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
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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
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