Table Of ContentFate of Yang-Mills black hole in early Universe
Łukasz Nakonieczny 1and Marek Rogatko 2
InstituteofPhysics
MariaCurie-SkłodowskaUniversity
20-031Lublin,pl.MariiCurie-Skłodowskiej1,Poland
3 Abstract. AccordingtotheBigBangTheoryaswegobackintimetheUniversebecomesprogressivelyhotteranddenser.
1 ThisleadsustobelievethattheearlyUniversewasfilledwithhotplasmaofelementaryparticles.Amongmanyquestions
0 concerning thisphase of historyof the Universethereare questions of existence andfate of magneticmonopoles and pri-
2 mordialblackholes.StaticsolutionofEinstein-Yang-Millssystemmaybeusedasatoymodelforsuchablackhole.Using
methodsoffieldtheorywewillshowthatitsexistenceandregularitydependcruciallyonthepresenceoffermionsaroundit.
n
a Keywords: primordialblackholes,fermions
J PACS: 04.50.+h
7
1 INTRODUCTION Dirac field. Using dimensionalreductionand bosoniza-
tionwewillanalyzetheinfluenceoffermionsonthecon-
]
c Recentobservationaldatastronglysuggesttheexistence sideredPBH.
q of dark matter (DM) [1]. One of a few remaining DM
-
r candidates compatible with the Standard Model (SM)
g areprimordialblackholes(PBH)[2].Theseblackholes DIRACEQUATIONIN MAGNETIC
[
were formed in the early Universe during gravitational YANG-MILLSBLACK HOLE
2 collapseofdensityfluctuation.TheexistenceofthePBH BACKGROUND
v was first proposed in [3, 4, 5] and their formation dur-
3 inginflationorphasetransitionintheearlyUniversewas
A general form of a spherically symmetric and time
3
discussedin[6,7,8].Anothertypeofobjectsthatcould
4 independentmetricis
be formedduringphase transition in early Universe are
2
1. tmopoonloopgoilceasl[9d]e.fIefctthsesliekoebcjeocstmsriecalsltyrifnogrsmeadnddumrinaggneevtoic- ds2=−A2(r)dt2+B−2(r)dr2+r2dW 2, (1)
0
lution of the Universe we should consider the possibil- where r is an usualradialcoordinateand dW 2 =dq 2+
3
1 ity of interactions among them. Within the framework sin2(q )df 2 is a standard metric on two dimensional
: offieldtheoryasimplemodelofthesystemthatallows sphere.Thelocationofaneventhorizonofablackhole
v
the existence of magnetic monopoles may be given by is given by the largest positive root of A2(r), which is
i
X Yang-Mills(YM)theorywithSU(2)gaugegroup.Parti- denotedasr .Actuallyforanalyzingfermionsequations
H
r clelikesolutionsofEinstein-Yang-Mills(EYM)system ofmotionitismoreconvenienttointroducetheso-called
a describingthemagneticmonopolewerefirstconstructed tortoise coordinate r . This coordinate is given by the
in [10]. On the otherhand,solutionsdescribinga black followingrelation: ∗
hole with magnetic monopole in this context were pre-
sentedin[11].Therewasalsoshownin[12]thatfinite- B(r)
dr = dr. (2)
nessoftheblackholemassdemandsabsenceofanelec- ∗ A(r)
tricpartofYMfield.
Ourmetricexpressedincoordinates(t,r ,q ,f )is
Another important component of the early Universe
∗
whose interaction with PBH we should consider is ds2=A2( dt2+dr2)+C2dW 2, (3)
fermionic matter. To describe these interactions we use − ∗
methodsoffieldtheory.AsatoymodelofPBHwetake where we introduce a function C to keep in mind that
the aforementioned spherically symmetric YM black now we have r =r(r ). From now on, to simplify our
hole. As a representation of fermionic matter we use notation we will omit∗explicit writing of arguments of
functions. During our computations we will use tetrad
formalism. The local ortogonal tetrad is defined by the
1 [email protected] relation gmn =eim emjh ij, where eim is an element of the
2 [email protected] tetrad, gmn and h ij are metric tensorsin curvedand flat
spacetimesrespectively.We use the followingsignature MASSLESS FERMIONS
convention:h = 1,h =h =h =+1.
00 11 22 33
−
The generalformof static and sphericallysymmetric Inthemasslescaseweseefrom(6)thatchiralitiesdecou-
Yang-Millspotentialmaybewrittenas ple and effective two dimensional theory contains two
fermionic fields connected to y and y . The effective
L R
Hm =eim [ainktk+1−2lwC(r)eijknjt k], (4) Lagrangianforthefieldy Ris
L = iG¯ g˜a(cid:209) G l B G¯ g˜ag˜3G +
whereai=(a0,a1)andwrepresentelectricandmagnetic FR−2d − R a R− a R R
partsofYMfield,niisaunitvectornormaltothesphere, +VG¯ g˜ G VG¯ g˜ G , (9)
R L R R R R
l is a coupling constant, e stands for a totally anti- −
ijk
symmetricLevi-Civitasymbol,andt k aregeneratorsof where V = Cw. In this formula GR is two dimensional
spinor field connected to y by rescaling and multipli-
SU(2)gaugegrouprepresentedbyPaulimatrices.Asthe R
cationbyappropriates matrices[15].Twodimensional
representationforfourdimensionalgammamatriceswe
flatspacetime gammamatricesaregivenbythe follow-
chooseWeylbasis
ingrelations:
0 I 0 s i
g 0=(cid:20)I 0(cid:21), g i=(cid:20) s i 0(cid:21), (5) g˜i,g˜j =2h ij, h 00= 1= h 11,
− { } − −
g˜0= is 3, g˜1= s 2,
whereIisaunittwodimensionalmatrixands iarethree − −
1
Paulimatrices.Gammamatricesincurvedspacetimeare g˜3=g˜0g˜1=s 1, g˜ = (I g˜3). (10)
givenbythestandardrelation:g m =em g k.Representing L/R 2 ±
k
y
aspinory asy = y L wemaywritetheDiracequa- TheLagrangianforthefieldconnectedtoy Ldiffersfrom
(cid:18) R(cid:19) (9)onlybyasignofatermproportionaltoB .
a
tionas
Analyzingequationsofmotionforfermionsincurved
i/D+y my =0, spacetimeisa highlynontrivialanddifficulttask.How-
R L
− ever, in case at hand since we can express our prob-
i/D−y L my R=0, (6) lem in form of effective two dimensional theory we
−
may use bosonizationtechnique [16]. This technique is
whereoperators/D±aregivenby[13,14] well defined for flat spacetime two dimensional prob-
w 1 lemsandalsoforasymptoticallyflatspacetimes[17,18].
/D±=A−1¶ t−il {[s 0a0±s 1a1]n¯·t¯± 2l−C n¯·s¯ ×t¯}+ Themeritofbosonizationisthatweexpressafermionic
sectorofourtheoryintermsofcomplexscalarfield.Ba-
1
s¯ n¯A−1¶ r s¯ n¯ A−1C−1¶ r C+ A−1A−1¶ r A + sicbosonizationformulasinourcasearegivenbelow
± · ∗± · { ∗ 2 ∗ }
±C−1DS2. (7) ja=y¯g˜ay = 1 e ab(cid:209) f , (11)
√p b
In the above formula a bar over a quantity represents
1
threedimensionalvector,adot–scalarmultiplication, j3a=y¯g˜ag 3y = (cid:209) af , (12)
–vectormultiplication,andD isDiracoperatoronS×2. √p
S2
We are mainly interested in fermions in s-wave sector, yg¯ Ly =be2i√pf , yg¯ Ry =be−2i√pf . (13)
whichisdescribedbythelowesteigenvalueofD .The
S2
actionofoperatorsn¯·t¯,n¯·s¯ ×t¯,s¯ ·n¯ andDS2 onthese After bosonization from (9) we obtain the following
statesmaybefoundin[13].Wemayusethisknowledge scalarLagrangian:
to integrate over the angles in four dimensional Dirac
actiontoobtaineffectivetwodimensionaltheoryin(t,r ) L = 1(cid:209) f (cid:209) af l B 1 (cid:209) af +
plane. ∗ BR −2 a R R− a√p R
On the otherhand, dimensionalreduction of the YM +Vb(e2i√pf R e 2i√pf R), (14)
actiongivesusthefollowingLagrangian: − −
C2 1 fromwhichwederiveanequationofmotionforf Rfield
L = f fab dw2 (w2 1)2, (8)
YM−2d − 4 ab −| |−2C2 | | − l
(cid:209) (cid:209) af + (cid:209) Ba+4ib√p Vcos(2√pf )=0. (15)
whered =(cid:209) iB , f =¶ B ¶ B ,andB =eia. a R √p a R
a a− a ab a b− b a a a i
Fromnowonlatinlettersfromthebeginningofalphabet
Then, we derive an equation of motion for scalar field
will label indexes from curved two dimensional (t,r )
spacetimeandthosefromthemiddlefromflatspacetim∗e. connected to the y L sector of our original fermionic
theoryanalogically: for electric and magnetic parts of this field in the pres-
enceoffermions.Afterusingbosonizationformulas(11)
l
(cid:209) (cid:209) af (cid:209) Ba+4ib√p Vcos(2√pf )=0. (16) theseequationsread[15]
a L−√p a L
l
(cid:209) [C2fab] 2 w2Bb= [(cid:209) bf (cid:209) bf ], (24)
Equations (15) and (16) are highly nonlinearand we a − | | √p R− L
werenotabletofindasolutionintermsofknownspecial
2
functionsinthewholespacetime.Nevertheless,wemay (cid:209) (cid:209) aw w(w2 1)+2wB Ba=
a −C2 | | − a
find some useful information about their solutions by
2b
analyzing them in two asymptotic regions, namely in i [sin(2√pf )+sin(2√pf )]. (25)
R L
C
nearhorizonregionandinlarger region.
Inthelarger regionwehavet∗hat Conclusions that stem from the above equations are
∗
the following. First, let us remind that equations (15)
A2 1, C=r, w 1, r r. (17)
≈ ≈± ∗∼ and(16)differonlybysignin frontofthetermpropor-
tional to B and we assume that initially a black hole
Takingthisintoaccounttheequation(15)turnsinto a
hasonlymagneticcharge(B =0).Second,fromequa-
a
w(¥ ) tion (24) we see thatinthis case contributionsofscalar
−¶ t2f R+¶ r2f R+4ib√p r cos(2√pf R)=0. (18) fields cancel each other and Ba =0 is still a valid so-
∗ lution to (24). Third, from equation (25) we see that
After dropping a term proportional to O(r 1) we get there is a nonzero contribution of scalar fields to mag-
−
afreewaveequation neticpartofYMfield.Butsincethesefieldsaretimede-
pendentsoshouldbetheresultingw.Ontheotherhand,
¶ 2f +¶ 2f =0. (19) through Einstein equations, this results in time depen-
− t R r R
∗ dence of metric tensor elements and ultimately means
A regularsolutionto thisequationmaybe expressedas thattheassumptionofstaticityofYang-Millsblackhole
aplanewave isdestroyedinthepresenceofmasslessfermions.
f =c e iw (t r ), (20)
R 0 − ±∗
MASSIVEFERMIONS
where c is an integration constant. On the other hand,
0
inthenearhorizonregionwehave
Formassivefermionsweseefrom(6)thatchiralitiesare
A2=2k (r r ), C=r , r r =e2k r , (21) mixed up. To use bosonization in this case we need to
− H H − h ∗ make an additional assumption about the form of y R
where k is surface gravity of a Yang-Mills black hole. andy L.Weusethefollowingansatz:
Theequation(15)inthisregiontakestheform
G =is 3G G, (26)
L R
≡
−¶ t2f R+¶ r2∗f R+i8b√prwh(rh)k e2k r∗cos(2√pf R)=0. wnehcetreedGwLitahnydGaRnadrey twroesdpiemcetinvseiloyn[a1l5s]p.inHoarvfiinegldtshcisonin-
L R
(22)
mindaneffectivetwodimensionalfermionicLagrangian
Becauseasweapproachablackholehorizonr ¥ , isasfollows:
we may drop a term proportional to e2k r∗ in th∗e→ab−ove L = iG¯g˜a(cid:209) G l B G¯g˜ag˜3G+
equation.Fromthisweseethataregularsolutionisagain GF−2d − a − a
atimedependentplanewavegivenby +(V+m)G¯g˜LG+(m V)G¯g˜RG, (27)
−
f R=c1e−iw (t±r∗), (23) where like in massless case V = Cw. Using the same
bosonizationformulasas inthe masslesscase we arrive
wherec issomeotherintegrationconstant.Onthebasis atthefollowingscalarLagrangian:
1
ofthisanalysiswemayconcludethataregularsolution
1 l
to the considered scalar field equation will be time de- L = (cid:209) f (cid:209) af B (cid:209) af +
pendent. But both scalar fields represent fermionic cur- GB −2 a −√p a
rents and their time dependence ultimately means that +(V+m)be2i√pf +(m V)be 2i√pf . (28)
−
fermionicfieldswillalsobetimedependent.Toseehow −
thismayinfluenceYMfieldweuseequationsofmotion
Theequationofmotionforscalarfieldinthiscaseis Firstwewilldiscussthemasslesscase.Byasymptotic
analysis we find an evidence that regular solutions to
l
(cid:209) (cid:209) af + (cid:209) Ba+ equations(15)and(16)shouldbetimedependent.These
a √p a equationsdescribebosonizedmasslessfermionsandgive
anonzerocontributiontoamagneticpartofYang-Mills
+4ib√p Vcos(2√pf )+imsin(2√pf ) =0. (29)
field.Becausetheirsolutionsaretimedependentthere-
(cid:26) (cid:27)
sulting magnetic part of YM field will be time depen-
Inthelarger limit,afterdroppingtermsproportionalto dentand,throughEinsteinequations,elementsofamet-
O(r 1),weo∗btainasine-Gordonequation rictensorwillalsobetimedependent.
−
In massive case bosonizedfermionsare describedby
−¶ t2f +¶ r2f −4b√p msin(2√pf )=0, (30) solutionstoequation(29).Thesametypeofanalysislike
∗ in massless case also revealsthe destructionof staticity
for which a regular time dependent and decaying at
ofourblackhole.Moreover,massivefermionswilllead
infinity(kinktype)solutionisgivenby
to the formation of a dyonic black hole. But, as was
shownin [12], the presenceof an electricpartof Yang-
f = √2p arctan(cid:18)e−q81b−pvm2(r∗−vt)(cid:19). (31) Mills field leads to infinite mass of the resulting black
hole.
In conclusion we may say that the presence of
On the other hand, the same type of analysis like in
fermionsand theirinteraction with the consideredPBH
themasslesscaserevealedthatinthenearhorizonlimit
will lead to its destruction through mechanisms de-
we againhave a free wave equationwith a regulartime
scribedabove.
dependentsolutioninformofaplanewave:
f =c e iw (t r ). (32)
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