The Spaetime of a Dira Magneti Monopole
Marelo Costa de Lima 1
and Ivano Dami~aoSoares 2
1
Departamentode Fsia, UniversidadeFederaldoPara,
Belem,CEP66075-900,PA,Brazil
2
Laboratorio deCosmologiae FsiaExperimentalde AltasEnergias,
Centro BrasileirodePesquisas Fsias,
Rua Dr. XavierSigaud150,Ura,Riode Janeiro, CEP22290-180, RJ,Brazil
Reeivedon9November,2001
WeonstrutaspaetimewhoseonlysoureofurvatureisaDiramagnetimonopole,andwhose
geometryinheritsthestrutureoflinesofsingularitiesofthemonopoleeletromagnetipotentials.
Thespaetimehasthe topologyS 3
R , isstationaryand asymptotiallyatbutnot
asymptoti-allyMinkowskian,withitsatnullinnityhavingthetopologyofS 3
. Thesemildpathologies,as
aausalityandstringstruture,allowthespaetimeongurationtohaveagravitationalmagneti
mass, whih results proportionalto the harge of the monopole. This suggests that theDira
monopolemaybethesoureofmagnetimassingravitationalongurations,whihhasno
Newto-niananalogue. AlsohastheroleofaNUTparameterinthemetriofthespaetime,suggesting
thatthehargeofthemonopoleanprovideaphysialrealizationoftheNUTparameter.
I Introdution
Magneti monopoles areeld ongurationsthat arise
naturally in gaugeeld theories, even at the lassial
level. The idea of a magneti monopole was put
for-ward originally by Dira in a lassial paper [1℄, and
more reentlyremadeby Wu and Yang [2℄ in its
gen-eralizationfornon-Abeliangaugetheories. Two
funda-mental onepts areintrodued in Dira'soriginal
pa-per: (i)eletromagnetismisagauge-invariant
manifes-tationofthenon-integrabilityofthephaseofthewave
funtion; (ii) nodal lines of the wave funtion
(spae-like lines along whih the wave funtion vanishes) are
singularity lines of the eletromagneti potential and
the end points of nodal lines are pointsof singularity
of the eletromagneti eld. At this singularity there
is a magneti monopole, suh that the total magneti
ux througha losed surfaeaboutthis singularityis
dierentfrom zero. In other words, althoughthe pair
ofMaxwellequationsdF =0implythatthetotal
mag-netiuxthroughanylosedsurfaeatagiveninstant
oftimemustbezero,thisisnolongertrueifamagneti
monopoleispresentinsidethelosedsurfae. Equation
dF =0mustbeviolatedin somepointof thesurfae.
Furthermore,itmustbeviolatedin somepointofany
losedsurfaeenlosingthemonopolesothatitis
vio-latedalongalineofpoints-whihwedenotebystring
-extendingfromthemonopoletoinnityortoanother
poleofsamestrengthandoppositesign. Atthispoint,
monopolegeneratesgravitation,doesthegravitational
eldinheritsthisstrutureoflinesofsingularities? And
ifitdoes,what arethephysialonsequenes? An
an-swer to these questions, and whih is in part a
mo-tivation for this paper, is onnetedto the possibility
of the physial realization of a magneti gravitational
massgeneratedbythehargeoftheDiramonopole.
As a eld theory, General Relativity also presents
eldongurationswhihmaybeassoiatedwith
mag-neti mass monopoles and strings. Weird
ongura-tionsofthistypeareknowninGeneralRelativitysine
the disovery [3℄ and analysis [4℄ of the two
parame-ter NUT vauum solution. The remarkablefeature of
thissolutionemergesfromanewparameter-theNUT
parameter - and, as we will disuss, there are strong
suggestionsthat this parameterisassoiatedwith the
existene of magneti gravitational harges and wire
strutures in the spaetime. The NUT solution was
generalizedbyBrilltoinludeeletromagnetieldsas
souresof urvature [5℄ but the physial origin of the
NUT parameter remained obsure. We note that, in
theNUT solution, thevanishing of theNUT
parame-teryieldstheShwarzshildsolutionwhileitsvanishing
intheNUT-Brillsolutionresultsintheknown
Reissner-Nordstromspaetime.
AlthoughthephysialoriginoftheNUTparameter
is obsure, there has been suggestionin the literature
thatthetwoparameterNUTmetrianbeonsidered
of vauum Einstein's equations, two funtions an be
introdued - the mass potential
M
and the angular
momentumpotential
J
denedonthemanifoldof
or-bits of the timelike Killing vetor - that haraterize
uniquely the loal struture of these spaetimes.
Ex-panding thesepotentialsin multipole moments,it an
be shown that, for the NUT solution, the monopole
moment assoiated with
M
is the mass, while the
monopole moment assoiated to
J
is the NUT
pa-rameter[7℄. TheNUTparameteristheninterpreted
asamonopolesoureof angularmomentum,anddual
tothemassinthesensethatthereisaduality
transfor-mation[6,8℄in theplane (
M ;
J
) suh that,bythis
transformationwemaytaketheShwarzshildsolution
(m; = 0) into the pure NUT solution (m = 0;).
Later, RamaswamyandSen [9℄ wereableto introdue
adenition of dual (or magneti) mass for stationary
spaetimesandestablishedthepreiseonditionsfora
spaetime to admit dual mass. They showed that for
stationaryspaetimeswhihhavethenullboundaryat
innity with the topology of S 3
(namely, a S 1
bun-dleoverS 2
),preiseexpressionsanbeonstrutedat
thisboundarywhihharaterizemassanddualmass,
based on the Weyl urvature of the spaetime.
How-ever the spaetime must obviously be aausal. This
is exatly the ase of the NUT spaetime, where the
dual massis proportionalto the NUTparameter. We
notethatspaetimeswithasymptotitopologyRS 2
have dual mass vanishing identially, what exludes
Reissner-Nordstromasaandidate. Themassandthe
dualmassareassoiatedwiththeeletriandmagneti
partoftheWeyltensor,respetively,andthereforethe
dualmassisdenotedmagneti,inthesenseofthe
dual-ityofeletromagnetism(adualitywhihisalsopresent
onneted with the deomposition of the Weyl
urva-tureinitseletriandmagnetiparts).
Inorder tounderstand the nature ofthe NUT
pa-rameterand its assoiationwith thedual ormagneti
mass, we here onstrut a spaetime with topology
S 3
Rhavingamagnetimonopoleastheonlysoure
ofurvature. Asitturnsout,themagnetihargeofthe
monopolebehavesinthesolutionasaNUTparameter,
thusprovidingaphysialsourefortheNUTparameter
ordualmass. Howeverpathologies inthesolutionwill
bepresentbeausegeometrialstruturesofthe
spae-timemustabsorbthesingularitiesofthemonopoleeld,
although these pathologies arein asense neessaryto
apreiseharaterizationofdual(magneti)mass. We
organize the paper as follows. In Setion II we
on-strutandanalyseaspaetimemanifoldwithtopology
S 3
R(abasionditionfortheexisteneofdualmass)
andeletromagnetieldsonthismanifold,solutionsof
Einstein-Maxwellequations. Thisonstrutionismade
in detail, using Hamilton quaternions and dierential
forms, whih onstitutea naturalframe for this
anal-ysis. In Setions III and IV we examine relevant
ge-ometrialpropertiesof thespaetime whoseurvature
is generated by the magneti monopole. The
urva-turetensor and theeletriand magnetiparts of the
Weyl tensor are examined, as well as the asymptoti
strutureofthespaetime;themagnetiomponentof
theWeylurvatureisfundamentalinharaterizingthe
dual gravitationalmass, asdisussed already. Wealso
examinehowthestrutureofwiresingularitiesbothin
the eletromagneti elds and in the geometry of the
spaetime areintrinsiallymerged,anduse ismadeof
the violation of the PoinareLemma in topologially
non-trivialspaetime domainsin order to haraterize
thenatureofthesesingularities.Wealsointrodue
Mis-ner'stypeoordinatestoproperlyonnetthe
eletro-magnetipotentialtothegeometry1-forms,andrelate
gaugetransformationsofthepotentialsto
transforma-tionsintemporaloordinatesdenedonS 1
. InSetion
V onlusions aremade, with somedisussionson the
pathologies of the solutionas aausality, and possible
interestoftheseobjetsforquantumgravity.
II The Spaetime with Topology
S 3
R Generated by the Dira
Monopole
ThemethodsusedinpartofthisSetionarepartially
borrowedfromRef.[10℄andarepresentedherefor
om-pleteness. Calulationsarenotgivenindetailbutthey
anbehekedwithoutdiÆulty.
LetE
4
bethefourdimensionalEulideanspaewith
Cartesianoordinates(a 0
;a 1
;a 2
;a 3
). Wedenethe
hy-persurfaeS 3
ofE
4
astheset ofpointswhihsatisfy
(a 0
) 2
+(a 1
) 2
+(a 2
) 2
+(a 3
) 2
=1: (1)
Forevery a = (a 0
;a 1
;a 2
;a 3
) and b =(b 0
;b 1
;b 2
;b 3
) 2
S 3
wedenethemultipliationlaw
ab=(a 0
b 0
a 1
b 1
a 2
b 2
a 3
b 3
;a 0
b 1
+a 1
b 0
+a 2
b 3
a 3
b 2
;
a 0
b 2
+a 2
b 0
+a 3
b 1
a 1
b 3
;a 0
b 3
+a 3
b 0
+a 1
b 2
a 2
b 1
This isthemultipliationlawofHamilton quaternions
[11℄. Under(2)S 3
beomesagroup,atingonitselfby
left multipliation; namely, for a given v 2 S 3
, aleft
motionofS 3
intoitselfisexpressedas
a0=va (3)
andwehavethata02S 3
foralla2S 3
. S 3
issaid
sim-plytransitivesineforeaha2S 3
thereexistsonlyone
leftmotionv fromatoagivena0,namely,v=a 1
a0.
S 3
atingonitselfbyleftmultipliation(3)isaLie
groupwiththethreeindependentleft-invariantvetors
elds
e
(0)
(a)=( a 1 ;a 0 ;a 3 ; a 2 ) e (1)
(a)=( a 2 ; a 3 ;a 0 ;a 1 ) (4) e (2)
(a)=( a 3 ;a 2 ; a 1 ;a 0 )
Theyareobtainedbyanarbitraryleftmotionaofthe
threeindependentunitvetors(0;1;0;0),(0;0;1;0)and
(0;0;0;1) whih dene the tangent spaeof S 3
at the
identity (1;0;0;0). We remark that a left-invariant
vetor eld (a) on the Lie group S 3
is dened by
v(a)=(va),forallv;a2S 3
.
Wehave ananalogous piture forrightmotionsof
S 3
onitself,namely
a0=av (5)
with theorrespondingright-invariantvetorelds on
S 3 , d (0)
(a)=( a 1 ;a 0 ; a 3 ;a 2 ) d (1)
(a)=( a 2 ;a 3 ;a 0 ; a 1 ) (6) d (2)
(a)=( a 3 ; a 2 ;a 1 ;a 0 )
Bases(4) and (6), expressed ase
(i) (a) = 1 2 e (i) a and d (i) (a) = 1 2 d (i) a
, onstitute two distint
repre-sentationsofthealgebraoftheLieGroupS 3 , [e (i) ;e (j) ℄=C
k ij e (k ) ; (7) [d (i) ;d (j) ℄= C
k ij e (k ) ; (8) where C k ij = ijk
. They are related by an
anti-isomorphisminduedbytheinversemapofS 3
onitself,
andobviouslysatisfy
[e
(i) ;d
(j)
℄=0; i;j=0;1;2: (9)
Introduing onS 3
the Euler angles (t;;)by the
a 0 =os 2 os t+ 2 a 1 =os 2 sin t+ 2 (10) a 2 =sin 2 sin t 2 a 3 =sin 2 os t 2
with t 2 [0;2℄, 2 [0;℄ and 2 [0;2℄, the
left-invariantvetorelds e
(i) areexpressed e (0) = t e (1) = sint +otgost t ost sin (11) e (2) = ost otgsint t + sint sin
withorrespondingdualleft-invariant1-forms
! 0
= dt+osd
! 1
= sintd ostsind (12)
! 2
= ostd+sintsind;
satisfyingd! i = i jk ! j ^! k
. Forfuture referenewe
givetheright-invariantvetorelds in thisoordinate
system, d (0) = d (1) = sin otgos + os sin t (13) d (2) = os otgsin + sin sin t
Takingon theone dimensional manifoldR the
o-ordinater(0r<1),withvetorelde
(3) = r and 1-form! 3
=dr,thegroupS 3
Ranbeharaterized
by the left-invariant vetor elds (e
(0) ;e (1) ;e (2) ;e (3) ),
whihsatisfy(7)and
[e
(i) ;e
(3)
℄=0; i=0;1;2; (14)
andwhihareabasisfortheleft-invariantvetorelds
onS 3
R ;orrespondinglythe invariantdual1-forms
(! 0 ;! 1 ;! 2 ;! 3
)areabasisfortheleft-invariant1-forms
onS 3
R .ThemanifoldS 3
Risthesimplytransitive
overinggroupofthealgebra(7)and(14).
ThegeometryisobtainedbyintroduingonS 3
R
the left-invariant metri g = g
ab !
a
! b
, where g
ab = diag(4A 2 (r); B 2 (r); B 2
(r); 1) suh that the
se-tions r = onst: have the topology of S 3
; it assumes
theform
IntermsoftheLorentzianCartanframedened by
0
= 2A(r)(dt+osd)
1
= B(r)d (15)
2
= B(r)sind
3
= dr
we have g =
AB
A
B
, with
AB =
diag(1; 1; 1; 1) , A;B = 0;1;2;3. By
onstru-tion,thisgeometryhastheright-invariantvetorelds
(13) as Killing vetors whih at transitively on the
surfaes r = onst: Expressing the Maxwelltensor in
this frame asF =F
AB
A
^ B
, Maxwell'sequations
inthis geometryaregivenby
dF =0; dF
=j
= 1
3 j
A
ABCD
B
^ C
^ D
(16)
where denotesthedual operation. Forapure radial
magnetieldintheframe(15),weobtainthesolution
F
12 =
B 2
; j 0
=
2A
B 4
(17)
other omponents zero, where the onstant is the
hargeofthemagnetimonopole,beingproportionalto
the magneti ux asshown unambiguously in Setion
III.TheassoiatedMaxwellenergy-momentum tensor
resultsin
T
AB =
2
2B 4
diag(1;1;1; 1); (18)
and,forthiseldonguration,Einstein-Maxwell
equa-tions yield the solution A(r) = =2 and B(r) 2
=
r 2
+ 2
=4or,byaproperresaleofthetimevariable,
g=(dt+ osd) 2
dr 2
(r 2
+ 2
=4)(d 2
+sin 2
d 2
): (19)
d
Thissolutionisstationary,withKillingvetors(13)
plus=t. Wemention that theorbitsofthe timelike
Killing eld =t,whih aredened on S 3
, arelosed
timelikegeodesis,haraterizingtheaausalnatureof
thespaetime. Thetimeoordinatetin(19)hasperiod
2.
Alsothespaetimeisasymptotiallyatforr!1.
However it is not asymptotially Minkowskian, sine
r ! 1 has the topology of S 3
, by onstrution, and
is therefore aausal. These harateristis allow the
spaetimeto beendowedwithadual (magneti)
grav-itational mass, a point to be disussed in Setion IV.
Furthermore, in (19), has the role of the NUT
pa-rameter, suggesting the harge of the monopole as a
physial realizationfortheNUTparameter. Inthe
lo-al Cartanframe (15),thenon-nullomponentsof the
urvaturetensoraregivenby
R
0101
= R
0202 =
1
4 R
2121
= R
3131
= R
2323 =
2
(r 2
+ 2
=4) 2
R
0123
= R
0231 =
r
(r 2
+ 2
=4) 2
d
showinglearlythatthemagnetimonopoleistheonly
soureof urvatureof thespaetimeand onsequently
responsibleforalltheweirdpropertiesofthespaetime.
If=0,thespaetimeisat. Notethattheurvature
tensor is regular everywhere and onlythe metri fails
to be regular along the stringsof the eletromagneti
potentialsA's assoiatedwith themagneti eld (17),
asshowninthenextSetion. Inthissensewesaythat
thespaetimeexhibitsonlymildsingularities.
TheeletriandmagnetipartsoftheWeyl
E
AB
=diag(0; E; E;2E);
H
AB
=diag(0;H ;H ; 2H ) (20)
where
E =
2
4(r 2
+ 2
=4) 2
; H=
r
2(r 2
+ 2
=4) 2
: (21)
The fat thatE O( 1
r 4
) andH(
2r 3
), as r!1,
is ruial for the haraterization of the spaetime as
havingzeroeletrigravitationalmass(theusual
New-tonianmassasdened asymptotially)andanon-zero
magnetigravitationalmass,onnetedtothemagneti
partof theWeylurvature tensorand proportionalto
thehargeofthemonopole.
III The Magneti Monopole
String Struture in the
Grav-itational Field. Integrability
Conditions
Let us now examinehow thespaetime inherits the
stringstruture of themagnetimonopole. From(15)
and (17) the monopole Maxwell tensor F an be
ex-pressedasthe2-form
F= sind^d: (22)
SinedF =0,wemayassumefrom Poinare'sLemma
[13℄ that there exists a potential 1-form A suh that
F =dA, namely, that the2-formF isexat. However
thePoinare'sLemmaholdsfortopologiallynon
om-pliateddomains,thatisnottheaseforthedomainof
denition of (22). Tosee this,letus integratedF =0
inthe3-dimsetionofthespaetimer=onst:thatby
onstrutionis aS 3
. Taking intoaountthat S 3
isa
Hopfber bundle[14℄, withbase spae S 2
oordinated
by (;) and Hopf bershomeomorphi to S 1
gener-ated bytheKillingeld=t,this volumeintegralan
bereduedto
Z
S 3
dF = Z
S 2
F =4 (23)
whih is dierent of zero for 6= 0. Sine the
man-ifolds S 3
and S 2
have no boundary, this implies that
in some point of S 3
and of the setion S 2
the
equa-tionsdF =0andF =dAareviolated. Toharaterize
topologiallythedomain ofdenitionofF,letus
on-sider the manifold S 3
whih has no boundary. Now
if we extrat one point of S 3
we obtain a boundary
whihisS 2
. Tosee this,letusinterset S 3
byaplane
2
thispartiularsetion,theboundaryistheunitsphere
S 2
. We an gradually diminish the radius of S 2
tak-ing setions of type a 2
=a 2
0
=onst: with inreasing
a 2
0
suh that the radius beomes very small, namely,
(a 0
) 2
+(a 1
) 2
+(a 3
) 2
=1 (a 2
0 )
2
=. Thisorresponds
tothelimitingsituationofextratingapoint from S 3
,
obtainingaboundarywhih isS 2
withradius <<1.
NowS 2
isalsoamanifoldwithoutboundary,and
there-foretheseondequalityin(23)impliesthatF =dAis
violatedatsomepointofS 2
. Thereforethetopologyof
thedomainofregularityofF isS 2
withonepoint&
re-moved,withorrespondingS 3
withonepointremoved.
AndEq.(23)shouldatuallybeexpressed
Z
S 3
& dF =
Z
S 2
&
F =4; (24)
thenon-nullresultof(23)omingfromtheontribution
oftheboundaries(S 3
&)and(S 2
&). By
ontin-uously varying r, we have in the spaetime a line of
pointswherethisviolationours,dening whatis
de-notedbyastring struture. Indeed,assoiatedto the
eldFgivenin(22)wemayhavethe1-formpotentials
A
N
=(1 os)d (25)
A
S
= (1+os)d (26)
whih aresingularrespetivelyat = and=0. In
==2 where both arewelldened, theyare related
bythegaugetransformation
A
N A
S
=d =d(2): (27)
Regularityandsingle-valuednessonditionsonthewave
funtion of a harged partile in interation with this
monopole magneti eld impliesthat the phasefator
S=e(2)intheoverlapregionmustbeamultiple n
of2, whih isDira's quantization, namely, 2e=n
(inunitssuhthat~==1). Herenisaharateristi
ofthepartiularstring,orrespondingtothequantized
magnetiuxthroughS 2
(f. [1℄-[2℄). Wenotethatthe
boundaryinS 2
& appearingin theseondintegralof
(24)isnothomotopiequivalentfordistintn's.
Going now to the gravitational eld of the
spae-time, dened from (15), let us onsider the 1-form
0
= (dt+osd). Its exterior derivative yields
d 0
= ( sind ^ d). A disussion based on
Poinare'sLemma,analogoustotheoneleadingto(24),
showsus that the domain of denition of 0
presents
a line of singularities where the identity d 2
0
= 0 is
violated (this being equivalent to the violation of the
identityfortheRiemanntensorR A
[BCD℄
=0). Whatis
therelation between these lines of singularities in the
theeletromagnetipotentialsA(f. (25))areinherited
bythespaetimeandoinideexatlywiththelinesof
singularitieswhihleadto theviolationofd 2
0
=0.
To see this, let us introdue oordinates whih
al-lowustoonnetthe1-formsofthegeometrywiththe
eletromagneti potentials(25). Weonsider rstthe
oordinatet in(19). >From(15)wean express
dt= 0
otg
B
2
;
uptoaresaleofthetimevariablein 0
analogousas
done in (19). In the metri (19), wehaveg(dt;dt) =
1
2
(otg=B) 2
. Thereforethistemporal oordinate
makesthe metri singular both in = 0 and = .
Howeverweandenetwonewdistinttemporal
oor-dinatesinsuhawaythatthegeometryissingularina
singlevalueoftheangle,either=0or=,ofS 3
.
Thisanbedonebyanalyzingthetopologyinduedby
S 3
onthespaeoftheEulerangles(t;;)introdued
in (10),giving us a pitureof S 3
as theunion of two
solidtoriwiththeirsurfaesidentied.
ConsiderEqs.(10)deningpointsinS 3
andtheube
of Fig.1 in the Cartesian spae (t;;). Let us now
makein (10)thetransformations
t!t+; !+ (28)
with = 2n. These transformations do not alter
thepointsofS 3
,andthereforetheplanest
0 and
0 of
the ubeof Fig.1 must be identied respetivelywith
the planes t
0
+ and
0
+. In this way, for eah
=
0
= onst: we have atorus T 2
obtained by the
identiationsshownin Fig.2.
Figure 1. Cube in the spae (t;;) whose points are in
1 1 orrespondene with the points of S 3
by the
trans-formations(10). ThisidentiationmakesS 3
topologially
equivalenttotheunionoftwosolidtoriwiththeirsurfaes
identied.
Figure2. Eetoftheidentiation(28) ontheplane=0 oftheubeofFig.1,resultinginatorusT 2
.
d
Nowweshowthat theplanes =0and = are
redued,byidentiations,toone-dimensionaltori T 1
,
that is, irles S 1
. Let us examine the ase = 0:
from (10) we have that (a 0
= os(t+ )=2;a 1
=
sin(t+)=2;a 2
=0;a 3
=0). ThesepointsofS 3
remain
invariantbythetransformations
t!t+Æ; ! Æ; (29)
for any arbitrary onstant Æ. Thus all points of the
plane=0relatedby(29)mustbeidentied. Inother
words, all points of the straight lines t+ = ,
an arbitrary onstant, must be identied; eah of the
straightlinest+=
0
is reduedto apointthat an
beharaterizedbytheonstant
0
. Thisonstantmay
thenbeusedasaoordinatefortheirleS 1
obtained
bytheidentiations(28)-(29),anddenotedbyt
N ,
t
N
0
=t+: (30)
Analogously, for = we have from (10) that
(a 0
=0;a 1
=0;a 2
=sin(t )=2;a 3
NowthepointsofS 3
remaininvariantbythe
transfor-mation
t!t+Æ; !+Æ; (31)
with Æ an arbitrary onstant. The points of =
related by (31) must be identied. These pointsnow
belongto thestraightlinest =
0 ,
0
anarbitrary
onstant, eah of whih is redued to apoint
hara-terized by
0
andthis onstantan beused asa
oor-dinatefor theirleS 1
obtainedby theidentiations
(28)-(31),anddenotedbyt
S ,
t
S
0
=t : (32)
We an now use these oordinates t
N and t
S ,
to-gether with (;), to haraterize S 3
as the union of
two solid tori identied byits boundary, givenby the
torusT 2
orrespondingto=
0 (0<
0
<),as
illus-tratedin Fig.3. Wex
0
==2 suh that eah solid
torusorrespondstoahemisphere ofS 3
.
Figure3. RepresentationofS 3
astwosolidtoriidentiedbytheirsurfaes ==2,resulting fromtheidentiations(28),
(29) and(31). tN andtS areMisner'soordinatesdenedontheinnerirlesofeahsolidtorus.
d
Notethatbyaresaleoftasdonein (19),wehave
atually
t
N
=t+; t
S
=t : (33)
Thesetwonewtimevariableshaveperiod2,and
arerelatedby
t
N =t
S
+2: (34)
Atrivialalulationgives,fortheCartanmetri
1-form 0
(f. (15))
0
N
= dt
N A
N
(35)
0
S
= dt
S A
S
(36)
Thus, asmentionedbefore, in thetime oordinate t
N ,
the metriis regularat =0and hasaline of
singu-larities along =, oinidingwith the string of the
eletromagnetipotentialA
N
. Analogously,fort
S ,the
metriisregularat=andhasalineofsingularities
along =0, oinidingwith the stringof the
eletro-magnetipotentialA
S
. Thesetimeoordinatesdisplay
learly howthe metriinherits the stringstruture of
themagnetimonopolepotentials. Notethatthegauge
transformations(27)whihrelateA
N andA
S
also
on-net 0
and 0
. Ingeneralthegaugetransformations
of the eletromagneti potentials A's are absorbed by
transformationsinthetemporaloordinatest
N andt
S
dened on the S 1
manifolds at the enter of the two
solidtori ofFigure3. Inthisinstane, Dira's
quanti-zationofthehargeofthemonopoledenethewinding
numbernoftheperiodioordinatest
N andt
S
dened
onS 1
,wherenisassoiatedtothequantizedmagneti
ux throughatwosphereenlosing the monopole. In
aequivalent way, from thepoint of viewof ber
bun-dlestrutureofS 3
,thewindingnumbernharaterizes
theperiodiityoftheintegralurvesofthevetoreld
=tdenedonthebershomeomorphitoS 1
.
Coor-dinates(33)wereoriginallyintroduedbyMisner[4℄in
hisanalysisofthetopologyoftheNUTgeometry.
IV The S
3
Null Innity and the
Gravitational Magneti Mass
Let us disuss onisely the idea of dual (or
mag-neti)massasdenedbyRamaswamyandSen[9℄. They
showedthatforstationaryspaetimeswhihhaveanull
boundaryatinnitywithtopologyS 3
(namely, having
the struture of a prinipal S 1
ber bundle over S 2
)
preise expressions an be onstruted at this
Weylurvatureof thespaetime. Thisonditionis
es-sentialsineanullinnitywithtopologyS 2
R(with
R being the null generators), e.g., in asymptotially
Minkowskianspaetimes, thedualmassvanishes
iden-tially.
Theintrodutionofastrutureofnullinnitywith
theS 3
topologyinthespaetimeof(19)anbe
imple-mentedasfollows. Tostart,letusnotethatthe
spae-timehastopologyS 3
R ,withthetimelikeKilling
ve-toreld=tgeneratingtheS 1
HopfbersofS 3
. This
impliesthatthereexistslosedtimelikeurves(infat,
losed timelikegeodesis) and noglobal spaelike
sur-faes in thespaetime. Although thenotionof spatial
innityannotbeintrodued,thenotionofnullinnity
exists. Let us dene newoordinates x a
=(u;;;x)
byu=t r,x=r 1
,withuhavingalsotheperiodof
2. Inthisnewoordinatesystem,thegeometry(19)
takestheform
g=(du+ dx
x 2
+ osd)
2 ( 1 x 2 + 2 4 )(d 2 +sin 2 d 2 ) dx 2 x 4 (37) d
The Penrose onformalompatiation of the
spae-time at innity an be implemented by introduing a
onformal transformation on the geometry, with
on-formal fator = x and to extend the oordinate
hart to inlude x = 0, namely points at innity
r ! 1 for all u;;. The spaetime manifold is
now endowedwith athree-sphere boundaryat r!1
innity,that is, at = 0. The geometrial
stru-ture of thenull innity J is givenby the vetor eld
n a = g ab r b j =0
= (=u) a
(whih is the timelike
Killingtranslation=tonJ)andthedegenerate
met-rig= 2
g,evaluated at x =0;dx=0. Namely,the
asymptotiallyatnullinnityJ isanullthree-sphere
oordinatizedbyx a
=(u;;),a=0;1;2,withthe
de-generate metrig
ab
=diag(0; 1; sin 2
), and vetor
eldn a
=(1;0;0)=(
u )
a
.
Let us onsider a BMS innitesimal
supertransla-tion generated by n a
on J; assoiated with this
su-pertranslation the total energy-momentum p a
an be
dened in agivengaugeby p a
=K ab
l
b , withl
b a
ov-etoronJ suhthat n a
l
a
= 1andK ab
isassoiated
withtheasymptotieletripartoftheWeylurvature
tensor. Fromp a
theeletrimass(theusualNewtonian
mass) M J = 1 32 Z S 2 ab p dS ab (38)
is dened,at theinstantrepresentedbythe ross
se-tionS 2
ofJ. Todenethedualmass,thenatural
an-didate to be onsidered is p a = K ab l b , where K ab
istheasymptotimagnetipartoftheWeylurvature
tensor,resultinganalogously
N J = 1 32 Z S 2 ab p dS ab : (39)
In thease of stationary spaetime (more speially,
in the absene of gravitational waves) both the
ele-trigravitationalmass(38)andthedual(ormagneti)
gravitationalmass (39)are well dened, gauge
invari-antand independentofthehoie ofl
a
, andalso
inde-pendentofthehoieoftherosssetion. Theduality
aspet isonnetedtothedualitybetweentheeletri
and magneti parts of the Weyl urvature[12℄,
analo-gous to theduality in eletromagnetism,leadingus to
denote theabovedened quantity(39)equivalentlyas
dualmassormagnetigravitationalmass.
Forourspaetime(19),themassandthedualmass
arealulatedthroughthequantitiesdenedonJ,
K ab =lim !0 (4 3 E pq g pa g qb ) K ab =lim !0 (4 3 H pq g pa g qb );
respetively. Inthelimit,onlythefollowingterms
sur-vive, K ab =lim r!1 (4r 3 E 33 )n a n b ; K ab =lim r!1 (4r 3 H 33 )n a n b ; whereE 33 andH 33
aretheeletriandmagneti
om-ponents of the Weyl urvature of thespaetime given
in (20)-(21). Note that the geometry g alulated at
= 0 in the oordinate system x a
= (u;;;x) is
g j
=0
= 2dudx 2 osddx (d
2 +sin 2 d 2 ). Itresults K ab =0; K ab =4n a n b ; (40)
yielding from (38) and (39) the gravitational eletri
massM
J
=0and thedual(ormagneti)-gravitational
mass
Inthisway,thehargeoftheDiramagnetimonopole
is the soure for the magneti gravitational mass of
the spaetime. The fat that the spaetime inherits
the struture of singularities of the monopole eld is
fundamental to turn itinto aandidate withthe mild
pathologies (as aausality and asymptotiallyatnull
innitywiththetopologyofS 3
)neessaryforthe
def-initionofmagnetigravitationalmass,withtheharge
of the monopole providing a possible physial
realiza-tionoftheNUTparameter.
V Conlusions an Final
Com-ments
In this paper we basially onstrut a spaetime
whose only soure of urvature is a Dira magneti
monopole, and whose geometry inherits the struture
of singularities of the monopole eletromagneti
po-tentials. The spaetime has topology S 3
R , is
sta-tionaryandasymptotiallyatbutnotasymptotially
Minkowskian, by onstrution. The urvature tensor
is regular everywhere and only the metri fails to be
regular along the stringsof themonopole
eletromag-neti potentials. The metridepends on one
parame-ter , the harge of the monopole; has the role of
the NUT parametersuggesting that thehargeof the
monopoleonstitutes apossiblephysial realizationof
theNUT parameter. Thetimelikelines aredened on
S 3
implying the existeneof losed timelike geodesis
and no globalspaelikesurfaes in the spaetime.
Al-though the notion of spatial innity annot be
intro-dued, the notion of null innity J exists, with the
topology ofS 3
. This struture plusthe mild
patholo-gies(asstringsandaausality)turnthespaetimeintoa
andidateforagravitationalongurationhaving
mag-neti gravitationalmass, assoiated with the non-null
magneti part of the Weyl urvature present due to
the monopole. In the framework introdued by
Ra-maswamyandSen[9℄,wherethespaetimeisstationary
(no gravitational waves are present), a preise
deni-tionof magnetigravitationalmassan beestablished
dualtothedenitionofeletrigravitationalmass(the
usualNewtonianmassasdenedasymptotially),both
denitions basedonBMSsupertranslationsdened on
thenullinnitywithtopologyS 3
(aS 1
Hopfber
bun-dle over S 2
). It turns out that the spaetime of the
Dira magneti monopole is a ongurationwith zero
eletrigravitationalmass,andmagneti gravitational
massproportionaltothehargeofthemonopole.
AshtekarandSen[15℄extendedtheframeworkof[9℄
to ases in whih gravitationalwavesmay be present
(more speially, in whih the Bondi news funtion
does not vanish). They showed that, while the
ele-tri gravitational mass an be radiated away in form
of gravitational waves (with balane equations given
massannotberadiatedaway. Inthisaspet,the
mag-netigravitationalmassofourmodel(whihis
propor-tional to the magneti harge of the monopole) is in
perfetaordanewithAbelianMaxwelltheorywhere
botheletriandmagnetihargesareabsolutely
on-served,andopposedtonon-AbelianYang-Millstheories
wherebothhargesmayberadiatedaway. Weremark
oneagainthat inourmodeltheeletrigravitational
massiszero.
A possible extension of our paper to be
exam-inedwouldbeto unoverthepossiblespaetime
stru-tureshavingmagnetimonopoleongurationsin
non-Abeliangaugetheories[17℄ astheonlysoureof urv
a-tureandhowtheyouldpossiblybethesoureof
ele-tri andmagneti gravitational massesof these
spae-timeongurations.
Wemustommentthatthespaetimestruture
dis-ussedmay not be of interest as amodel for
gravita-tionalongurationsofmarosopiobjets,beauseof
itspathologies asausalityviolation. Howeverthe
in-terestinthese ongurationsmayomefrom quantum
gravitywhere dualmassvauumsolutionsare
station-ary points of the gravitational ation funtional and
mayplayanimportantroleinEulideanquantum
grav-ity.
Finallywementionthatatentativeforestablishing
theequationofmotionofpointpartileswithmagneti
gravitationalmasswereproposedin [18℄,basedonthe
Jaobi equation for geodesi deviation where, by the
sameproessofdualityanalogy,theeletripartofthe
Weyltensorissubstituted bytheitsmagnetipart.
Aknowledgments
One of us (IDS) would liketo thank Dr. Eugenio
RamosB.deMelloforsomeusefulommentsonthe
pa-per,andalsothankCNPqofBrazilforpartialnanial
support.
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