations are satisfied ∇µ,∇ν∇µfR=Rαν∇αfR, (178) Particles 2022,5329 5. Bak, D.; Cangemi, D.; Jackiw, R. Energy-momentum conservation in gravity theories. Phys. Rev. D 1994 ,49, 5173. [CrossRef] [PubMed] 6. Lee, D.L.; Lightman, A.P.; Ni, W.T. Conservation laws and variational principles in metric theories of gravity. Phys. Rev. D 1974, 10, 1685. [CrossRef] 7. Rosen, N. The Energy of the Universe. Gen. Rel. Grav. 1994,26, 319. [CrossRef] 8. Lessner, G. Møller’s energy-momentum complex—Once again. Gen. Relativ. Gravit. 1996,28, 527. [CrossRef] 9. Palmer, T.N. Gravitational energy-momentum: The Einstein pseudotensor reexamined. Gen. Relat. Gravit. 1980 ,12, 149. [CrossRef] 10. Ferraris, M.; Francaviglia, M. Covariant first-order Lagrangians, energy-density and superpotentials in general relativity. Gen. Relat. Gravit. 1990,22, 965–985. [CrossRef] 11. Capozziello, S.; De Laurentis, M. Extended Theories of Gravity. Phys. Rept. 2011,509, 167. [CrossRef] 12. Capozziello, S.; Francaviglia, M. Extended Theories of Gravity and their Cosmological and Astrophysical Applications. Gen. Rel. Grav. 2008,40, 357. [CrossRef] 13. Bogdanos, C.; Capozziello, S.; De Laurentis, M.; Nesseris, S. Massive, massless and ghost modes of gravitational waves from higher-order gravity. Astropart. Phys. 2010,34, 236. [CrossRef] 14. Canfora, F.; Vilasi, G.; Vitale, P. Nonlinear gravitational waves and their polarization. Phys. Lett. B 2002,545, 373. [CrossRef] 15. Mikhail, F.I.; Wanas, M.I.; Hindawi, A.; Lashin, E.I. Energy-Momentum Complex in Møller’s Tetrad Theory Of Gravitation. Int. J. Theor. Phys. 1993,32, 1627. [CrossRef] 16. Multamaki, T.; Putaja, A.; Vilja, I.; Vagenas, E.C. Energy-momentum complexes in f(R) theories of gravity. Class. Quant. Grav. 2008,25, 075017. [CrossRef] 17. Capozziello, S.; Capriolo, M.; Transirico, M. The gravitational energy-momentum pseudo-tensor of higher order theories of gravity. Ann. Phys. 2017,525, 1600376. [CrossRef] 18. Capozziello, S.; Capriolo, M.; Transirico, M. The gravitational energy-momentum pseudo-tensor: The cases of f(R) and f(T) gravity. Int. J. Geom. Meth. Mod. Phys. 2018,15, 1850164. [CrossRef] 19. Capozziello, S.; Capriolo, M.; Caso, L. Weak field limit and gravitational waves in f(T, B) teleparallel gravity. Eur. Phys. J. C 2020 , 80, 156. [CrossRef] 20. Capozziello, S.; Capriolo, M.; Caso, L. Gravitational waves in higher order teleparallel gravity. Class. Quantum Grav. 2020 ,37, 235013. [CrossRef] 21. Xulu, S.S. The Energy-Momentum Problem in General Relativity. arXiv 2003, arXiv:hep-th/0308070. [CrossRef] 22. Weinberg, S. Gravitation and Cosmology; Wiley: New York, NY, USA, 1972. 23. Capozziello, S.; Stabile, A. Gravitational waves in fourth order gravity. Astrophys. Space Sci. 2015,358, 27. [CrossRef] 24. Stelle, K.S. Classical Gravity with Higher Derivatives. Gen. Rel. Grav. 1978,9, 353. [CrossRef] 25. Capozziello, S.; Stabile, A.; Troisi, A. The Post-Minkowskian Limit of f(R)-gravity. Int. J. Theor. Phys. 2010,49, 1251. [CrossRef] 26. Greiner, R.; Reinhardt, J. Field Quantization; Springer: Berlin, Germany, 1996. 27. Capozziello, S.; Faraoni, V. Beyond Einstein Gravity; FTP 170; Springer: New York, NY, USA, 2011. 28. Misner, C.W.; Thorne, K.S.; Wheeler, J.A. Gravitation; Freeman and Co.: New York, NY, USA, 1971. 29. Straumann, N. General Relativity; Springer: New York, NY, USA, 2013. 30. Pauli, W. Theory of Relativity; Pergamon Press: Oxford, UK, 1958. 31. Dirac, P.A.M. General Theory of Relativity; Princeton University Press: Princeton, BJ, USA, 1996. 32. Hawking, S.W.; Ellis, G.F.R. The Large Scale Structure of Spacetime; Cambridge University Press: Cambridge, MA, USA, 1973. 33. Schutz, B. A First Course in General Relativity; Cambridge University Press: Cambridge, MA, USA, 2009. 34. Ohanian, H.C.; Ruffini, R. Gravitation and Spacetime; Cambridge University Press: Cambridge, MA, USA, 2013. 35. Stephani, H. General Relativity; Cambridge University Press: Cambridge, MA, USA, 1990. 36. Modesto, L.; Rachwal, L. Universally finite gravitational and gauge theories. Nucl. Phys. B 2015,900, 147. [CrossRef] 37. Giaccari, S.; Modesto, L. Classical and Quantum Nonlocal Supergravity. arXiv 2016, arXiv:1605.03906. 38. Briscese, F.; Pucheu, M.L. Palatini formulation of non-local gravity. Int. J. Geom. Methods Mod. Phys. 2017 ,14, 1750019. [CrossRef] 39. Modesto, L.; Shapiro, I.L. Super-renormalizable quantum gravity with complex ghosts. Phys. Lett. B 2016,755, 279. [CrossRef] 40. Modesto, L. Super-renormalizable or finite Lee-Wick quantum gravity. Nucl. Phys. B 2016,909, 584. [CrossRef] 41. Quandt, I.; Schmidt, H.J. The Newtonian limit of fourth and higher order gravity. Astron. Nachr. 1991,312, 97. 42. Capozziello, S.; Capriolo, M.; Caso, L. Weak field limit and gravitational waves in higher order gravity. Int. J. Geom. Methods Mod. Phys. 2019,16, 1950047. [CrossRef] 43. Abedi, H.; Capozziello, S.; Capriolo, M.; Abbassi, A.M. Gravitational energy–momentum pseudo-tensor in Palatini and metric f(R) gravity. Ann. Phys. 2022,439, 168796. [CrossRef] 44. Allemandi, G.; Capone, M.; Capozziello, S.; Francaviglia, M. Conformal aspects of Palatini approach in extended theories of gravity. Gen. Rel. Grav. 2006,38, 33–60. [CrossRef] 45. Dick, R. Covariant conservation laws from the Palatini formalism. Int. J. Theor. Phys. 1993,32, 109–120. [CrossRef] 46. Barraco, D.E.; Dominguez, E.; Guibert, R. Conservation laws, symmetry properties, and the equivalence principle in a class of alternative theories of gravity. Phys. Rev. D 1999,60, 044012. [CrossRef] Particles 2022,5330 47. Koivisto, T. Covariant conservation of energy momentum in modified gravities. Class. Quant. Grav. 2006 ,23, 4289–4296. [CrossRef] 48. Barragan, C.; Olmo, G.J.; Sanchis-Alepuz, H. Bouncing Cosmologies in Palatini f(R) Gravity. Phys. Rev. D 2009 ,80, 024016. [CrossRef] 49. Szydłowski, M.; Stachowski, A.; Borowiec, A.; Wojnar, A. Do sewn up singularities falsify the Palatini cosmology? Eur. Phys. J. C 2016,76, 567. [CrossRef] 50. Goheer, N.; Larena, J.; Dunsby, P.K.S. Power-law cosmic expansion in f(R) gravity models. Phys. Rev. D 2009 ,80, 061301. [CrossRef] 51. Buoninfante, L.; Lambiase, G.; Petruzziello, L. Quantum interference in external gravitational fields beyond General Relativity. Eur. Phys. J. C 2021,81, 928. [CrossRef] 52. Buoninfante, L.; Lambiase, G.; Miyashita, Y.; Takebe, W.; Yamaguchi, M. Generalized ghost-free propagators in nonlocal field theories. Phys. Rev. D 2020,101, 084019. [CrossRef] 53. Buoninfante, L.; Lambiase, G.; Yamaguchi, M. Nonlocal generalization of Galilean theories and gravity. Phys. Rev. D 2019 ,100, 026019. [CrossRef] 54. Buoninfante, L.; Ghoshal, A.; Lambiase, G.; Mazumdar, A. Transmutation of nonlocal scale in infinite derivative field theories. Phys. Rev. D 2019,99, 044032. [CrossRef] 55. Buoninfante, L.; Cornell, A.S.; Harmsen, G.; Koshelev, A.S.; Lambiase, G.; Mazumadra, A. Towards nonsingular rotating compact object in ghost-free infinite derivative gravity. Phys. Rev. D 2018,98, 084041. [CrossRef] 56. Buoninfante, L.; Lambiase, G.; Mazumdar, A. Ghost-free infinite derivative quantum field theory. Nucl. Phys. B 2019 ,944, 114646. [CrossRef] 57. Buoninfante, L.; Koshelev, A.S.; Lambiase, G.; Mazumdar, A. Classical properties of non-local, ghostand singularity-free gravity. J. Cosmol. Astropart. Phys. 2018,9, 034. [CrossRef] 58. Capozziello, S.; Capriolo, M. Gravitational waves in non-local gravity. Class. Quantum Grav. 2021,38, 175008. [CrossRef] 59. Capozziello, S.; Capriolo, M.; Nojiri, S. Considerations on gravitational waves in higher-order local and non-local gravity. Phys. Lett. B 2020,810, 135821. [CrossRef] 60. Capriolo, M. Gravitational radiation in higher order non-local gravity. Int. J. Geom. Methods Mod. Phys. 2022. [CrossRef] 61. Gottlober, S.; Schmidt, H.J.; Starobinsky, A.A. Sixth Order Gravity and Conformal Transformations. Class. Quant. Grav. 1990 ,7, 893. [CrossRef]