Theory of Gravitation
S.Ragusa and D. Bosquetti
Departamento deFsiaeInformatia,
InstitutodeFsiadeS~aoCarlos,
UniversidadedeS~aoPaulo,C.P.369,
13560-250 S~aoCarlos,SP,Brazil
e-mail: ragusaif.s.usp.br
Reeived29Marh,2000. Revisedversionreeivedon5June,2000
The eld equations of a proposed nonsymmetri theory of gravitation are derived when
eletro-magnetieldsare present,byadoptinga nonminimalouplingwhihensuresthevalidity ofthe
equivalenepriniple. Thestatiandspheriallysymmetrisolutionoftheeldofahargedpoint
partileisobtained.
I Introdution
In a previous work [1℄ a theory of gravitation based
on anonsymmetri metriwasformulated(pure
grav-itation with no assoiation of the antisymmetri part
of the metri to the eletromagneti eld strength).
The souresof themetrield are thematter
energy-momentum tensor T
and the matter fermioni
par-tile number urrent density S
. This is aonserved
urrent with partile number fermioni harge F =
R
p
gS 0
d 4
x,aonstantwhih measuresthe oupling
oftheurrenttothegeeometry. Thetheorywasshown
to be free of non-physial radiative negative-energy
modes evenwhen it is expanded about aRiemannian
bakground, being outside of the lass of ill-behaved
nonsymmetritheoriesanalyzedbyDamour,Deserand
MCarthy[2℄. Onlythesymmetripartofthe
onne-tionispresentintheeldequations,makingthetheory
asloseaspossibleto generalrelativity. A solutionof
theeldequationsforaspheriallyneutralpoint
parti-lehasbeenobtained[3℄,togetherwithitsimpliations
for the motionof light and testpartiles. The theory
is shown[3℄ to be onsistent with allfour general
rel-ativity solartests. Inthefollowingwe shallstudy the
eldequationswheneletromagnetieldsarepresent.
TheeletromagnetieldLagrangianthatweshalluse
istheonegivenbyMann,PalmerandMoat[4℄,whih
resues the validity of the weak equivalene priniple,
whih Will [5℄ showed to be violated if only a
mini-mal eletromagnetioupling(same formasingeneral
relativity) were adopted. This is beause with suh
aminimal ouplingthe gravitationalaelerationa of
aneletrially neutralbodyoftotalmassmomposed
of harged partiles turns out to depend on its
inter-naleletrostati energy E
e
. Spedially, the
aelara-tion is related to the one of gravity g by [6℄ a =g
(1+E
e =m
2
), where depends on the metri
oe-ients. Will thenshowsthat in thease ofa
nonsym-metritheorywithaminimaleletromagnetioupling
is not zero. With the oupling proposed by Mann
et al, aswritten in Eq. (1) below, turns out to be
null,ensuringthentheequivalenepriniple. Theeld
equationsontainonefreeparameter,Z. However,the
solution of the eld equations for a stati and
spher-iallysymmetri eld turns out to be independent of
Z.
InSe. IIweestablishtheexpressionofthe
gravita-tionallymodiedMaxwellinhomogeneousequationand
oftheeletromagnetienergy-momentum-stresstensor.
InSe. IIIwedisplaytheeld equationsofthetheory
whentheeletromagnetieldispresentandinSe. IV
wedeterminethesolutionoftheoupledeldequations
fortheaseofastatiandspheriallysymmetripoint
partile. WedrawouronlusionsinSe. V.
II The eletromagneti eld
equations
Mann et al [4℄ write the eletromagneti Lagrangian
densityL = p
L
em =
16 fg
g
[ZF
F
+(1 Z)(F
F
+F
F
) ℄; (1)
where,asusual,F
=A
; A
;
istheeletromagneti eldstrengthtensorand
f = p
g
p
g
s
: (2)
Thematrixg
istheinverseofthenonsymmetrigravitationaleldg
denedby
g
g
=g
g
=Æ
; (3)
g=detg
andg
s =detg
()
. Theminimalouplingwouldorrespondtohavef andZbothequaltoone. Equation
(2)analsobewrittenas
L
em =
1
16 fF
F
h
g ()
g ()
+(2Z 1)g [℄
g [℄
+(1 Z)g [℄
g [℄
i
; (4)
whereround(square)braketsstandforsymmetri(antisymmetri)part.
VaryingtheLagrangiandensitywithrespetwithrespettoA
oftheeldplustheinterationLagrangiandensity,
L
em =
p
gJ
A
,weget,usingEq. (4)
h
p
gf
g ()
g ()
+(2Z 1)g [℄
g [℄
+(1 Z)g [℄
g [℄
F
i
= 4
p
gJ
; (5)
whih is the inhomogeneous Maxwell equation in the presene of the nonsymmetri eld. Next we onsider the
variationwithrespettog
. Weget
ÆL
em =
1
2 p
gÆg
E
(6)
where
E
=
1
4
1
4
fg
2
f
g
g
g
(ZF
F
+(1 Z)(F
F
+F
F
))
fg
( ZF
F
+(1 Z)(F
F
+F
F
))
(7)
d
istheenergy-momentumtensoroftheeletromagneti
eld. This is atraeless tensor, g
E
=0, beause
wehavetherelation
g
f
g
=0: (8)
This an be proved by diret alulation from the
relations g 1
= "
Æ g
0
g 1
g 2
g 3Æ
and g 1
s =
"
Æ g
(0)
g (1)
g (2)
g (3Æ)
, or from the relations g =
" Æ
g
0 g
1 g
2 g
3Æ and g
s = "
Æ
g
(0) g
(1) g
(2) g
(3Æ)
together with g
( g
=g
)= g
(to obtainthis
last relationjust write itsrighthand sideasag
and
thenontratwithg
toobtaina= 1).
III The gravitational eld
equa-tions
Whentheeletromagneti eldispresent,the
gravita-tionaleld equationsofthetheory[1℄beome
U
() +g
()
=8(T 0
() +E
()
); (9)
(g
[℄;
+:p:)=8(E
[℄; +T
0
[℄;
+:p); (10)
where T 0
=T
(1=2)g
T with T = g
T
; .p.
standfortheylipermutationoftheindies; and
,istheosmologialonstantand
U
=
();
(); +
()
()
()
() ; (11)
involvingonlythesymmetripartoftheonnetion,is
theanalogueoftheRiemannianRiitensor,and
p
gg [℄
; =4
p
gS
: (12)
Sine at innity the eld of loalized matter tends to
thatofatspae-time,g [℄
aswellasg
[℄
mustsatisfy
theboundaryonditionofvanishingatinnity.
Wealsohavetherelation
() =
1
2 g
()
(s
; +s
; s
; )
+ 1
4
ln s
g
g ()
s
Æ
Æ
Æ
Æ
inverseofg ()
,denedbyg ()
s
=Æ
.
It is to be noted that T
, dened [1℄ by ÆL
m =
( p
g=2)Æg
T
where L
m
is the matter Lagrangian
density, is related to the ontravariant matter tensor
T
, dened by L
m = (
p
g)=2)Æg
T
as in
gen-eral relativity,by[1℄
T
=g
g
T
: (14)
ThisfollowsfromtherelationÆg
= Æg
g
g
,
re-sulting from the variation of Eq. (3). Therefore, T
will have a symmetri and an antisymetri part even
forasymmetriT
.
IV The eld of a harged point
partile
The statiand spheriallysymmetri metritensor in
spherialpolaroordinatesisoftheform
g
00
=(r);g
11
= (r);
g
22 = r
2
;g
33 = r
2
sin 2
;
g
01
= !(r)= g
10
; (15)
and allother omponentsequalto zero. Thenon-zero
elementsomponentsoftheinversematrixare
g 00
=
( !
2
) ;g
11
=
( !
2
) ;
g 22
= 1
r 2
;g 33
= 1
r 2
sin 2
;
g 01
= !
!
2 = g
10
: (16)
Outsidethesoure,invauum,thesolutionofEq. (12)
is [3℄,for=0,!r 2
( !
2
) 1
2
=F ,whereF isthe
onservednumberfermioniharge. Then,
! 2
=
F 2
F 2
+r 4
: (17)
As g =(! 2
)r 4
sin 2
and g
s
= r
4
sin 2
, we
getfrom Eq. (2),
f = s
1 !
2
=
s
1 g
01
g 10
g 00
g 11
: (18)
Theseondwayofwritingthevalueoffmakesiteasier
01
for=0,
r
r 2
E
p
=0; (19)
independentlyofZ. Uponintegrationweget
E= Q
r 2
p
; (20)
where the onstant of integration has been put equal
tothe hargeQof thepartile to reproduethe usual
Reissner-Nordstrom (RN) result when F = 0; whih
implies,from Eq. (27)below, =1. From Eqs. (7),
(18)and(20)weobtainthefollowingnon-zero
ompo-nentsofE
:
4E
00 =
1
2 Q
2
r 4
q
1 !
2
1+ !
2
; (21)
E
11 =
E
00
; (22)
4E
22 =
Q 2
2r 2
1
q
1 !
2
; (23)
E
33
= sin 2
E
22
; (24)
4E
01 =
Q 2
r 4
!
q
1 !
2
= 4E
10
: (25)
Wesee that theenergy-momentum tensor is
indepen-dent of Z and, therefore, the same will our for the
gravitationaleldequations.
FromnowonthealulationproeedsasinRef. [3℄.
FromEqs. (15)and(25)wesee thatEq. (10)is
iden-tially satised. From Eqs. (9) and (22) we obtain
U
00 +U
11
= 0; as in Ref. [3℄. It then follows the
samerelation
0
+
0
=
2
r F
2
F 2
+r 4
; (26)
derivedin[3℄,whihintegratesto
=
1+ F
2
r 4
1=2
: (27)
ReallingEq. (17)wethenobtain
!=
Fr
(F 2
+r 4
) 3=4
: (28)
From now one we shall negleted the small
ontribu-tionof theosmologialonstant in Eq. (9). TheU
( r
)
0
1+ r
2
0
+
0
+
4F 2
r( F 2
+r 4
) =
Q 2
r 4
F 2
+r 4
1=2
: (29)
UsingEq. (26)weget
( r
)
0
1+ 1
3F
2
( F 2
+r 4
) =
Q 2
r 4
F 2
+r 4
1=2
: (30)
Choosingtheonstantofintegrationin suhawaythat theRN resultisobtainedwhenF =0,weget
1
=
1+ F
2
r 4
1+ F
2
r 4
3=4
2m
r +
Q 2
r f(r)
; (31)
wheremisthemassofthehargedpartileand
f(r)= Z
dr
r(F 2
+r 4
) 1=4
; (32)
whihgoesto r 1
whenF vanishes. Then,fromEq. (27)weobtain
=
1+ F
2
r 4
1=2
1+ F
2
r 4
1=4
2m
r +
Q 2
r f(r)
: (33)
Equation(32)anbeputin losedform:
f(r)= 1
2jFj 1=2
2
6
4tan 1
1+ r
4
F 2
1=4
+ 1
2 ln
0
B
1+ r
4
F 2
1=4
1
1+ r
4
F 2
1=4
+1 1
C
A
2 3
7
5
; (34)
withtheterm =2togivetherightlimit, r 1
;whenF vanishes.
d
Theeletrieldis,from Eqs. (20)and(27),
E(r)= Q
r 2
1+ F
2
r 4
1=4
: (35)
Atlargedistanes,r>>jF j 1
2
;E(r)goesintotheRN
Coulomb eld but for small values of r, it behaves as
r 1
.
V Conlusions
Byadoptinganonminimalouplingthatensuresthe
va-lidity ofthe equivalenepriniple wehavederivedthe
eld equations of a proposed nonsymmetri theory of
gravitation[1℄wheneletromagnetieldsarepresent.
The nonminimal oupling ontains one free
parame-ter, Z. However, it is shown that for a stati
spher-ially symmetri eld the eletromagneti eld
equa-tions and the energy-momentum-stress tensor are
in-dependent of Z. Therefore, from this last fat, it
fol-equations. The solution of the eld equations for the
aseof hargedpointpartileis obtained. Apartfrom
destrongdeviationofthemetritensorfromtheusual
Reissner-Nordstrom(RN)solution,theeletrield
de-partsstronglyfromtheCoulombeldvalueobtainedin
theRN ase,to whihit approahesonlyat large
dis-tanes. Atsmalldistanes itbehavesasr 1
:
Referenes
[1℄ S.Ragusa,Phys.Rev.D56,864(1997).
[2℄ T. Damour, S. Deser and J. MCarthy, Phys.Rev. D
47,1541(1993).
[3℄ S.Ragusa,Gen.Relat.Gravit.31,1(1999).
[4℄ R.B.Mann,J.H.PalmerandJ.W.Moat,Phys.Rev.
Lett.62,2765(1989).
[5℄ C.M.Will,Phys.Rev.Lett.62,369(1989).
[6℄ A. P. Lightman and D. L. Lee, Phys. Rev. D 8, 364
