Regular and conformal regular cores for static and rotating solutions

0.08 MB·

by  Mustapha Azreg-Aïnou

Regular and conformal regular cores for static and rotating
solutions
MustaphaAzreg-A¨ınou
Bas¸kentUniversity,DepartmentofMathematics,Bag˘lıcaCampus,Ankara, Turkey
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Abstract

n
a Usinganewmetricforgeneratingrotatingsolutions,wederiveinageneralfashionthesolutionof
J animperfectfluidandthatofitsconformalhomolog.Wediscusstheconditionsthatthestress-energy

0 tensorsandinvariantscalarsberegular. Onclassicalphysicalgrounds,itisstressedthat,conformal
1 fluids used as coresfor static or rotatingsolutions, are exemptfrom any maliciousbehaviorin that

theyarefiniteanddefinedeverywhere.
]
c
q PACSnumbers:04.70.Bw,04.20.-q,97.60.Lf,02.30.Jr
-
r
g
[ 1 Introduction

3
v The quest for rotating solutions has always been a fastidious task. It took more than two decades to
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discovertherotatingsolutionofVanStockum[1]andmorethanfortyyearstoderivethatofKerr[2]since
8
7 thefoundation ofGeneralRelativityin1916. Severalpartialmethodshavebeenputforwardtoconstruct
0 rotating solutions [1]- [15] but no general method seems to be available. This work is no exception and
.
1 presents a novel partial method for generating rotating solutions from static ones. However, the method
0
willallowus(1)togeneraterotatingsolutionswithoutappealingtolinearapproximations [16]and(2)to
4
1 applythematchingmethods[17–19]toregularblackholecoresaswellastowormholecores[15,20,21].
: The excellent paper by Lemos and Zanchin offers an up-to-date classification of the existing matching
v
i methods, discusses the types of regular black holes derived so far and presents new electrically charged
X
solutions with a regular de Sitter core [19]. The present method reduces the task of finding a rotating
r
a solutiontothatoffindingatwo-variablefunctionthatisasolutiontotwosecondorderpartialdifferential
equations.
We work with Rm nrs = ¶ s G m nr + (m =1 4) and a metric gmn with signature (+, , , ).
− ··· → − − −
Wemakeallnecessary conventions suchthatthefieldequations taketheformGmn =Tmn .
Weconsiderafluidwithoutheatflux,thestress-energy tensor(SET)ofwhichadmitsthedecomposi-
tion
Tmn =e um un +p em en +p em en +p em en (1)
2 2 2 3 3 3 4 4 4
where e is the mass density and (p , p , p ) are the components of the pressure. We have preferred
1 2 3
m m
the notation u , instead of e , which is the four-velocity of the fluid. The four-vectors are mutually
1 m m
tpheernpetnhdeiccoumlarplaentednneosrsmreallaiztieodn:,ugmnum==u1m,ueni eim(e=m e−n 1+(ei=m e2n +→e4m)e.nI)f,tlheeadflsuitdoiTsmpn er=fec(te,+p2p=)upm3u=n p4p≡gmn p,.
− 2 2 3 3 4 4 −
Givenastaticspherically symmetricsolution tothefieldequations inspherical coordinates:
dr2
ds2=G(r)dt2 H(r)(dq 2+sin2q df 2) (2)
−F(r)−
1
wegenerate astationary rotating solution, themetricofwhich, writteninBoyer-Lindquist (B-L)coordi-
nates,wepostulate tobeoftheform
G(FH+a2cos2q )Y Y √F√GH FGH
ds2= dt2 dr2+2asin2q − Y dtdf
(√FH+a2√Gcos2q )2 −FH+a2 h(√FH+a2√Gcos2q )2i
2√F√GH FGH+a2Gcos2q
Y dq 2 Y sin2q 1+a2sin2q − df 2, (3)
− − n h (√FH+a2√Gcos2q )2 io
by solving the field equations for Y (r,q ), which depends also on the rotating parameter a. More on the
derivation andgeneralization of(3)willbegivenelsewhere[22]. Forfluidsundergoing onlyarotational
motion about a fixed axis (the z axis here), Trq 0 leading to Grq =0, which is one of the very two
equations to solve to obtain Y (r,q ). From now≡on, we use the following conventions and notations:
m : 1 t,2 r,3 q ,4 f )and(u,e2,e3,e4)=(u,er,eq ,ef ).
↔ ↔ ↔ ↔
2 The solutions
Toeasethecalculations, weusethealgebraiccoordinatey=cosq andreplacedq 2bydy2/(1 y2)in(3).
−
Forthe sake of subsequent applications (to regular black holes and wormholes), wewillassume H =r2
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unless otherwise specified. Setting K(r) √FH/√G and using an indexical notation for derivatives:
≡
Y ,ry2 ¶ 2Y /¶ r¶ y2,K,r ¶ K/¶ r,etc,theequationGrq =0yields
≡ ≡
(K+a2y2)2(3Y Y 2YY )=3a2K Y 2 (4)
,r ,y2 ,ry2 ,r
−
This hyperbolic partial differential equation may possess different solutions, but a simple class of solu-
tionsismanifestlyoftheformY (r,y)=g(K+a2y2)whereg(z)issolutionto
2z2gg 3z2g 2+3g2=0 (5)
,zz ,z
−
wherez=K(r)+a2y2. AgeneralsolutiondependingontwoconstantsisderivedsettingA(z)=g /gand
′
leadstoY =c z/(z2+c )2. However,thissolutiondoesnotexhaustthesetofallpossiblesolutionsof
gen 2 1
theformg(z) to(5)which, being nonlinear, admitsother moreinteresting power-law solutions g(z)(cid:181) zn
leadingto
Y =K(r)+a2y2 or Y =[K(r)+a2y2] 3 (6)
1 2 −
where Y is included in Y taking c = 0 and c = 1. A consistency check of the field equations
2 gen 1 2
Gmn =Tmn andtheformofTmn [Eq.(1)]yieldsthepartialdifferential equation
Y [K 2+K(2 K ) a2y2(2+K )]+(K+a2y2)(4y2Y K Y )=0, (7)
,r ,rr ,rr ,y2 ,r ,r
− − −
which is solved by Y (but not by Y ) provided K =r2+p2 where p2 is real. We have thus found a
1 2
simplecommonsolutiontobothEqs.(4)and(7)givenby
Y =r2+p2+a2y2. (8)
Wedonotknowthesetofallpossible solutions toEqs.(4)and(7),however,wecanstilldistinguish
twofamilies ofrotating solutions. Depending onG(r),F(r)andH(r),arotating solution givenby(3)is
called a normal fluid, Y , if the static solution (2) is recovered from the rotating one in the limit a 0:
n
→
Thisimplieslim Y =H. Otherwisetherotating solution iscalled aconformal fluid,Y . GivenG(r),
a 0 c
F(r)andH(r),th→enormalds2 andconformalds2 fluidsareconformally related
n c
ds2 =(Y /Y )ds2. (9)
c c n n
2
Now,sincelim Y =H (bydefinition)andlim ds2=ds2 [Eq.(2)],thisimpliesthatlim ds2=
a 0 c a 0 n stat a 0 c
ds2 ,andthusl→im 6ds2 isanewstaticmetriccon→formaltods2 . → 6
stat a 0 c stat
→
For the remaining part of this work, we shall explore the properties of both the normal (Sect. 3)
and conformal (Sect. 4) rotating solutions that can be constructed using the unique simple solution Y
availabletous,whichisgivenby(8). Fromnowon,weshallusetheprimenotationtodenotederivatives
offunctions.
3 Physical properties of the model-independent normalinterior core: G=
F
The constraints G=F and K =r2+p2 yield H =K, so we deal with a normal fluid since lim Y =
a 0
H [Eq. (8)]. The invariants R and Rmnab Rmnab are proportional to r −6 and r −12, respectivel→y, with
r 2 K+a2y2 =H+a2y2. Thus, the static and rotating solutions (3) are regular if H(r) is never zero
≡
(p2 = 0), which is the case for wormholes and some type of regular phantom black holes [15,21]. If
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H=r2 (p2=0),thentherotatingsolution(3)mayhavearingsingularityintheplaneq =p /2(y=0)at
r=0(more details are given in[22]). Asweshall see below, there are cases wherethe numerators ofR
and Rmnab Rmnab also vanish for r=0and q =p /2 tothe same order, leading to aring-singularity free
solution(3). Whenthisisthecase,thecomponentsoftheSETaswellasthetwoinvariantsremainfinite,
butundefined, ontheringr 2=0.
Setting2f(r) K FH,D (r) FH+a2 andS (K+a2)2 a2D sin2q ,thesolution(3)reducesto
≡ − ≡ ≡ −
2f r 2 4afsin2q S sin2q
ds2= 1 dt2 dr2+ dtdf r 2dq 2 df 2 (10)
n h −r 2i − D r 2 − − r 2
D sin2q r 2
= (dt asin2q df )2 [adt (K+a2)df ]2 dr2 r 2dq 2. (11)
r 2 − − r 2 − − D −
Wefixthebasis(u,er,eq ,ef )by
m (K+a2,0,0,a) m √D (0,1,0,0) m (0,0,1,0) m (asin2q ,0,0,1)
u = , e = , e = , e = . (12)
r 2D r r 2 q r 2 f − r 2sinq
p p p p
The components of the SET are expressed in terms of Gmn as: e = um un Gmn , pr = grrGrr, pq =
qq m n −
g Gqq , pf =ef ef Gmn . Wefind:
−
2(rf f) p2 2p2(3f a2sin2q )
e = ′− − + − (13)
r 4 r 6
2p2D f 2p2a2sin2q
pr =−e − r 6 , pq =−pr− r ′2′, pf = pq + r 6 . (14)
Thus, for wormholes and some type of regular phantom black holes [15,21] where always r 2 >0 (H
nevervanishes), thecomponents ofthe SETarefinite inthestatic androtating cases. Eqs. (13)and(14)
will be used in [22] to derive the rotating counterpart of the stable exotic dust Ellis wormhole emerged
in a source-free radial electric or magnetic field [29]. If H = r2, corresponding to regular as well as
singularblackholes,theaboveexpressions reducetothosederivedin[6,18]: e = p =2(rf f)/r 4,
r ′
− −
pq = pf =e f′′/r 2. InthiscasethecomponentsoftheSETdivergeontheringr 2=0unless f (cid:181) r4as
−
r 0,resultingin(1 F)(cid:181) r2asr 0,whichcorrespondstothe(anti)deSittercaseandtoregularblack
→ − →
holes. Infact,mostofregularblackholesderivedsofarhavedeSitter-likebehaviornearr=0[17,19,20].
3
From the third Eq. (14), on sees that the tangential pressures, (pq , pf ), are generally nonequal and
are equal only if p2 =0 or/and if a=0 (the static case). Hence, in the general rotating case, the tensor
mn
T hasfourdifferent eigenvalues representing thusatotallyimperfectfluid.
Itisstraightforwardtoverifythevalidityofthecontinuityequation: (e um );m =0,wherethesemicolon
mn m n
denotes covariant derivative. Theconservation equation, T ;n =0,isconsistent withu ;n u =0which
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showsthatthemotionofthefluidelementsisnotgeodesic. Thisisattributable tothenonvanishing ofthe
r-andq -components ofthepressuregradient.
Thepurposeofconstructingrotatingandnonrotatingsolutionswithnegativepressurecomponents,as
might be the case in (13) to (14), is, as wasmade clear in [18], two-fold, in that, following a suggestion
bySakharovandGliner[23,24],(1)thecoreofcollapsingmatter,withhighmatterdensity,shouldhavea
cosmological-type equation ofstatee = p, (2)theproblem ofthering singularity, whichcharacterizes
−
Kerr-type solutions, could be addressed if the interior of the hole is fitted with an imperfect fluid of the
type derived above. Fitting the interior of the hole with a de Sitter fluid is one possible solution to the
ring singularity [18,19]. Another possibility is to consider a regular core or a conformal regular one as
weshallseeinthecaseG=F (Sect.4).
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3.1 Rotatingimperfect L -fluid—de Sitter rotating solution
Instances of application of (3) to re-derive the Kerr-Newman solution from the Schwarzschild solution
and to generate a rotating imperfect L -fluid (IL F) from the de Sitter solution are straightforward. To
derive the Kerr-Newman solution, we take F =G=1 2m/r+q2/r2 and H =r2, the solution is then
−
given by (10)with 2f =2Mr q2, D =r2+a2 2Mr+q2, r 2 =r2+a2cos2q and S =(r2+
KN − KN − KN KN
a2)2 a2D sin2q .
KN
−
ConsiderthedeSittersolution
ds2L =(1 L r2/3)dt2 (1 L r2/3)−1dr2 r2(dq 2+sin2q df 2) (15)
− − − −
where F =G=1 L r2/3 and H =r2. The metric ds2L of the rotating IL F is given by (10) with 2fL =
−
L r4/3,D L =r2+a2 L r4/3,r L2 =r2+a2cos2q andS L =(r2+a2)2 a2D L sin2q . Exceptfromashort
− −
descriptionmadein[25],therotatingIL Fhasneverbeendiscusseddeeplyinthescientificliterature. The
componentsoftheSETaree =L r4/r L4, pr= e , pq =pf = L r2(r2+2a2cos2q )/r L4. Thelimita 0
− − →
leadstodeSittersolution wherethefluidisperfectwithe =L and pr = pq = pf = L .
−
TherotatingIL Fisonlymanifestlysingular ontheringr 2 =0[(q ,r)=(p /2,0)or(y,r)=(0,0)].
L
Infact,thecurvatureandKretchmann scalars
4L r2 8L 2r4(r8+4a2y2r6+11a4y4r4 2a6y6r2+6a8y8)
mnab
R= , Rmnab R = − (16)
−r2+a2y2 3(r2+a2y2)6
do not diverge in the limit (y,r) (0,0). Despite the fact that the limits do not exist, wecan show that
→
theydonotdiverge. LetC: r=ah(y) andh(0)=0beasmoothpath through thepoint (y,r)=(0,0)in
theyrplane. Wechooseapaththatreaches(y,r)=(0,0)obliquelyorhorizontallybutnotvertically,that
is, we assume that h(0) is finite [for paths that may reach (y,r)=(0,0) vertically we choose a smooth
′
path y=g(r)/a and g(0)=0 where g(0) remains finite]. On C, the limits of the two scalars as y 0
′
→
read
4L h(0)2 8L 2h(0)4[6 2h(0)2+11h(0)4+4h(0)6+h(0)8]
′ ′ ′ ′ ′ ′
, − , (17)
−1+h(0)2 3[1+h(0)2]6
′ ′
which are nonexisting [for h(0) depends on the path] but they remain finite. Thus, the rotating IL F is
′
regular everywhere, however, the components ofthe SETare undefined onthe ring r 2=0. Pathsof the
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form: y=g(r)/aandg(0)=0,whereg(0)remainsfinite,leadtothesameconclusion. Theotherscalar,
′
mn
Rmn R ,behavesinthesamewayasthecurvature andKretchmannscalars.
Notice that the Kerr solution (q=0) and the rotating IL F one are derived from each other on per-
formingthesubstitution 2M L r3/3,sothatmostoftheKerrsolution properties, wherenoderivations
↔
with respect to r are performed, are easily carried over into the rotating IL F properties. For instance,
m
thestaticlimit,whichisthe2-surface onwhichthetimelikeKillingvectort =(1,0,0,0) becomesnull,
corresponds to g (r ,q ) = 0 leading to 2L r2 = 3+√9+12L a2cos2q . Thus, observers can remain
tt st st
static only for r <r . Similarly, the cosmological horizon, which sets a limit for stationary observers,
st
correspondstoD L (rch)=0leadingto2L rc2h=3+√9+12L a2. Hence,thestaticlimitisenclosedbythe
cosmologicalhorizonandintersectsitonlyatthepolesq =0orq =p (incontrastwiththeKerrsolution
wherethestaticlimitencloses theeventhorizon).
m
The four-velocity of the fluid elements may be expressed, in terms of the timelike t and spacelike
f m =(0,0,0,1) Killingvectors,asum =N(tm +W f m ),withN=(r2+a2)/ r 2D L andW =a/(r2+a2)
isthedifferentiable (W =constant)angular velocity ofthefluid. Sincethenporm ofthevectortm +W f m ,
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1/N2, is positive only for D L >0, which corresponds to the region r < rch, the fluid elements follow
timelike world lines only for r<r . As r r , W approaches the limit a/(r2 +a2) that is the lowest
ch → ch ch
angularvelocityofthefluidelementswhichwetakeastheangularvelocityofthecosmological horizon:
W =a/(r2 +a2). At the cosmological horizon, tm +W f m becomes null and tangent to the horizon’s
ch ch
nullgenerators, sothatthefluidelementsaredragged withtheangular velocityW .
ch
4 Physical properties of the conformal interior core: G = F
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InthiscaseH =K =r2+p2,unless p2=0,leading tolim Y =H. WithY =K+a2y2 [Eq.(8)],the
a 0
conformalrota6tingsolution ds2c isagaingivenby(10)to(11→)and6thebasis(u,er,eq ,ef )by(12)butthis
timer 2 K+a2y2=H+a2y2. ThecomponentsoftheSETaredifferentduetothenon-covarianceofthe
≡ 6
fieldequationsunderconformaltransformations[26]. TheSETrelatedtods2isonlypartlyproportionalto
c
thatrelatedtometricds2 andincludestermsinvolving firstandsecondorderderivativesoftheconformal
n
factor (K+a2y2)/(H+a2y2), which are the residual terms in the transformed Einstein tensor. Finally,
theSETrelatedtods2 takestheform
c
p2[6f r2 p2 a2(2 cos2q )] 2(rf f) 2p2(r2+p2+a2 2f)
e = − − − − + ′− , p = e − (18)
r 6 r 4 r − − r 6
2(r2+a2cos2q )f p2+2rf f 2a2p2sin2q
′ ′′
pq =− r 6 + r 4 − r 2, pf = pq + r 6 (19)
which is finite and defined everywhere if p2 =0. If p2 =0, the SET if finite, but undefined on the ring
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r 2=0,if f (cid:181) r4 asr 0((anti)deSitterbehavior forF =G). Thecurvature scalar
→ 6
2 p2[r2+p2+a2(2 cos2q )] 2p2f 2f
′′
R= { − − } (20)
r 6 − r 2
isalsofiniteforall p2. TheKretchmannscalariscertainlyfiniteeverywhereforall p2. Conclusionsmade
earlier concerning the continuity and conservation equations apply to the present case of the conformal
fluid.
4.1 Examples ofstaticand rotating conformal imperfect fluids
Consider a static regular black hole or a wormhole of the form (2) where G=F are finite at r=0 and
H(r)= r2+q2. In the (t,u,q ,f ) coordinates, where u is the new radial coordinate, G(u) = G(r(u)),
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F(u)=G(u)/r (u)2 and H(u)=r(u)2+q2. Since we want K(u)=u2+p2 [Eq. (8)], we have to solve
′
thedifferential equation: dr/du=[r(u)2+q2]/(u2+p2),yielding
r(u)=qtan[(q/p)arctan(u/p)] (21)
where p2=0andq2=0. In(t,u,q ,f ) coordinates, theequivalent staticsolution takestheform
6 6
r(u)2+q2 2 du2
ds2 =G(u)dt2 [r(u)2+q2](dq 2+sin2q df 2). (22)
(s) −(cid:16) u2+p2 (cid:17) G(u)−
Whilemetrics (2)and (22)are equivalent, their rotating counterparts are not. Themetricds2 of thecon-
c
formalrotating core fluid,thatistherotating counterpart of(22),isgivenby(10)with2f =u2+p2
(s)
−
F(u)H(u),F(u)H(u)=(u2+p2)2G(u)/[r(u)2+q2],D =F(u)H(u)+a2,r 2 =u2+p2+a2cos2q and
(s) (s)
S =(r2+p2+a2)2 a2D sin2q . Since p=0theSETandcurvature scalar, givenby(18)to(20)on
(s) (s)
− 6
replacingrbyu, f by f andr byr ,arefiniteeverywhere. Onecanthusfollowoneoftheprocedures
(s) (s)
intheliterature [17–20],asthe oneperformed in[18], tomatchtherotating metric ds2 totheKerrblack
c
hole.
Itisstraightforward tocheckthatlim ds2 doesnotyieldds2 ;rather, thelimityieldsanewstatic,
a 0 c (s)
→
conformalimperfectfluid,solution.
5 Conclusion
AmastermetricinB-Lcoordinatesthatgeneratesrotatingsolutionsfromstaticoneshasbeenputforward.
Thefinalformofthegenerated stationary metricdepends onatwo-variable function thatisasolution to
two partial differential equation ensuring imperfect fluid form of the source term in the field equations.
Only one simple solution of the two partial differential equations has been determined in this work and
appearstoleadtostationary, aswellasstatic, normalandconformalimperfectfluidsolutions.
OnapplyingtheapproachtothedeSitterstaticmetricandtoastaticregularblackholeorawormhole,
two regular rotating, imperfect fluid cores, normal and conformal respectively, with equation of state
nearinge = pinthevicinity oftheorigin(r 0),havebeenderived.
− →
ConformalfluidcoreshaveeverywherefinitecomponentsoftheSETandofthecurvatureandKretch-
mannscalars.
Wehave not examined any energy conditions and related constraints on the mass density since even
violations of the weak energy condition, not to mention the strong one, have become custom to issues
pertaining to regular cores [18,27,28]. These violations worsen in the rotating case as was concluded
in[18].
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8