On the Loalization of the Gravitational Energy
N. PintoNeto and P.I. Trajtenberg
CentroBrasileirode PesquisasFsias-CBPF
RuaXavierSigaud,150-ep22290-180-Riode Janeiro, RJ-Brazil,
E-mailnelsonpnlafexsu1.lafex.bpf.br
Reeived26August,1999
UsingaformalismintroduedbyFeynman,Deser, Grishhuk,Petrov andPopova,the
pseudoten-sors, suh as dened by Einstein, Tolman, Landau/Lifshitz and Mller, are expessed as gauge
dependenttensors inabakgroundspae,asthe gravitationalenergy-momentumtensorofDeser,
Grishhuk, Petrov and Popova. Using a resultobtained by Virbhadra for the energy density in
the Reissner-Nordstromspaetime,it is shownthat theation ofthesegauge transformations on
theabovetensorial expressionsisthesameasthe ationoftheoordinatetransformationsonthe
equivalentpseudotensorialexpressions,meaningthatthesetensorsanbesettozeroatapointby
asuitablehoieofgaugetransformation.
I Introdution
From the advent of General Relativity (GR) theory,
variousmethodshavebeenproposedtodeduethe
on-servation lawsthat haraterize the gravitational
sys-tems. In an initial phase, whih extended until the
middleofthe50's,predominatedintheliteraturetheso
alledpseudotensorialformalism[1℄. Inthisformalism,
theonservationlawsare expressed,asusual, through
theontinuityequations
T
;
=0; (1)
wheretheonservedquantityT
referstotheloalux
and densityof energyand momentum of gravitational
systems whih mayinlude, in general,boththe
mat-ter ontribution through itsenergy-momentum tensor
T
, and the gravitational eld ontribution, here
rep-resentedbythetermt
:
T
=
p
g(T
+t
): (2)
An essential property of this formalism is that,
ontraryto thematter ontribution, the gravitational
term isnotatensor andadmits nohomogeneouslaws
of transformation under oordinate transformations,
whih justies its designation, like T
itself, as the
energy-momentum pseudotensor.
Ineet,aordingtoagenerallyaepted
interpre-tation by the authors whih built this formalism, the
onservationlaws(1)representafundamentalphysial
proess,whosedynamialdesriptiondoesnotdepend
on the hoie of the oordinate system. However, the
invarianeoftheonservationlaws(1)impliesthatthe
transformation laws of T
must not be homogeneous
and this inhomogeneity is related to the gravitational
term,sinethematterontributionisatruetensor
den-sityterm.
On the other part, the presene of the
inhomoge-neousterm in the transformation law of T
makes it
possibletoannulthegravitationaltermloally,i.e., at
onegiven point. This property is onsistent with the
Equivalene Priniple aordingto whih the
gravita-tionaleld anbeloally eliminatedifweadopt a
o-ordinatesystem adapted to an observerin freefall at
apoint. Reiproally,inatspae-time,wherethereis
nogravitationaleld,thegravitationaltermwillnotbe
zerowhenexpressedin anarbitraryoordinatesystem
whih is not artesian. Hene, in the pseudotensorial
formalismit does not make sense to ask if the
gravi-tationalenergydensityiszeroornotat asinglepoint,
sinethisoneptdependsontheoordinatesystem. In
otherwords,thepseudotensorialformalismannot
at-tributeameaningtotheloalizationoftheenergyand
momentumofagravitationalsystemwhihisabsentof
ambiguities.
Furthermore,the onservation laws (1) do not
de-terminetheonservedquantityT
univoally. Ineet,
theonservationondition(1)allowsusto expressT
througha given anti-symmetri potentialfuntion, or
superpotential H [℄
, aording to the following
rela-tion:
T
=H
[℄
;
; (3)
However, if U
is a new funtion (not
neessar-ilyanti-symmetri),whihdiersfromH [℄
byaterm
, whose divergene or double divergene is
; 0or
;;
0 (4)
then,thequantity
denedthroughthisnew
super-potentialisalsoonserved:
=U
;
)
;
=0 (5)
Using this freedom on the hoie of the
superpoten-tial, authors like Einstein, Tolman, Landau, Lifshitz
and Mller arrived through dierent methods at the
followingsuperpotentials:
H [℄
=
1
2 ~ g
(~g
~ g
~ g
~ g
)
;
(Einstein) (6)
T
= H [℄
+
1
2 (Æ
~ g
Æ
~ g
)
;
(Tolman) (7)
L
= 1
2 [~g
~ g
~ g
~ g
℄
;
(Landau=Lifshitz) (8)
M [℄
=
1
~ g
g
(g
; g
;
)(Mller) (9)
d
where ~g
= p
gg
is theinverse ofg~
andisthe
Einsteinonstant. Beause oftheinherentambiguities
ofthepseudotensorapproah,manyotherproposalsfor
thedenitionofthegravitationalenergy-momentum
lo-al density havebeenmade. Amongthem weanite
theKomardenition[2℄,theDeser,Grishhuk,Petrov
andPopova(DGPP)denition[3℄andtheBrown-York
denition 1
[4℄. TheKomardenition seemstobemore
adequatelyrelated to thenotionof gravitationalmass
thentothenotionofenergy(seeRef. [6℄)The
Brown-York denition is not in fat aloal but aquasi-loal
denition. TheDGPPdenitionsueedstobuilda
lo-al gravitationalenergy-momentumtensor by
desrib-ing gravitation as a spin two eld propagating in a
Rii-at xed spaetime, but it suers from a gauge
ambiguity.
In this paper we will examine in more detail the
gaugeambiguityoftheDGPPdenition. Wewillshow
thatthisambiguityisnothingbuttheoordinate
trans-formation ambiguity of the pseudotensors translated
to the DGPP language. In fat, the DGPP
energy-momentum tensor itself is a new energy-momentum
pseudotensorifwewriteitinthegeometriallanguage.
Reiproally,weanwriteallpseudotensorsastensors
intheDGPPapproah,allofthempossessingthesame
gaugeambiguity.
This paperisorganizedasfollows: Insetion Ithe
DGPPtheoryisbrieydisussedanditsexpressionfor
the energy-momentum tensor is dedued. In setion
II we show how the pseudotensors an be expressed
astensorelds, beinghowever,liketheDGPP
energy-momentum tensor, gauge dependent. In setion III,
using aresultobtainedbyVirbhadra, itisshownthat
the gauge dependene of the pseudotensors is
equiva-lenttothedependenewithrespettothehoieofthe
oordinate system in GR. We onlude in setion IV,
withsomeommentsanddisussions.
II The DGPP formalism
IntheDGPPformalismthegravitationaleldK
and
the gravitationalpotentialh
anbe treatedas
ten-sor elds dened in aertain Riemannian bakground
spaetime previouslyxed, haraterized bythe
bak-groundmetri
andbythebakgroundonnetions
C
sothat
p
gg
= p
(
+h
); (10)
and
=C
+K
: (11)
ThebakgroundspaemustbeRii-at,i.e.,R (o)
=0,
in orderforthetheorybeequivalenttoGR
Intermofthetensoreldsdenedon(10)and(11),
the Einstein/Hilbert ation for the gravitational eld
anbeexpressedas:
S= 1
2 Z
R p
gd 4
x= 1
2 Z
~
h
(K
; K
; )+(~
+ ~
h
)(KK)
; (12)
1
where
~
p
; ~
h
p
h
; (13)
K
K
; (14)
and
(KK)
K
K
K
K
: (15)
Thesemi-olon(;)representstheovariantderivativewithrespettothebakgroundonnetions. Wehaveomitted
in thelagrangian(12)theterms
L
0 =
p
R (o)
; L
1 =
~
h
R (o)
andL
2 =[~
K
~
K
℄
;
; (16)
whihdonotontributetothevariationoftheation: L
0 andL
1
beauseofRii-atness,and L
2
beauseitisa
totaldivergene.
The dynamial equations for the gravitational eld an be obtained by varying the DGPP ation (12) with
respettotheelds ~
h
andK
resultingin thefollowingequationsfor ~
h
and K
:
G L
= (KK)
+
1
2
(KK)
+Q
;
; (17)
and
~
h
; +(~
+ ~
h
)K
(~
+ ~
h
)K
(~
+ ~
h
)K
=0; (18)
where
2G L
=[
h
+
h
Æ
h
Æ
h
℄
;;
; (19)
and
2Q
=
h
K
+h
K
h
K
h
K
+
+ h
(K
K
)+h
(K
K
)+
+ h
(K
+K
):
(20)
Theenergy-momentumtensorofthegravitationaleld desribedbytheation(12)anbealulatedasusual,
byvaryingtheDGPPlagrangianwithrespettothebakgroundmetri,yielding
t
=
1
p
ÆL
Æ
= (KK)
+
1
2
(KK)
+Q
;
; (21)
d
whihisjust theright-hand-sidetermofequation(17)
for thegravitationaleld. Combiningthe above
equa-tions, it is possibleto write the energy-momentum of
thegravitationaleld as:
2t
=[
h
+
h
Æ
h
Æ
h
℄
;; : (22)
Hene, in the DGPP presription, the gravitational
term anbe expressed as atensorwith respet to the
oordinatetransformationsonthebakgroundspae.
Thetheory anbeextended to inlude the matter
ontribution. IfL (m)
istheLagrangianassoiatedwith
thematter eldstheorrespondingenergy-momentum
tensorwill bedened,inanalogywith(21)as:
T
=
1
p
ÆL
(m)
Æ
: (23)
Admittingthehypothesisofminimalouplingbetween
the matter and the gravitational eld, the free eld
equation(17)anbegeneralizedtotheform:
G L
=(t
+T
): (24)
withG L
andt
dened by(19)and(21).
ThesymmetrypropertiesofG L
implyinthe
iden-tity
G L;
0; (25)
whih results, aording to (24), in the ovariant
energy-momentum onservation laws for the matter
plusthegravitationaleld:
T ;
=0)T
=t
+T
: (26)
Hene, in the DGPP formalism the invariane of the
onservationslawswithrespettooordinate
transfor-mationsisompatiblewiththetensornatureofthe
on-servedquantities,unlikethepseudotensorialformalism.
III The pseudotensors in the
DGPP formalism
Itwillbeshowninthissetionhowthepseudotensors,
denedoriginallyintermoftheEinstenmetrig
itsderivativesmaybefullyexpressedintermofthe
ele-mentarytensoreldsh
andK
denedintheDGPP
formalism.
Considerrst the original expressionsof the
pseu-dotensorsintermofthemetrig
:
E
=
1
2 [~g
(~g
~ g
~ g
~ g
)
; ℄
;
(Einstein=Tolman); (27)
L
= 1
2 [~g
~ g
~ g
~ g
℄
;;
(Landau=Lifshitz); (28)
M
=
1
[~g
g
(g
; g
; )℄
;
(Mller): (29)
d
DenedaseldsontheDGPPbakgroundspaethe
pseudotensors,at eahpoint,anbewritten inaloal
inertialoordinatesysteminwhihtheonnetionsare
zeroatapointandwheretheovariantderivativesare
ordinaryderivatives. To express the pseudotensors in
an arbitrary oordinate system we an use the
bak-ground onnetion C
(see eq. (11)) to replae
ordi-naryderivativesbyovariantderivatives.
FortheLandau/Lifshitzpseudotensor oneobtains,
inanarbitraryoordinatesystem,theovariantform:
L
= 1
2 [~g
~ g
~ g
~ g
℄
;;
(30)
Replaing in the aboveexpression the denition (10),
itresultsthat
L
=( )fT
+ 1
2 [h
h
h
h
℄
;;
g; (31)
where
T
= 1
2 [
h
+
h
h
h
℄
;; :
(32)
is theDGPPenergy-momentum tensorof matter plus
the gravitational eld (see equations (19), (24) and
(26)). Hene, in the DGPP formalism, the
Lan-dau/Lifshitz pseudotensor an be written in tensorial
formaswasalreadyreognizedin referene[3℄.
For the pseudotensors of Einstein/Tolman and
Mller,somediÆultiesanappearduetothepresene
ofthe ovariantmetridensity~g
,whihanonlybe
expressedin termoftheDGPPeld h
=
h
bymeansofaninniteseries
~ g
=
1
p
g g
=
1
p
f
h
+h
h
h
h
h
+::::g: (33)
d
SuhadiÆultyanbeoveromebyusingtherelation:
~ g
; =K
~ g
+K
~ g
K
~ g
: (34)
Then,weobtainfortheEinstein/Tolmanpseudotensor
theexpression:
E
=
p
2 [A
; K
A
+K
A
℄
;
; (35)
whereA
isgivenby
A
=Æ
(
+h
) Æ
(
+h
): (36)
The Einstein/Tolman pseudotensor beomes a tensor
densityofweight1intheDGPPformalism. Similarly,
fortheMllerpseudotensorweget
M
=
p
[K
A
℄
;
; (37)
whihisalsoatensordensity.
Hene,allthepseudotensorsadmit apurely
tenso-rialrepresentationin theDGPPformalism. Wewould
liketo remark that theproedure of replaingthe
or-dinaryderivativebytheovariantoneyieldsnontrivial
results(nonnulltensors)onlyintheDGPPframework,
where the bakgroundonnetion C
, insteadof
is used,sothat in thisasethe ovariantderivativeof
g
is notzero.
IV Gauge dependene in the
DGPP formalism
If the gravitational energy-momentum pseudotensors
the DGPP energy-momentum tensor itself, where do
residetheambiguitiesinthedenition ofloalized
en-ergyinGRintheDGPPapproah? Wemustremember
thattheDGPPframeworkisjustanalternativeviewof
thesametheory,i.e.,GR.Theansweromesifwenote
that theoordinatetransformationinvarianeofGRis
translated to invariane under gauge transformations
onh
and K
intheDGPPformalism.
Ineet,onsidertheinnitesimaloordinate
trans-formation x 0
=x
+
(x), whih hanges the
fun-tional formofg~
as
~ g
0
(x)=g~
(x)+$ (1)
~ g
(x); (38)
wheretheLie derivative$ (1)
~ g
isgivenby
$ (1)
~ g
(x)= ~g
;
+g~
; +~g
; ~ g
; : (39)
Intheaseofanitetransformation,thehangeing~
isgivenby
~ g
0
(x)=g~
(x)+ 1
X
k =1 1
k! $
(k )
~ g
; (40)
where$ (k )
istheLiederivativeoforderk denedas
$ (k )
=$
(1)
[$
(k 1)
℄: (41)
Substitutingin (40)thedenition(10)weget
~ g
0
(x)=~
(x)+ ~
h
(x)+ 1
X
k =1 1
k! $
(k )
(~
+ ~
h
): (42)
d
Thetransformedmetridensityanbedeomposedin
twodistintways:
~ g
0
(x)=~
(x)+ ~
h 0
(x) (43)
and
~ g
0
(x)=~
(x)+ ~
h
(x); (44)
where, byomparisonwith(42)onegets
~
h 0
(x)= ~
h
(x)+ 1
X
k =1 1
k! $
(k )
(~
+ ~
h
); (45)
~
h
(x)= ~
h
(x)+ 1
X
k =1 1
k! $
(k )
~
h
; (46)
and
~
(x)=~
(x)+ 1
X
k =1 1
k! $
(k )
~
: (47)
On the form (43) the transformation ats only in
the eld ~
h
, letting invariantthe bakground metri
(x). Inthissense,oneaninterpretthe
transforma-tion(45)asapuregaugetransformation.
Analogouslythepure gaugetransformation forthe
eld K
(x)is
K 0
(x)=K
(x)+
1
X
k =1 1
k! $
(k )
(C
+K
); (48)
whilefortheasewhihorrespondtothe
transforma-tion(46)wehave
K
(x)=K
(x)+
1
X
k =1 1
k! $
(k )
K
: (49)
It is possible to show that the dynamial equations
forthe gravitationaleld (17) areinvariantunder the
transformations(45)and(48)supposingthatthe
bak-groundspaeisRii-at. [3℄
On the form (44), (46),(47) and (49)however, the
transformation is not a pure gauge transformation ,
sineitats ontheeld aswellason thebakground
metri. Infat,thesearetheusualtransformationson
tensorialeldsresultingfromageneralmappingofthe
manifoldonwhihtheyaredened. Hene,alltensors
in the manifold, in partiular the energy-momentum
tensorsdenedinthelastsetion,willtransforminthe
usualhomogeneousway:
T 0
(x)=T
(x)+ 1
X
k =1 1
k! $
(k )
T
(x) (50)
However, the situation is ompletely dierent for the
aseoftransformations (45)and(48). The tensorsdo
nottransform in the usual way (50)but ontains
ex-tra inhomogeneous terms whih brings the possibility
ofannullingthem. TheDGPPenergy-momentum
ten-sor,forexample,transformin thisaseaordingto:
T
(h
0
;K 0
)=T
(h;K)+ 1
^
G L
"
1
p
1
X
k =1 1
k! $
(k )
(~
+ ~
h
) #
where ^
G L
is the operator whih when ating on h
yields expression (19). Hene, it is always possible to
ndgaugetransformations(45)and(48)whihmakes
theenergy-momentumtensorsdenedpreviouslytobe
null. This is the analoguein the DGPP approah to
whathappenswiththepseudotensorsin GR.
To illustrate in more detail what happens in the
DGPP approah, we will investigate a spei
exam-ple, taking the DGPP and Landau/Lifshitz
energy-momentum tensors, and omparing their
transforma-tions under gauge and oordinate transformations in
theeldandgeometrialframeworks,respetively.
Ini-tially, let us express the Reissner-Nordstrom solution
in theasymptotiallyMinkowiskianoordinatesystem
(T;x;y;z)as:
ds 2
= dT 2
+dx 2
+dy 2
+dz 2
+(1 B)(dT +dr) 2
(52)
d
whereB=1 2m
r +
Q 2
r 2
andr 2
=x 2
+y 2
+z 2
>r 2
0 ,with
r
0
representingtheReissner-Nordstromeventhorizon.
Usingthisoordinatesystem,thepseudotensorsare
al-uledin Ref. [9℄givingtheresult
E 0
0 =L
00
=1=2M 0
0 =
Q 2
8r 4
; (53)
where theoriginalsexpressions (27),(28) and(29)for
the pseudotensors were used. Performing the
oordi-natetransformationonthetimevariable
T =t+ Z
(B 1
1)dr; (54)
the Reissner-Nordstrom solution an be expressed in
thenewoordinatesystem(t;x;y;z)intheform
ds 2
= Bdt 2
+dx 2
+dy 2
+dz 2
+(B 1
1)dr 2
: (55)
Inthis oordinatesystem thepresriptionsof
Ein-stein/TolmanandMllerfurnishthesameprevious
re-sults, while theLandau/Lifshitzpseudotensor isgiven
by
L 00
= Q
4
+Q 2
r(r+4m) 4m 2
r 2
8r 4
[Q 2
+r(r 2m)℄ 2
; (56)
revealing,onthisform,theambiguitiesthat
harater-ize the pseudotensorial formalism with regard to the
physialmeaningoftheenergydensityingravitational
systems.
We an write the two asymptotially at line
ele-mentused aboveinspherialoordinates(r;;). For
thelineelement(52)weobtain
ds 2
= dT 2
+dr 2
+r 2
(d 2
+sen 2
d 2
)+(1 B)(dT +dr) 2
; (57)
d
whereasfortheseondline element(55)is givenby
ds 2
= Bdt 2
+B 1
dr 2
+r 2
(d 2
+sen 2
d 2
): (58)
Oneobservesthat,asbefore,themetris(57)and(58)
arerelatedbytheoordinatetransformation(54).
Let us now alulate how the energy-momentum
tensors hanges in the DGPPformalism. The metri
ofthebakground,whihisatspaetime,isgivenby
ds 2
= dT 2
+dr 2
+r 2
(d 2
+sen 2
d 2
): (59)
From equations (10)and (57)weobtain thefollowing
non-nullomponentsofthegravitationaleld:
h 00
=h 11
= h 01
= h 10
=B 1: (60)
Considering now the transformed metri (58), and
maintaining the line element (59) for the bakground
metri,thenewnonullpotentialsh 0
(x)willbegiven
by:
h 000
=1 B 1
;h 011
=B 1: (61)
Itmustbestressedthatthesamebakgroundat
met-ri (59) was used to determine the new potential, in
agreement with the form (43) where the bakground
metri is not hanged. In this form, the oordinate
transformation(54)orrespond,intheDGPPlanguage,
toagaugehangeinh
(x)givenby(60)toh 0
(x)
de-ned in (61)where, byhypothesis, h
(x)andh 0
(x)
arerelatedbythetransformation(45).
Completing the alulations by Virbhadra, we
now determine the energy density of the
Reissner-Nordstromsolution in theDGPP presriptiondened
in (22)by
t 00
(;h)= 1
[ 00
h
+
h 00
2 0
h 0
℄
Usingrstthepotential(60)andthebakground
met-ri(59)oneobtainforthispresription
t 00
(;h)= Q
2
8r 4
; (63)
in agreement with the results obtained by Virbhadra
using thepseudotensorial formalism. Forthe
gravita-tional eld(61)weget
t 00
(;h 0
)= Q
2
r 2
4m 2
r 2
+6mQ 2
r 3Q 4
8(r 2
2mr+Q 2
) 3
: (64)
Hene, just as the Landau/Lifshitz pseudotensor,
the DGPP energy-momentum tensor is strongly
af-feted by the transformation. Performing the same
alulations forthe Landau/Lifshitzenergy density in
DGPPform(seeeq. (31)),weget
L 00
(;h)=t 00
(;h)= Q
2
8r 4
(65)
forh
and
L 00
(;h)= Q
4
+Q 2
r(r+4m) 4m 2
r 2
8r 4
[Q 2
+r(r 2m)℄ 2
(66)
for h 0
. These expressions are idential to the
ex-pressions obtained by Virbhadra for the
Reissner-Nordstromenergy density in theLandau/Lifshitz
pre-sription,beforeandaftertheapliationofthe
oordi-natetransformation(54).
Thus,undertheationofthegaugetransformation
h
(x) ! h 0
(x), with h
(x) and h 0
(x) given by
( 60 ) and (61), the pseudotensor of Landau/Lifshitz
transforms itself exatly as if it have suered the
a-tionoftheoordinatetransformation(54)revealing,in
thatform,anequivalenebetweentheoordinate
trans-formations in the GR spaetime and this pure gauge
transformationdened ontheDGPPformalism.
Wewould liketoemphasizethatthehangein L 00
from equation (65)to equation (66)is not due to the
fatthatwearedealingwithaomponentofatensor.
Thishangeanbealulatedbyusingthe
transforma-tions(46)and(47)yielding
L 00
(
;h
)=t 00
(
;h
)= 2Q
4
+Q 2
r 2
4mQ 2
r
8r 2
(r 2
2mr+Q 2
) 2
; (67)
d
whihisdierentfrom equation(66).
V Conlusion
Inthislastsetionwesummarizetheonlusionsabout
thepseudotensorialandDGPPformalismsbasedonthe
resultswehaveobtainedintheprevioussetions.
First, from the relations obtained in setion II,
and from the denition (32) of the DGPP
energy-momentum tensor one onlude that, although
hav-ingexhibitedagravitationalenergy-momentumtensor,
whihisaneessaryonditiontoexpressaloal
quan-tityofenergyin aform independent oftheoordinate
system, the DGPP formalismdoes not solve, at last,
theproblem oftheloalizationofthegravitational
en-ergy beause the DGPP energy-momentum tensor is
notinvariantunderthepuregaugetransformation(45),
and it an be put to zero under a suitable hoie of
thiskindoftransformation. Infat,in theDGPP
for-malismboththepseudotensorsandtheDGPP
energy-momentumtensoranbedenedfromtensorial
super-potentials U
, whih are gauge dependent and dier
betweenthembyertainanti-symmetritensorialterm
sothat theonservationlawsarenotviolated.
Theexpressions below showthe tensorial
superpo-tentialsin thevariouspresriptionsaswellasthe
rela-tionsthat anbeestablishedbetweenthem:
U
= 1
2 [
h
+
h
h
h
℄
;
(DGPP); (68)
L
= ( )U
+ 1
2 [
~
h
~
h
~
h
~
h
℄
;
(Landau=Lifshitz); (69)
E
=
p
U
+
p
2 [K
A
K
A
(
h
h
)
;
℄(Einstein); (70)
M
= 2E
+Æ
E
Æ
E
Hene,thefat that theDGPPenergy-momentum
is areal tensoris notimportant, sinethe
pseudoten-sors an also be written as tensors in this formalism.
Whatisruialisthedependeneofallthesequantities
under thepure gaugetransformations (45),(48). F
ur-thermore, aordingtothe resultsofsetion IV,these
gauge transformations are equivalent to the general
oordinate transformations dened on the GR
spae-time. Hene, one onlude that in spite of their
in-trinsioneptualsdierenes,thepseudotensorialand
DGPPformalismshaveanalogouspropertiesand
there-foreonstituteessentiallyequivalentformstorepresent
the onservation laws of gravitational systems. This
does not mean that the DGPP formalism is useless.
Note that the manifold mappinggroup (MMG)is, at
thesametime, theovariantgroupandthesymmetry
groupofGR[10℄. Inthis sense,theroleof theDGPP
formalism is to split these two distint aspets of the
MMGbydening abakgroundmetrioverwhihthe
eldspropagate. TheovarianeroleoftheMMG
man-ifestsitselfinthetensorialnatureofeldsandequations
ofmotionintheDGPPformalism,whihistrivialsine
anytheoryanbeset ovariant. Thesymmetryroleof
theMMG,whihisoneofthemostimportantfeatures
ofGR,appearsin theDGPPformalismasthe
symme-try of the equations of motion under the pure gauge
transformation (45). Oneshould also note that ifthe
eldh
doesnotalwaysappeartogetherwith
(and
alsoK
withC
)in theirself-ouplingandoupling
withmatter,thetheoryloosesitsgaugesymmetry,and
there is no sense in nullifying the energy-momentum
tensor bymeans of agaugetransformation: the
grav-itational energy turns out to be loalizable. This is
beausethebakgroundmetri
is nowobservable,
not onlyits ombination g~
=~
+ ~
h
, and hene
the symmetry group of the theory will no longer be
the MMG but the symmetry group of
(e.g., the
Poinaregroupifitistheatmetri). Thetheoryan
notbeformulatedingeometrialterms,apreferred
ref-ereneframe is present (the oneadapted to
), and
the Equivalene Priniple is not satised (there is at
leastonetypeofpartilewhihfollowsthegeodesisof
,notofg
astheothers). Thisisthe onlywayto
haveanotionofloalizablegravitationalenergyinsome
alternativetheoryofgravitywhihneessarilywill not
satisfythe CovarianePriniple (inthe sense that the
MMGisnomorethesymmetrygroupofthetheory[10℄)
andtheEquivalenePriniple.Foranexampleofa
the-oryonstrutedontheselinesseeRef.[11℄.
Hene,theDGPPformalism,althoughequivalentto
GR, notonlylarify someimportantaspets ofit but
alsohelps usto envisagealternativeroutesto desribe
thegravitationaleld.
Referenes
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[2℄ A.Komar,Phys.Rev.113,934(1959).
[3℄ S.Deser,Self-InterationandGaugeInvariane,
Gen-eral Relativity and gravitation 1, 9 (1970), and
L.P.Grishhuk,A.N. Petrov andA.D.Popova,
Com-mun.Math.Phys.94,379(1984).
[4℄ J.D.BrownandJ.W.YorkJr.,Phys.Rev.D47,1407
(1993).
[5℄ J.W. Maluf, \Sparling Two-Forms, The Conformal
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Teleparallel Equivalent ofGeneral Relativity", GRG
30,413(1998).
[6℄ N.Pinto-NetoandI.Dami~aoSoares,Phis.Rev.D52,
5665(1995).
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