Bounds on the Conial Defet
Parameter from Solar System Tests
Wilson H. C. Freire 1;2
, V.B. Bezerra 2
, and J.A. S. Lima 3
1
UniversidadeRegionaldoCariri,Departamento deMatematia
63100-000 Crato,Ce,Brazil
2
UniversidadeFederaldaParaba,
Departamentode Fsia, CaixaPostal5008,
58059-970J.Pessoa,Pb,Brazil
3
UniversidadeFederaldoRioGrandedoNorte,
Departamentode Fsia, CaixaPostal1641,
59072-970Natal,RN,Brazil
Reeived25Otober,1999
WeanalysetheplanetaryperihelionpreessionandthedeetionoflightintheShwarzshildeld
modied by a onial defet. By using observational results from solar system tests of general
relativity, we obtain limits onthe onial defet parameter. For the physial ase of a osmi
stringthequotedlimitsareoneandthreeordersweakerthantheosmologialbounds.
I Introdution
The general theory of relativity (GTR) is believed to
be the best theoretial framework presently available
to desribethe gravitationalinteration. At the time
ofitsformulation,themajorahievementsofthetheory
wasthedeetionofstarlightandtheperiheliumshift
ofMeruryplanetinthesuneld. Theremaining
las-sialtest(gravitationalredshift),isreallypreditedby
awiderangeoftheoriessineitisadiretonsequene
oftheequivalenepriniple.
Thebendingoflightandtheperiheliumshiftagree
withtheEinsteinvalueswithanauranyofone
per-ent (an overview is given by Will[?℄). Both eets
are usually analysed assuming a negligible
ontribu-tion from solaroblatness or whatever eet whih
de-parts the metri from exat spherial symmetry.
A-tually, with some few exeptions, all the alulations
are done in the ontext of the spherially symmetri
Shwarzshildmetri. Asapreliminarypointof
prini-ple,iftheSundeviateslightlyfromexatspherial
sym-metry, either due to an appreiable solar quadrupole
moment[?℄ orevensomeunexpetedtopologial
prop-erty of the gravitational eld (matter distribution), a
moreompletetreatmentofsuheets requirea
gen-eralizationof thestandardShwarzshildline element.
Intheaseofanonzerosmallquadrupolemoment,the
main physial onsequeneshave been disussed with
some detail in the literature either in the Newtonian
approximationorinrelativistiframework(see[?℄ and
Refs. therein). However, as far as we know, an
ex-tendedtreatmentisnotonlymatterofaademi
inter-est, beause suh apossibilitybasially remains asan
openquestion[?℄.
Inthepresentworkwearemoreinterestedon
pos-sible distortionsontheSuneld provokedby
topolog-ialeets. Someauthors havesuggestedthatmostof
simpleexatsolutionsof Einstein'sequationsan
eas-ily be generalizedto inlude a onial defet[?℄. Suh
spaetimes are geometriallyonstruted by removing
awedge,thatis,byrequiringthat theazimuthalangle
aroundtheaxisruns overtherange 0<<2b. For
verysmalleets,thebparameteritselfmaybewritten
asb=1 ,whereisasmalldimensionlessparameter
quantifying theonialdefet. In partiular,for =0
theShwarzshildlineelementisreoveredwhereasfor
a onial defetgenerated by aosmi string one has
= 4G, where is the mass perunit length of the
string[?, ?℄. We believe that it is worthwhile to
on-siderwhatlimitssolarsystemobservationsouldplae
onasuhmetrimodiedbytheonialdefetandto
investigate ifthese observations an support the
exis-teneofaosmistringthreading theSun.
TheShwarzshildspaetime endowedwith a
on-ial defetwasonsidered previously[?℄ and some
as-petswerestudied. Thelineelementorrespondingto
ds 2 =(1 2M r )dt 2 (1 2M r ) 1 dr 2 r 2 d 2 b 2 r 2 sin 2 d 2 ; (1) d
where M = Gm is the geometri mass of the entral
body.
In this paper, studying the orbits of masslessand
massivepartilesin theabovegeometry,wederivetwo
limits on the value of the onial defet parameter
from perihelionshift of theplanetary motionand
de-etionoflight. Parenthetially,wereallthateetsof
nontrivialtopologiesareusuallydisussedinthe
osmo-logial framework, as forinstane, in onnetion with
the oneptof small universes[?℄. The physial
onse-quenes of osmi stringsregarding toits impliations
on the struture formation problem has also been
ex-tensivelyinvestigated[?℄-[?℄. Morereently,therst
2-years of COBE/DMRdata measuring thequadrupole
anisotropies of the 3K reli radiation have been used
to plae limits on the possible topologies of the
uni-verse[?℄. In priniple, through a onvenient
general-ization of the Friedmann-Robertson-Walker metri, a
similarapproahwouldbeappliedforonstrainingthe
parameter. However,itis important toderivelimits
onthis keyparameterfrom dierentkindsof
phenom-ena,mainlyinadomainwheretheobservationsenvolve
lenghtsalesdiversefromthat onesusuallyonsidered
in theosmologialontext.
Let us now onsider the motion of a test partile
in the Shwarzshild spaetime with a onial defet.
Its trajetoryaneasily beestablishedthroughthe
rel-ativistiHamilton-Jaobiequation
g S x S x =m 2 (2)
where m is the partile mass and S is the generating
funtion.
Following standard lines, we assume an S in the
form
S= Et+R (r)+()+L
(b);z
'; (3)
where E is the energyand L
(b);z
is the azimuthal
an-gularmomentum ofthepartileasmeasuredatspatial
innityinthepreseneoftheonialdefet.
As one may hek, in the spaetime given by (1)
the partial Hamilton-Jaobiequation is separableand
redues totwoordinarydierentialequations
d d 2 =K L 2 (b);z sin 2 ; (4) dR dr 2 = r 2 2 E 2 (m 2 r 2 +K) r 4 ; (5)
whereKisCarter'sfourthonstantofmotion[?℄,whih
is givenby K =L 2
(b)
, with L
(b)
beingthe total
angu-larmomentumof thepartile in suh bakgroundand
=r 2
2Mr. ThesquareofL
(b) is L 2 (b) =g r 2 p 2 +g '' r 2 p 2 ' =p 2 + p 2 ' b 2 sin 2 ; wherep andp '
aretheomponentsofthegeneralized
momentum. In partiular, for a motion in equatorial
plane(=
2 ;p
=0) theaboveequationreads
L 2 (b) = p 2 ' b 2 = L 2 b 2 ;
and sine 0 <b 1, wesee that L 2 (b) L 2 (b=1) as it shouldbe.
The geodesi equations are easily determined by
ombining the above equations with the denition of
thefour-momentum p =m dx ds = g S x ; (6)
wheresisaparameteralongthetrajetory. Onends
m dt ds =E 1 2M r 1 ; (7) m 2 dr ds 2 =E 2 m 2 + L 2 (b) r 2 ! r 2 ; (8) m 2 d ds 2 = L 2 (b) r 4 L 2 (b);z r 4 sin 2 ; (9) m d' ds = L (b);z br 2 sin 2 : (10)
Notethatthepreseneoftheonialdefet
parame-terthroughb()istheuniquedierenebetweentheset
(7)-(10)and theorresponding equations in the
stan-dardShwarzshildeld,whiharereadilyreoveredin
theb!1limit.
Toanalysetheinueneoftheonialdefetonthe
deetionoflightraysandtheperihelionshiftwe
on-sidertheorbitequation
dr d' 2 = b 2 r 4 L 2 (b) " E 2 1+ L 2 (b) r 2 ! (1 2M=r) # ; (11)
whih is aonsequeneof (8) and (10),with
L
(b) and
rameterfromtheperihelionshift. Itisonvenienteto
makeahangeofvariablesu= 1
r
intermsofwhihEq.
(11)anbereastinthefollowingform
du d' 2 = b E L (b) 2 b 2
( 1 2Mu) u 2 + 1 L 2 (b) ! : (12)
Thestandardproedureinsearhingforgeneral
rel-ativisti eets in the perihelion shift is to assume a
nearly irularorbitandonsider theu 3
term
appear-ingin(?? ).
For non-irular orbits and massive partiles,
Eq.(??) turnsinto
d 2 u d' 2 +b 2 u= M L 2 +3Mb 2 u 2 : (13)
The rst term on the right-hand side of Eq.(??)
leadstoNewtonianorbits. Ifweonsideronlythisterm
niansolutionwhih isgivenby
u 0 = 1 r = M b 2 L 2
[1+eos(b(' '
0
))℄; (14)
where'
0
andeareonstantsofintegration,ebeingthe
eentriityoftheorbit.
Now,letus alulatetherstorretionby
pertur-bation expansion. Writtingu = u 0 +u 1
, where u
0 is
given by Eq.(??) and onsidering orbitsof small
ex-entriityweobtainthefollowingequationforu
1 (') d 2 u 1 d' 2 +b 2 u 1 = 6M 3 b 2 L 4 eos[b(' ' 0 )℄; (15)
whosesolutionisgivenby
u 1 = 3M 3 b 4 L 4 eb'sin[b(' ' 0 )℄: (16)
Inludingthisorretion,wehavethat
u = M b 2 L 2
1+e os[b(' ' 0 )℄+ 3M 2 b 2 L 2 b'sin[b(' ' 0 )℄ : (17) d Now,taking 4' 0 =3 M bL 2 '; (18)
thesolutionu(')maybewrittenas
u= 1 r = M b 2 L 2
f1+eos[b(' '
0 4'
0
)g: (19)
From Eq.(??) we onludethat the required shift
perrevolutionis
4' 0 = 6 b 3 M L 2 +2(b 1 1): (20) where2(b 1 1)
=2istheontributionofthe
oni-aldefetfortheperihelionshiftintheabseneofloal
gravitationaleet.
Now,expandingsuhexpressiontorstorderinthe
onialdefetparameteritfollowsthat
4' 0 4' S 2; (21) where 4' S
is the standard deviation in the
Shwarzshildeld. FortheplanetMeruryitisknown
that 4'
0
agree with 4'
S
= 510 7
to betterthan
half perent. ExpandingEq.(??)to rst order in we
onludethattheonialdefetparameterisbounded
by
<10 9
: (22)
If we assume that this onial defet orresponds
to aosmi string,thenits linearmassdensity willbe
boundedby
<10 19
g=m; (23)
whih is three orders weakerthan thevalue predited
forgranduniedstrings.
Now,letusomputethetotalhange'='
+1
'
1
,intheangularoordinate'ofalightray
inom-ingfrominnitywithimpatparameterlandesaping
toinnity. From(11)weonludethat 'isgivenby
'=2 L (b) b Z 1 rmin dr (E 2 r 4 L 2 (b) ) 1=2 : (24)
In terms of the variable u previously dened,
Eq.(??)beomes '= 2 b Z r 1 min 0 du (l 2 u 2
+2Mu 3
) 1=2
; (25)
wherel= L
(b)
E
. Naturally,theturningpointintheorbit
ofthelightrayisalsodened by
R 3 0 l 2 (R 0
2M)=0 (26)
Thus, ifM =0(onlytheonialdefetis present)
onehasl=R
whihisawellknownresult[?,?℄. Therefore,thereisa
deetion duetothedefetevenwhen M=0.
When M 6= 0, ' will not be equal to
b
, as
ex-peted. As usual,torstorderin M,theontribution
to light deetion due exlusively to the mass M is
readilyobtainedby inserting(?? )into (?? )and
dier-entiating the resultingequation with respet to M[?℄.
Onends
(')
M j
M=0 =
2
b Z
1=R
0
0
(R 3
0 u
3
)du
(R 2
0
2MR 3
0 u
2
+2Mu 3
) 3=2
j
M=0
: (28)
d
InthelimitM!0andl!R
0
,Eq. (?? )beomes
(')
M j
M=0 =
2
b Z
1=l
0 (l
3
u 3
)
(l 2
u 2
) 3=2
du; (29)
andresultsin
(')
M j
M=0 =
4
lb
: (30)
Thus,torstorderinM,thenetdeetionoflight
( inluding the ontributions due to the mass and to
theonialdefet)is givenby
Æ'=' M
(')
M j
M=0 +(b
1
1)= 4M
lb +(b
1
1): (31)
d
Naturally, as remarked before, we are looking for
small deviations from Shwarzshild spaetime.
Ex-panding (?? )to rstorderinweobtain
Æ' Æ'
S
; (32)
where Æ'
S
= 1:75 00
is the defetion of light with no
onial defet ( =0). Therefore, the defetauses a
deetion of size 2, in addition to the gravitational
deetion.
Observations from photographplates in totalsolar
elipses points systematiallyto light-deetion in the
neighbourhood of the Sun. However, the
experimen-tal resultsare largelysatteredand presentnotieable
disrepanies[?℄. Measurementsofdeetionusingvery
longbaselineinterferometry(VLBI)forradiowaves
em-mited byquasars(QSO)giveauraiesof about10 3
.
Therefore, using this fat and using the expansionof
(?? ), we obtain that the onial defet parameter is
bounded, inthisase,by
<10 7
: (33)
Ifweassumethatthisonialdefetorrespondsto
aosmi string,this limitisoftheorderof magnitude
oftheorrespondingboundspreditedbygrandunied
strings. Thislimitimpliesthatthelinearmassdensity
of thestringissuhthat
<10 21
g=m; (34)
whih is approximately of the same order of the
on-strainimposed byobservations and bythe string
se-narioforgalaxyformationintheontextofgrand
uni-edtheories.
Thelimitthatomesfromtheperihelionshift
anal-ysisis less preise than that one established from the
deetionoflightandthreeordersofmagnitudelarger
thantheosmologialbounds. Naturally, ifsome
por-tionoftheperihelionshiftis duetoothers eets,like
thequadrupolemomentumoftheSun,thisupperlimit
wouldbehigherbytheorrespondingamount.
Dierently from the limits on the onial
parame-terpreditedusingtheobservationalresultsfromthe
solar systemtests of generalrelativity onerning the
perihelion shift of Merury, in the ase of deetion
oflightthe limitsplaedonthis parameter isin good
agreementwiththepreditedboundsforgrandunied
strings. Certainly, this last resultdoesnot meanthat
thesolarsystemobservationsansupporttheexistene
ofreallyweirdmatterlikeaosmistringwhihthreads
theSun. Infat, we arenot addressingourquestions
tothispoint,buttotheuseoftheobservationalresults
fromthesolarsysteminordertogetboundsonthe
on-ialdefetparameterwhihmodiestheShwarzshild
metri. This approah provides, at least, an
indepen-dentalternativewaytogetlimitsontheparameterthat
deformstheShwarzshildsolutionusingdiret
exper-imentaltestinsteadofosmologialonsiderations.
NotethatfromEqs.(??)and(?? ),wemayonlude
Shwarzshildeldwithaonialdefet(whih anbe
assoiatedwithaosmistring)doesnotdependon(
or ),suheets are uniquelydueto topologial
fea-turesorequivalently,due tothelakofspherial
sym-metryproduedbytheonialdefetorbytheosmi
string.
Finally, we stress that our results are ompletely
generalinthesensethatanydeviationoftheb
param-eterfromunityanbeassoiatedwithothereetsand
notneessarilytothepreseneofaosmistring.
Aknowledgments
ItisapleasuretothankRobertBrandenbergerand
A.R.Plastinoforaritialreadingof themanusript.
This work is partially supported by the the projet
Pronex/FINEP(No. 41.96.0908.00)andConselho
Na-ional de Desenvolvimento Cient
io e Tenol
ogio-CNPq(BrazilianResearhAgeny).
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