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

Ci a ion: Capozziello, S.; Cap iolo, M.; Lambiase, G. Ene gy-Momen um Complex in Highe O de Cu a u e-Based Local G a i y. Pa icles 2022,5, 298–330. h ps://doi.o g/10.3390/ pa icles5030026 Academic Edi o : A men Sed akian Recei ed: 15 July 2022 Accep ed: 1 Augus 2022 Published: 10 Augus 2022 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2022 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). Re iew Ene gy-Momen um Complex in Highe O de Cu a u e-Based Local G a i y Sal a o e Capozziello 1,2,3,4,*,† , Mau izio Cap iolo 2,5,† and Gae ano Lambiase 5,6,† 1 Dipa imen o di Fisica “E. Pancini”, Uni e si à di Napoli “Fede ico II”, Complesso Uni e si a io di Mon e S. Angelo, Edi icio G, Via Cin hia, I-80126 Napoli, I aly 2Is i u o Nazionale di Fisica Nuclea e, Sezione di Napoli, Complesso Uni e si a io di Mon e S. Angelo, Edi icio G, Via Cin hia, I-80126 Napoli, I aly 3Scuola Supe io e Me idionale, La go S. Ma cellino 10, I-80138 Napoli, I aly 4Depa men o Ma hema ics, Facul y o Ci il Enginee ing, VSB-Technical Uni e si y o Os a a, Lud ika Podes e 1875/17, 708 00 Os a a, Czech Republic 5Dipa imen o di Fisica “E. R. Caianiello”, Uni e si à degli S udi di Sale no, Via Gio anni Paolo II 132, I-84084 Fisciano, I aly 6 Is i u o Nazionale di Fisica Nuclea e, Sezione di Napoli, G uppo Collega o di Sale no, Via Gio anni Paolo II 132, I-84084 Fisciano, I aly *Co espondence: [email p o ec ed] † These au ho s con ibu ed equally o his wo k. Abs ac : An unambiguous de ini ion o g a i a ional ene gy emains one o he un esol ed issues o physics oday. This p oblem is ela ed o he non-localiza ion o g a i a ional ene gy densi y. In Gene al Rela i i y, he e ha e been many p oposals o de ining he g a i a ional ene gy densi y, no ably hose p oposed by Eins ein, Tolman, Landau and Li shi z, Papape ou, Mølle , and Weinbe g. In his e iew, we i s ly explo ed he ene gy–momen um complex in an n h o de g a i a ional La- g angian L=Lgµν,gµν,i1,gµν,i1i2,gµν,i1i2i3,··· ,gµν,i1i2i3···in and hen in a g a i a ional Lag angian as Lg= (R+a0R2+∑p k=1akRkR)√−g . I s g a i a ional pa was ob ained by in a iance o g a i- a ional ac ion unde in ini esimal igid ansla ions using Noe he ’s heo em. We also showed ha his enso , in gene al, is no a co a ian objec bu only an a ine objec , ha is, a pseudo- enso . The e o e, he pseudo- enso τη α becomes he one in oduced by Eins ein i we limi ou sel es o Gene al Rela i i y and i s ex ended co ec ions ha e been explici ly indica ed. The same me hod was used o de i e he ene gy–momen um complex in (R) g a i y bo h in Pala ini and me ic app oaches. Mo eo e , in he weak ield app oxima ion he pseudo- enso τη α o lowes o de in he me ic pe u ba ion h was calcula ed. As a p ac ical applica ion, he powe pe uni solid angle Ω emi ed by a localized sou ce ca ied by a g a i a ional wa e in a di ec ion ˆ x o a ixed wa e numbe k unde a sui able gauge was ob ained, h ough he a e age alue o he pseudo- enso o e a sui able space ime domain and he local conse a ion o he pseudo- enso . As a cosmological applica ion, in a la F iedmann–Lemaî e–Robe son–Walke space ime, he g a i a ional and ma e ene gy densi y in (R) g a i y bo h in Pala ini and me ic o malism was p oposed. The g a i a ional ene gy–momen um pseudo- enso could be a use ul ool o in es iga e u he modes o g a i a ional adia ion beyond wo s anda d modes equi ed by Gene al Rela i i y and o deal wi h non-local heo ies o g a i y in ol ing −k e ms. Keywo ds: ene gy–momen um complex; pseudo- enso ; g a i a ional ene gy 1. In oduc ion A widely accep ed de ini ion o g a i a ional ene gy densi y and i s localiza ion in cu ed space ime a e se ious p oblems ha a lic he gene al ela i i y. Se e al p esc ip- ions o g a i a ional con ibu ion o ene gy–momen um densi y and mo e gene ally o ene gy–momen um complex ha e been sugges ed by Eins ein, Tolman, Landau and Pa icles 2022,5, 298–330. h ps://doi.o g/10.3390/pa icles5030026 h ps://www.mdpi.com/jou nal/pa icles Pa icles 2022,5299 Li shi z, Papape ou, Mølle , and Weinbe g [ 1 – 10 ]. These a emp s a e based on he in- oduc ion o a supe -po en ial o h ough he expansion o he Ricci enso in he me ic pe u ba ion h . Thus, he g a i a ional pa o he ene gy–momen um densi y ans o ms as an a ine enso no as a co a ian enso , and o his eason, i is no eally a enso bu a pseudo- enso . This a ine p ope y o he g a i a ional s ess–ene gy enso makes he g a i a ional ene gy–momen um densi y no localizable. Howe e , in eg a ing he densi y o e a sui able spa ial egion a a ce ain ime such as o e an asymp o ically la space ime, iable o isola ed sys ems, he g a i a ional ene gy–momen um becomes a ou - ec o , as meaning ha changes in igh way unde asymp o ically la coo dina e ans o ma ions. O e all space i becomes quasi independen o he coo dina e sys em, ha is, he g a i a ional ene gy–momen um o he space ime exis s, bu i canno be local- ized. In his e iew a gene aliza ion o Eins ein’s pseudo- enso o Ex ended Theo ies o G a i y [ 11 , 12 ] is p oposed by imposing he in a iance o he highe o de g a i a ional Lag angian unde an in ini esimal igid ansla ion and by using Noe he ’s heo em. Then, hanks o a con inui y equa ion, a Noe he cu en and a Noe he cha ge we e de i ed ha co espond o a g a i a ional ene gy–momen um pseudo- enso and g a i a ional ene gy–momen um, espec i ely, bo h locally conse ed. By weakly pe u bing he me - ic enso a ound he Minkowskian me ic, a weak- ield limi , in a sui able gauge, he g a i a ional ene gy–momen um pseudo- enso o a Lag angian o n h o de appea s an objec easie o handle. Then, by a e aging o he pseudo- enso o e a sui able space ime domain, i is possible o calcula e he powe emi ed by some localized as ophysical sou ce ca ied away by he g a i a ional wa es. This app oach could be ele an o sea ching o pola iza ion s a es o g a i a ional wa es in addi ion o he wo s anda ds o gene al ela- i i y [ 13 , 14 ]. Finally, a e de i ing he g a i a ional ene gy–momen um pseudo- enso in F(R) g a i y o mula ed in Pala ini and me ic o malism, some cosmological applica ions we e discussed, whe ein a la FLRW me ic he o al ene gy densi y was ob ained in bo h app oaches [15,16]. Fo mo e de ails on he issue o ene gy–momen um localiza ion in modi ied he- o ies o g a i y such as (R) , (R,R, . . . , kR) [ 16 , 17 ], elepa allel g a i y and i s ex- ended e sion (T) , see Re . [ 18 ]. Meanwhile, o a s udy o wa elike solu ions o modi ied elepa allel g a i y necessa y o u u e applica ions o he pseudo- enso , see e e ences [19,20]. The e iew is o ganized as ollows. Fi s ly in Sec ion 2some de ini ions o g a i a ional pseudo- enso s in gene al ela i i y a e lis ed. In Sec ion 3.1 we de i ed he g a i a ional ene gy–momen um pseudo- enso o a gene al Lag angian o n h o de h ough wo p ocedu es: he i s me hod uses a a ia ional p inciple unde igid ans o ma ions ia Noe he ’s heo em and he second adop s he Landau–Li shi z p ocedu e [ 1 ] wi hou he in oduc ion o he supe -po en ial. Hence, in Sec ion 3.2, we p o ed ha a s ess– ene gy objec is ans o med in he co ec manne unde linea ans o ma ions bu no unde di eomo phisms and, he e o e, is a pseudo- enso and no a co a ian enso . In Sec ion 3.3, we calcula ed he Eule –Lag ange equa ions and he g a i a ional ene gy– momen um pseudo- enso o (R) g a i y, always using Noe he ’s heo em applied o a pa icula one-pa ame e g oup o di eomo phisms gi en by igid ansla ions. The e o e, in all models o g a i y we ob ained he con inui y equa ion o an ene gy–momen um complex. In Sec ion 3.4, we de i ed he g a i a ional ene gy–momen um pseudo- enso o a g a i a ion ield o a pa icula Lag angian Lg= (R+a0R2+∑p k=1akRkR)√−g . Sec ion 3.5, is de o ed o he weak- ield limi o he g a i a ional s ess–ene gy pseudo- enso expanded o lowes o de in a small pe u ba ion h , i.e., up o h2 o de , and we ha e shown wo simple cases whe e he index p is equal o ze o and one. Hence, in Sec ion 4.1, we a e aged he pseudo- enso o e an sui able egion con aining he isola ed sou ces and hen we ound he emi ed powe ca ied by he g a i a ional adia ion. A e wa d, in Sec ion 5.1, in Pala ini (R) g a i y, ela ed ield equa ions and ela ed g a i a ional ene gy–momen um pseudo- enso we e ob ained. The e o e in Sec ion 6, by adop ing a la FLRW space ime, an explici calculus o an ene gy densi y complex o Pa icles 2022,5300 powe law cosmological solu ions was pe o med, also in he me ic o malism o (R) . Conclusions a e summa ized in Sec ion 7. Finally in Appendix A.1, we p o ed ha he addi i e e ms ela ed o he symme ies o gµν and i s de i a i es yield a mean o ze o, i.e., hApη αi=hBpη αi= 0. While in Appendix A.2, we explici ly showed he six pola iza ion enso s associa ed wi h he g a i a ional wa es p esen in highe -o de heo ies. 2. Se e al De ini ions o G a i a ional Ene gy–Momen um Pseudo-Tenso in Gene al Rela i i y He e a e some o he mos impo an de ini ions o g a i a ional ene gy–momen um pseudo- enso in gene al ela i i y in he scien i ic li e a u e, o de ails see [21]. 2.1. Eins ein Ene gy–Momen um Complex In special ela i i y he law o conse a ion o ene gy and momen um is gi en by ∂Tµν ∂xµ=0 , (1) wi h Tµν he ene gy–momen um enso o ma e and non-g a i a ional ields. In gene al ela i i y his p inciple becomes o gene al co a iance ∇µTµν =0 , (2) which does no co espond o any law o conse a ion o physical quan i ies. Eins ein he e o e o mula ed he conse a ion law in he ollowing way ∂θ ν µ ∂xν=∂ ∂xνp−gTν µ+ ν µ=0 , (3) whe e ν µ is an pseudo- enso . So wha is conse ed is no only he enso o non- g a i a ional ields and ma e Tν µ bu a pseudo- enso ν µ mus be added o i . This pseudo- enso added can be in e p e ed as associa ed wi h he g a i a ional ield and he ene gy due o he sum o he con ibu ions o he g a i a ional ields plus hose due o he ma e is conse ed. Howe e , he pseudo- enso iali y beha iou o ν µ makes i dependen on coo dina es and he g a i a ional ene gy becomes non localizable. In o de o w i e he Equa ion (2) in he o m o an o dina y di e gence equa ion Equa ion (3) , Eins ein s a ing om he ollowing Lag angian densi y which is a non-co a ian scala densi y L=p−ggµνΓσ µνΓρ σρ −Γσ µρΓρ νσ, (4) in oduced a pseudo- enso de ined by he ela ion p−g ν µ=1 16π δν µL−∂L ∂gρσ,ν gρσ,µ!. (5) 2.2. Landau–Li shi z Ene gy–Momen um Pseudo-Tenso The g a i a ional ene gy–momen um pseudo- enso de ined by Landau–Li shi z has he g ea ad an age o being symme ic unlike Eins ein’s, which in gene al is no .This allows de ining he angula momen um and he e o e he ela ed conse a ion law. We adop a sys em o geode ic coo dina es whe e he i s de i a i es o he me ic enso gµν anish. Then, he Equa ion (2) is educed o (1) which can be w i en in e ms o he ollowing an isymme ic quan i y in he las wo indices ηµνσ =−ηµσν Tµν =∂ηµνσ ∂xσ. (6) Pa icles 2022,5301 Since he Le i–Ci i a connec ion Γ anishes a one poin , in such coo dina e sys em i is possible using Eins ein’s equa ions in he p esence o ma e w i en in such coo dina es, o exp ess he s ess–ene gy enso o ma e Tµν as Tµν =1 (−g) ∂ ∂xσ1 16π ∂ ∂xρ[(−g)(gµνgσρ −gµσgνρ)], (7) whe e indica ing he e m in b aces wi h he an isymme ic quan i y in he las wo indices hµνσ =−hµσν, we ge ∂hµνσ ∂xσ−(−g)Tµν =0 . (8) Re u ning o an a bi a y coo dina e sys em he p e ious di e ence does no cancel any- mo e so we can indica e i wi h (−g) µν o (−g)(Tµν + µν)=∂hµνσ ∂xσ. (9) Quan i ies µν a e symme ic bu a e no he componen s o a co a ian enso bu a ine. Using Eins ein’s ield equa ions again i is possible om Equa ion (9) o ge an explici exp ession o µν , de ined as he ene gy–momen um pseudo- enso o he g a i a ional ield, by means o he de i a i es o he componen s o he me ic enso , ha is 16π(−g) µν =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) whe e gµν =√−ggµν. 2.3. Mølle Ene gy–Momen um Complex The ene gy–momen um pseudo- enso s µν o bo h Eins ein and Landau–Li shi z besides ha ing he law o being enso s only a ine and no co a ian also depend on he choice o coo dina es. Then, Mølle looked o an exp ession o ene gy and g a i a ional momen um independen o he pa icula coo dina e sys em. To do his Mølle exploi ed he ac ha he pseudo- enso including ma e plus g a i y θµν =Tµν + µν can be de ined a less han a magni ude Sµν a ze o di e gence ∂µSµν = 0. In 1958 Mølle p oposed he ollowing complex enso o ene gy–momen um complex Tν µ=θν µ+Sν µ looking o he Sν µsuch ha Tν µ ans o med as a enso o only spa ial ans o ma ions Tν µ=1 8π∂ρhp−ggµσ,λ−gµλ,σgλνgσρi, (11) whe e he exp ession in squa e b acke s is he an isymme ic supe -po en ial Uνρ µ=−Uρν µ such ha ∂νTν µ=0 . (12) 2.4. Papape ou Ene gy–Momen um Pseudo-Tenso Papape ou in 1948 used he gene alized Beli an e me hod o de i e his pseudo- enso due o he complex o ene gy–momen um con ibu ions using Tolman’s exp ession o Eins ein’s o al pseudo- enso θν µ(3), i.e., θν µ=1 8π ∂ ∂xρ −gνσ ∂L ∂gµσ,ρ +1 2δν µgαβ ∂L ∂gαβ ,ρ , (13) Pa icles 2022,5302 whe e L is Eins ein Lag angian gi e by (4) and gνσ ha e been de ined in Equa ion (10) . Beli an e’s me hod consis s in inding a symme ic quan i y Ωµν =Ωνµ di e gence ee which di e s by ηµρθν ρ only o an an isymme ic quan i y di e gence in he i s wo indices Bµνρ =−Bµνρ i.e. Ωµν =ηµρθν ρ+∂ ∂xρBµνρ , (14) such ha ∂ ∂xνΩµν =0 , (15) wi h ηµν =diag(1, −1, −1, −1). Exp essing Bµνρ in e ms o o he ield spin densi y Sµνρ Bµνρ =−1 2(Sµνρ +Sρµν +Sρνµ), (16) you ge a e a ew coun s he exp ession o he o al pseudo- enso Ωµν Ωµν =1 16π ∂2 ∂xρxσp−g(gµνηρσ −gµρηνσ −gρσηµν −gνσηµρ). (17) This geome ic objec is symme ic wi h espec o he i s wo indicesµeν. 2.5. Weinbe g G a i a ional Ene gy–Momen um Pseudo-Tenso Weinbe g [22] de i ed he g a i a ional ene gy–momen um pseudo- enso by adop - ing a quasi-minkowskian coo dina e sys em. In his sys em he me ic enso gµν ends o ha o Minkowski ηµν a g ea dis ances om a localized ma e ial sys em. We w i e he me ic gµν as he sum o he me ic o Minkowski ηµν plus hµν ha goes o ze o o in ini y gµν =ηµν +hµν . (18) We linea ize Eins ein equa ions Gµν =− 8 πGTµν , expanding Ricci enso Rµν in e ms o powe s o hµν as R(1) µν −1 2ηµνR(1)=−8πGTµν + µν, (19) whe e µν =1 8πGRµν −1 2gµνR−R(1) µν +1 2ηµνR(1), (20) is he g a i a ional ene gy–momen um pseudo- enso . So in he Equa ion (19) you see ha eading he equa ion om igh o le , µν assumes he meaning o he sou ce o he linea ized cu a u e oge he wi h he enso o he non-g a i a ional ields and o he ma e Tµν . F om he linea ized Bianchi law o which quan i y R(1) µν mus sa is y, we ge he ollowing local conse a ion law ∂ ∂xν(Tµν + µν)=0 . (21) The pseudo- enso µν o he second o de in his µν =1 8πG−1 2hµνR(1)+1 2ηµνηρσR(1) ρσ +R(2) µν −1 2ηµνηρσR(2) ρσ+Oh3, (22) whe e Rµν o i s o de in his R(1) µν =1 2 ∂2hλλ ∂xµ∂xν−∂2hλµ ∂xλ∂xν−∂2hλν ∂xλ∂xµ+∂2hµν ∂xλ∂xλ!, (23) Pa icles 2022,5303 while o second o de 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. Ene gy–Momen um Complex in Cu a u e Based G a i y 3.1. The g a i a ional Ene gy–Momen um “Tenso ”’ o n h O de Lag angian Le us examine he ene gy–momen um complex o a ou h o de g a i a ional La- g angian, ha is, which depends up o ou h de i a i es o he me ic enso gµν as L=L(gµν , gµν,ρ , gµν,ρλ , gµν,ρλξ , gµν,ρλξσ) , whose ield equa ions, in gene al, a e o eigh h o - de in me ic o malism (see also [ 23 , 24 ]). In his manne we include all possible cu a u e in a ian s, no only  ope a o s, in o he g a i a ional ac ion. Then, we will gene alize he app oach o a g a i a ional Lag angian o n - h o de , i.e., which depends up o n h de i a- i es o me ic enso . We will de i e he ene gy–momen um enso using he Noe he ’s heo em, imposing ha g a i a ional ac ion is in a ian unde global ansla ions [ 1 ]. In his e iew he me ic signa u e o gµν adop ed is (+ , − , − , −) , while Ricci enso is de ined as Rµν =Rρµρν and Riemann enso as Rαβµν =Γα βν,µ+. . .. Le us a y he g a i a ional ac ion wi h espec o me ic gµν and coo dina es xµ[11,25,26] I=ZΩd4xL →˜ δI=ZΩ0d4x0L0−ZΩd4xL =ZΩd4xδL+∂µ(Lδxµ), (25) whe e ˜ δ s ands o he local a ia ion while δ means he o al a ia ion, ha is, keeping he alue o coo dina e x ixed. By in ini esimal ans o ma ions as x0µ=xµ+eµ(x), (26) he o al a ia ion o he me ic enso eads δgµν =g0 µν(x)−gµν(x)=−eα∂αgµν −gµα∂νeα−gνα∂µeα. (27) Unde global ans o ma ion, ∂λeµ= 0, he unc ional a ia ion o he me ic becomes δgµν =−eα∂αgµν . I we also equi e ha he ac ion o be in a ian unde his ans o ma- ion, ha is, ˜ δI=0, om a bi a iness o domain o in eg a ion Ω, we ha e 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) whe e he explici exp ession o g a i a ional ene gy–momen um enso , ha we will see being a pseudo- enso o a ine enso , is Pa icles 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) I he me ic enso gµν sa is ies he Eule –Lag ange equa ions o ou g a i a ional La- g angian 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) o an a bi a y eα, we ge a local con inui y equa ion o ou Noe he cu en ∂ηp−gτη α=0 . (31) In a mo e compac o m, he g a i a ional ene gy–momen um enso akes he ollow- ing o m 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) whe e we used he ollowing no a ion (),i0=I;(),i0···im=           (),i1i m=1 (),i1i2i m=2 (),i1i2i3i m=3 and so on ;(),ikik=(),ik Le us now gene alize ou app oach conside ing a gene al Lag angian densi y depending up o n h de i a i e o gµν , ha is, L=Lgµν,gµν,i1,gµν,i1i2,gµν,i1i2i3,··· ,gµν,i1i2i3···in . To al a ia ion o Lag angian Land i s Eule –Lag ange equa ions 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) whe e δ/δgµν is he unc ional de i a i e, while i is possible o exchange he a ia ion δ wi h he de i a i es δgµν,i0···im=∂i0···imδgµν , because we a e a ying keeping x ixed. So, we can ind a mos gene al local con inui y equa ion which allows us o de ine he ene gy–momen um pseudo- enso (which is an a ine enso as i will be p o ed la e ) o he g a i a ional ield o 2n h o de g a i y Pa icles 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) whe e Θis he Hea iside unc ion Θ[a,+∞[(n)=(1 i n∈[a,+∞[ 0 o he wise . (36) I ields and i s de i a i es anish on bounda y o ou spa ial egion o apidly dec eas- ing o he spa ial in ini e on an in ini y spacelike hype su ace, he g a i a ional ene gy– momen um enso is o ally conse ed and sa is ies a mo e gene al conse a ion law. An al e na i e way o ob ain he enso (35) is he p ocedu e de eloped by Landau [ 1 ]. Fo example, we s a by de i ing he enso (32) , because i s gene aliza ion o highe o de Lag angians is he same. Fi s o all, le us impose he s a iona y condi ion and a y he ac ion wi h espec o he me ic o ind he ield equa ions unde he hypo hesis ha bo h δgµν and he a ia ion o de i a i e δ∂ng anish on he bounda y o in eg a ion domain, canceling he su ace in eg als. Hence, he ollowing occu s: 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 pe o m he de i a i e o Lag angian espec o me ic enso and hen we pu i in o he ield equa ions (37). We ob ain ∂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) Pa icles 2022,5306 G ouping oge he e ms and enaming dumb indices, we ob ain ∂ηp−gτη α=0 , (40) ha is, he pseudo- enso is locally conse ed, whe e τη αis he enso de ined in (32). The ene gy–momen um complex, ins ead, can be de i ed conside ing he ma e ial Lag angian Lm=2χ√−gLmwi h s ess–ene gy enso gi en by Tηα =2 √−g δ(√−gLm) δgηα . (41) Thus, we use he ield equa ions in p esence o ma e , namely Pηα =χTηα , (42) whe e Pηα =−1 √−g δLg δgηα wi h he coupling χ=8πG c4. (43) By ield Equa ion (42), we ob ain 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 om symme y o enso Tη α, one ge s p−gTη α;η=p−gTη α,η−1 2gρσ,αTρσp−g. (47) The ela ion (45) ells us ha he conse a ion law o he ene gy–momen um complex, i.e., he sum o wo s ess–ene gy enso s due o ma e plus g a i a ional ields, is ela ed o he co a ian de i a i e o he only ma e pa . F om con ac ed Bianchi iden i ies we ge he o al conse a ion law and con e sely Gηα ;η=0↔Pηα ;η=0↔Tηα ;η=0↔∂ηhp−gτη α+Tη αi=0 , (48) whe e Gηα =Rηα −1 2gηαR is he Eins ein enso and he locally conse ed ene gy–momen um complex is gi en by Tη α=p−gτη α+Tη α. (49) In a nu shell, he con ac ed Bianchi iden i ies lead o he local conse a ion o ene gy– momen um complex o , ice e sa, he local conse a ion o ma e and g a i a ional ields in ol es he con ac ed Bianchi iden i ies (see also [ 27 ] o a de ailed discussion in modi ied g a i y). F om he local con inui y Equa ion (48) , i is possible o de i e some conse ed quan i- ies, Noe he cha ges, such as he o al 4-momen um o ma e plus g a i a ional ield. I we Pa icles 2022,5313 An impo an ex ension o local Lag angian (81) o non-local Lag angian is possible allowing p→∞. Le Dpbe a linea di e en ial ope a o de ined by Dp= p ∑ k=0 akk. (89) I he weak o s ong con e gence is gua an eed unde sui able assump ions o he co- e icien s ak (e.g. ∑∞ k=0|ak|<∞ ) and o he domain o he ope a o Dp , we ob ain he ollowing non-local ope a o F() lim p→∞ p ∑ k=0 akk=F()(90) and also ou local ac ion becomes non local, i.e. I=ZΩd4xR+RF()Rp−g. (91) Acco dingly in eg al ope a o ac s as Φ(x)=ZΩd4yF(x−y)R(x)=F()R(x). (92) Le us ca y ou now he limi n→∞ o he ene gy–momen um pseudo- enso o n -o de Lag angian (35), we may ob ain he non-local pseudo- enso , ha is lim n→∞τη α(x)=τη α(x). (93) Whe eas τη α(x) ans o ms as an a ine enso , we could show ha also i s limi o n→∞ , i.e., τη α(x), is an a ine enso . Fo an linea ans o ma ion x0µ=Λµ νxν|Λ| 6=0 (94) he ollowing a ine pseudo- enso changes as τη α(x)=Λ−1η σΛτ ατ0σ τx0. (95) Subs i u ing (95) in (93), we ha e 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 ha τσ τ(x) ans o ms as an a ine objec also in he limi n→∞. 3.5. The Weak-Field Limi o Ene gy–Momen um Pseudo-Tenso The g a i a ional ene gy–momen um pseudo- enso (87) ela ed o Lag angian (81) in weak ield app oxima ion can be pe o med pe u bing weakly space ime me ic a ound he Minkowski me ic ηµν as gµν =ηµν +hµν being |hµν|  1 , (97) whe e h=ηµνhµν is he ace o pe u ba ion. Thus, we expand he ene gy–momen um pseudo- enso o lowe o de in h , namely, e aining e ms up o h2 . Le ’s see wha becomes he weakly pe u bed pseudo- enso (88) in ha monic coo dina es whe e gµνΓσ µν = 0. The quad a ic pa o he Ricci scala Ryields R=−gµνΓρ µσΓσ νρ, (98) Pa icles 2022,5314 ha is R=−1 4gµνgσλgρegeµ,σ+geσ,µ−gµσ,egλν,ρ+gλρ,ν−gνρ,λ. (99) Keeping e ms up o second o de in h2, we ge ∂R ∂gαβ,γ!(1) gαβ,δ(1)h2 =1 2hαβ γ ,hαβ,δ−hγα β ,hαβ,δ, (100) acco ding o ∂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 pu hese e ms in o (88) , he s ess–ene gy pseudo- enso in gene al ela i i y up o o de h2 akes he o m 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 ha e o expand o second o de in h he co ec ions o he pseudo- enso (87) due o ex ended g a i y e ms. To lowe o de in hwe conside he ollowing 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 ake in o accoun only he e ms up o h2in ha monic 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 Pa icles 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 Equa ions (110) , (111) and (107) , we ha e dis ega ded he index pe mu a ions ( µν ) and (ηi1···i2h+1) because Apη α and Bpη α e ms, a e aged on a sui able space ime egion, anish, acco ding o Appendix (A.1) . Hence we calcula ed only he e m de i ing om (A1) wi hou conside ing he index pe mu a ions ( µν ) and (ηi1···i2h+1) . This because, aking in o accoun e ms ob ained om pe mu a ions in Apη α and Bpη α , a e aged on a sui able space ime egion, we ob ain ha a e equal o ze o as we will see below in Appendix A.1. This ma hema ical ick is essen ial o calcula ed he a e aged g a i a ional ene gy–momen um pseudo- enso and he powe emi ed by a sou ce. So, by inse ing equali ies (108) , (109) , (110) and (111) in o (87) , we ind he ex a e m o pseudo- enso τη α o second o de owing o ex ension o gene al ela i i y , ha we call ˜ τη α, ha is ˜ τη αh2 =1 2χ(1 4 p ∑ k=0 akk+1h!h,ηα+1 2 p ∑ =0 a  +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) whe e con en ions used a e (),αi0=(),αhi0 ,λ=h,λ. In summa y, we can spli he g a i a ional ene gy–momen um pseudo- enso in he gene al ela i i y pa and in he Ex ended G a i y pa , ha is τη αh2 =τη α|GR +˜ τη α. (113) Now in he pa icula case when p is equal o 0 and 1, ex ended co ec ions o he pseudo- enso ˜ τη α was de i ed. Then, o p= 0, ha is, Lg=R+a0R2√−g as in he case discussed in [25], we ob ain τη αh2 =τη α|GR +˜ τη α, Pa icles 2022,5316 wi h ˜ τη αh2 =a0 2χ1 2h,ηαh+hη λ,αh,λ−h,αh,η−1 4(h)2δη α. (114) While o p=1, ha is Lg=R+a0R2+a1RR√−g, one has τη αh2 =τη α|GR +˜ τη α, whe e ex ended co ec ions o pseudo- enso a e ˜ 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 i e a ion can be pe o med o e e y pin oducing new con ibu ions in o dynamics. 4. Powe Emi ed Ca ied by a G a i a ional Wa e We wish o calcula e he powe emi ed in he o m o g a i a ional wa es by an isola ed massi e sys em conside ing he local conse a ion o he ene gy–momen um pseudo- enso (40). 4.1. The A e age o he Ene gy–Momen um Pseudo-Tenso Le us now ega d he wa elike solu ions o he linea ized ield equa ions in acuum associa ed wi h Lag angian (81) , o de ails see Re . [ 42 ]. G a i a ional wa es solu ions can be exp essed as hµν(x)= p+2 ∑ m=1ZΩ d3k (2π)3(Bm)µν(k)ei(km)αxα+c.c. , (116) whe e (Bm)µν(k)=   Cµν(k) o m=1 1 3ηµν 2+(km)µ(km)ν k2 (m)Am(k) o m≥2, (117) wi h Cµν(k) ela ed o ans e se- aceless pola iza ion enso ypical o gene al ela i i y and Am(k) he ampli ude o wa e a k ixed. He e “c.c.” s ands o he complex conjuga e. The ace o enso (117) is (Bm)λ λ(k)=(Cλ λ(k) o m=1 Am(k) o m≥2, (118) and he kµ m=(ωm,k) is he wa e ec o wi h k2 m=ω2 m−|k|2=M2 whe e k2 1= 0 and k2 m6=0 o m≥2. Keeping k ixed, we de i e he ollowing ela ions 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) Pa icles 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 ∑ =2k2 nA eik x). (124) Now, we choose a domain o he space ime Ω such ha |Ω|  1 |k| [ 22 ]. Then, we can pe o m he a e age o he g a i a ional ene gy–momen um pseudo- enso τη α o e ou egion and all in eg als, including e ms such as ei(ki−kj)αxα , end o ze o, by means o ollowing iden i ies Re{ }Re{g}=1 2Re{ g}+1 2Re{ ¯ g}, (125) (kl)λh(Bl)ηλ −ηηλ(Bl)ρ ρi=−(kl)η 2Al. (126) In he ha monic gauge, a e a e aging and some algeb aic manipula ions, we ind (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 ∑ =2 (k )α(k )ηk2 (m+q)|A |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 se o pola iza ion enso s o ming a basis o he linea ized solu ions hµν is gi en in Appendix A.2. Acco ding o equali ies (127) , we can calcula e he a e age alue o he ene gy–momen um pseudo- enso 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) wi h g a i a ional coupling χ=8πG c4 . In TT gauge o he i s mode associa ed wi h k1 and only in ha monic gauge o esidual modes km, in he momen um space, i ge s ((k1)µCµν =0∧Cλ λ=0 i m=1 (km)µ(Bm)µν =1 2(Bm)λ λkνi m≥2. (129) Pa icles 2022,5318 We now explo e a g a i a ional wa e p opaga ing in he +z -di ec ion a k ixed, wi h 4-wa e ec o gi en by kµ=(ω, 0, 0, kz) whe e ω2 1=k2 z i k2 1= 0 and k2 m=m2=ω2 m−k2 z o he wise wi h kz> 0. Acco dingly he a e aged ime-space enso ial componen which can be seen as lux o g a i a ional ene gy along he z axis h ough he su ace ha delimi s ou domain Ω, eads 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 calcula e he emi ed powe pe uni solid angle Ω , adia ed by he localized sou ces, in a di ec ion ˆ x a k ixed. By choosing o he sui able gauge, o he local conse a ion o he ene gy–momen um pseudo- enso (40), he powe is gi en by dP dΩ= 2ˆ xiDτi 0E. (131) By anging he index p o he pseudo- enso (130) o e { 0, 1, 2 } , we ob ain he ollowing h ee cases o p= 0 Dτ3 0E=c4ω2 1 8πGhC2 11 +C2 12i+c4 16πG−1 6ω2|A2|2kz+2a0ω2m2 2|A2|2kz, (132) o 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 o 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) whe e he g a i a ional coupling χ has been explici ly indica ed. By Fo mulas (132) – (134) i is ob ious ha he i s e m comes ou o gene al ela i i y and he co ec ions s ongly depends on p . In any con ex whe e co ec ions o gene al ela i i y can be in es iga ed, his app oach could cons i u e a pa adigm o sea ch o highe o de e ec s. Pa icles 2022,5319 5. Ene gy–Momen um Complex o (R)G a i y in Pala ini App oach 5.1. The G a i a ional Pseudo-Tenso o (R)G a i y in Pala ini Fo mula ion In Pala ini app oach he me ic enso gµν and he connec ion Γα µν a e independen , ha means ha we do no assume any ela ion be ween he me ic and he connec ion, and Riemann and Ricci enso s a e, in gene al, de ined as Rµν(Γ) =∂αΓα µν −∂νΓα µα +Γα µν Γσ ασ −Γα νλ Γλ µα, (135) R(g,Γ) =Rµν(Γ)gµν. (136) So, he Pala ini g a i a ional ac ion o (R)appea s as [43] S=1 2κ2Zd4xp−g (R), (137) wi h he coupling κ2= 8 πG/c4 and g he de e minan o me ic enso gµν . By a ying he me ic gµν and he connec ion Γα µν , o a gene al in ini esimal ans o ma ion coo dina e xµ i ge s 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) whe e ˜ δ is he local a ia ion and δ is he a ia ion ha keeps he coo dina es x ixed. The a ia ion o he g a i a ional ac ion wi h espec o he me ic gµν and he connec ion Γα βγ yield ˜ δS=1 2κ2Zd4x(p−g RRµν −1 2gµν δgµν + Rgµν δRµν+∂µp−g δxµ), (141) whe e R:=d (R)/dR. Acco ding o he ollowing Pala ini iden i y δRµν =∇αδΓα µν−∇νδΓα αµ. (142) he ac ion (141) akes he o m ˜ δS=1 2κ2Zd4x(p−g RRµν −1 2gµν δgµν +δΓλνµh−∇λp−ggνµ R+∇αp−ggµαδν λ Ri +∂λhp−g Rgµνδλ α−gµλδν αδΓαµν +p−g δxλi). (143) By he p inciple o leas ac ion o s a iona y ac ion (137) , by imposing ha he a ia ion o me ic and i s de i a i es anish a he bounda y, we ob ain ield equa ions o he me ic enso and he connec ion in acuum, i.e., RR(µν)−1 2gµν =0, (144) ∇λp−ggνµ R=0. (145) Pa icles 2022,5320 Gi en ha we adop ing an a bi a y non-compa ible connec ion, he symme ic pa o he Ricci enso , R(µν) , en e in he Equa ion (144) and hen he Ricci enso is non symme ic, ha is Rµν =Rνµ +Rλλµν , (146) being Riemann enso Rσλµν no longe an isymme ic on i s i s wo indices, i.e., he e m Rλλµν does no anishes. Fo a gene ic in ini esimal ans o ma ion, he me ic enso and he connec ion 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) whe e we ha e neglec ed e ms o highe o de in ξµ in he se ies expansion. Unde a igid in ini esimal ansla ion, ha is, ∂µξν=0, we ob ain g0µν(xλ)≃gµν(xλ)−ξλ∂gµν ∂xλ, (150) Γ0α µν(xλ)≃Γαµν(xλ)−ξλ∂Γαµν ∂xλ. (151) The e o e, he Pala ini ac ion (143) becomes ˜ δSg=1 2κ2Zd4x(−p−g RRµν −1 2gµν ξλgµν ,λ −ξλΓβ νµ,λh−∇βp−ggνµ R+∇αp−ggµαδν β Ri +∂λh−p−g Rgµνδλ α−gµλδν αξβΓαµν,β+p−g ξλi). (152) I he me ic gµν and he Pala ini connec ion Γα βγ a e solu ion o Equa ions (144) and (145) , he s a iona y o he local a ia ion o he ac ion (152) , gi es he local conse a ion o g a i a ional ene gy–momen um pseudo- enso τλβo Pala ini (R)g a i y, namely ∂λp−gτλβ=0, (153) whe e τλβis de ined as τλβ=1 2κ2h (R)δλ β− R(R)gµνδλ α−gµλδν αΓαµν,βi. (154) I is wo h no ing ha he pseudo- enso de ined in Equa ion (154) has he opposi e sign o he one de ined abo e. In o de o de i e he ene gy–momen um complex, le us analyze he ac ion con aining he ma e pa , ha is Sm=Zd4xp−gLm. (155) Gene ally, he ma e Lag angian Lm depends on he connec ion as, o example, occu s in p esence o e mion ields. He e, we conside only ma e ial Lag angian which does no depend on he a ine connec ion Γ. Then, he ma e ene gy–momen um enso is de ined Pa icles 2022,5321 as in (75) . Hence, ield equa ions o me ic and connec ion, i.e., Equa ions (144) and (145) , in p esence o ma e yield RR(µν)−1 2gµν =κ2Tµν, (156) ∇λp−ggνµ R=0. (157) As al eady poin ed ou abo e he connec ion can be non compa ible wi h he me ic gµν , i.e., ∇λgµν 6= 0. In compac o m, we can de ine a new me ic, con o mally ela ed o he me ic gµν, as hµν := Rgµν. (158) so ha Equa ion (157) becomes ∇λ√hhµν=0. (159) Thus he Pala ini connec ion Γαµν appea s as he Ch is o el connec ion o he new me ic hµν, i.e., Γαµν =1 2 R(R)gαβ∂µ R(R)gνβ+∂ν R(R)gµβ−∂β R(R)gµν. (160) The Pala ini connec ion Γαµν and Le i–Ci i a connec ion ◦ Γαµν a e ela ed as Γαµν =◦ Γαµν +δα µAν+δα νAµ−gµν Aα, (161) whe e he ou - ec o Aµis de ined as Aµ:=1 2 R∇µ R. (162) Fo (R) = R, we eco e he Ch is o el symbols cons uc ed by he me ic gµν, ha is Γαµν =◦ Γαµν=1 2gαβgβµ,ν+gβν,µ−gµν,β(163) his means ha in gene al ela i i y no di e ence esul s in me ic and Pala ini o malism. The Ricci enso Rµν in Pala ini o malism and ha in me ic o malism Rµν , a e ela ed as ollows Rµν =Rµν +3 2 1 ( R(R))2◦ ∇µ R(R)◦ ∇µ R(R) −1 R(R)◦ ∇µ◦ ∇ν−1 2gµν ◦  R(R), (164) whe e ◦ :=◦ ∇µ◦ ∇µ and ◦ ∇ deno es he co a ian de i a i e associa ed wi h he Le i–Ci i a connec ion. Con ac ing enso ial equali y (164) wi h gµν , we ob ain he ela ion be ween Rand R, ha is, he Ricci scala in bo h app oach R=R+3 2( R(R))2◦ ∇µ R(R)◦ ∇µ R(R)+3 R(R)◦  R(R). (165) Pa icles 2022,5322 Adop ing he Pala ini connec ion Γαµν (160) , he symme y o Ricci enso is es o ed on accoun o he ela ion Γλ=∂λ 2 R√−g 2 R√−g, (166) which implies R[µν]=∂[µΓν]=0. (167) Fu he mo e he connec ion is non compa ible wi h me ic gµν being ∇λgµν =−gµν R∇λ R. (168) Despi e his, he co a ian de i a i es associa ed wi h Pala ini connec ion commu e each o he , as displayed below ∇ρ,∇λgµν =0 . (169) Thus, we es o e he an isymme y on he i s wo indices o Riemann enso , namely Rµνλρ =−Rνµλρ. (170) by he de ini ion o Riemann enso o an a bi a y enso Jµν ∇ρ,∇λJµν =−Rαµρλ Jαν −Rανρλ Jµα. (171) In addi ion, he con ac ed Bianchi iden i ies a e ul illed, ha is ∇µRµν −1 2gµνR=0. (172) Acco ding o he Pala ini connec ion Equa ion (160) and om he symme y o ene gy– momen um enso Tµν, aking in o accoun ha o he new me ic hµν we ha e Γλ=∂λ√−h √−h, (173) and Γµνλ +Γνµλ =1 R∂λhµν, (174) so we de i e he ollowing use ul exp ession √−h∇σTσν=∂σ√−hTσν−1 2 RTλρ∂νhλρ√−h. (175) Field equa ions in ma e (156) lead o 0=√−h 2 2 R Tµνgµν,βξβ+∂λp−g1 2κ2h (R)δλ β− Rgµνδλ α−gµλδν αΓαµν,βiξβ, (176) and om Equa ion (175) , a e some algeb aic manipula ions, we ge he ollowing 4- di e gence o ene gy–momen um complex no anishing ∂σhp−gTσβ+ σβi=√−h 2 R∇λTλβ+2√−h 3 R Tλβ∇λ R−√−h 2 3 R T∇β R. (177) F om con ac ed Bianchi iden i ies and he ield equa ions, he ollowing ela ions a e sa is ied ∇µ,∇ν∇µ R=Rαν∇α R, (178) Pa icles 2022,5329 5. Bak, D.; Cangemi, D.; Jackiw, R. Ene gy-momen um conse a ion in g a i y heo ies. Phys. Re . D 1994 ,49, 5173. [C ossRe ] [PubMed] 6. Lee, D.L.; Ligh man, A.P.; Ni, W.T. Conse a ion laws and a ia ional p inciples in me ic heo ies o g a i y. Phys. Re . 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