We investigate the properties of global monopoles with an event horizon. We
find that there is an unstable circular orbit even if a particle does not have
an angular momentum when the core mass is negative. We also obtain the
asymptotic form of solutions when the event horizon is much larger than the
core radius of the monopole, and discuss if they could be a model of galactic
halos.Comment: 5 pages, 7 figure

C. Wetterrich

D. Harari

D. Maison

H. Watabe

J.D. Bekenstein

M. Barriola

Nobuyuki Sakai

S.L. Liebling

S.W. Hawking

Takashi Tamaki

U. Nucamendi

W.A. Hiscock

English

arXiv

Tamaki, T

Sakai, N

Kyoto University Research Information Repository

Title Properties of global monopoles with an event horizonAuthor(s)Tamaki, T; Sakai, NCitationPHYSICAL REVIEW D (2004), 69(4)Issue Date2004-02URL http://hdl.handle.net/2433/49870RightCopyright 2004 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityProperties of global monopoles with an event horizonTakashi Tamaki*Department of Physics, Kyoto University, 606-8501, JapanNobuyuki Sakai†Faculty of Education, Yamagata University, Yamagata 990-8560, Japan;Graduate School of Science and Engineering, Yamagata University, Yamagata 990-8560, Japan;and Osservatorio Astronomico di Roma, Via Frascati, 00040 Monte Porzio Catone, Italy~Received 18 September 2003; published 25 February 2004!We investigate the properties of global monopoles with an event horizon. We find that there is an unstablecircular orbit even if a particle does not have an angular momentum when the core mass is negative. We alsoobtain the asymptotic form of solutions when the event horizon is much larger than the core radius of themonopole, and discuss if they could be a model of galactic halos.DOI: 10.1103/PhysRevD.69.044018 PACS number~s!: 04.70.2s, 97.60.Lf, 98.62.GqI. INTRODUCTIONIn unified theories, various kinds of topological defectshave been predicted and may appear in cosmological phasetransitions in the early Universe. Therefore it is important toinvestigate such defects both theoretically and observation-ally. Concerning global monopoles, important facts werefound recently: ~i! Solutions with an event horizon exist@1–3#; ~ii! regular global monopoles coupled nonminimallyto gravity have a stable circular orbit and may explain rota-tion curves in spiral galaxies, as shown by Nucamendi, Sal-gado, and Sudarsky ~NSS! @4#.The result ~i! is interesting since the static solutions ofregular global monopoles are always repulsive @5#. More-over, black hole solutions are stable though topologicalcharge is lost in the strict sense @6# and there are solutionswith zero mass which are somewhat pathological @2#. Thus,we need to understand their properties. In particular, it wouldbe interesting to investigate a particle motion around the ho-rizon. This is one of our main concerns in this paper.Possibility ~ii! shows that global monopoles can be locallyattractive in the nonminimally coupled theory of gravity. Al-though there have been many attempts to explain rotationcurves in spiral galaxies, there is no definite one at present.Among solitonic objects, global monopoles have the remark-able property that energy density decreases with the distancer22 @7#, which may be desirable to explain the flatness ofrotation curves. To remove the unprefereble repulsive prop-erty of global monopoles, NSS introduced nonminimal cou-pling and succeeded to obtain locally attractive solutions.Taking both ~i! and ~ii! into consideration, we notice apossibility that global monopoles with an event horizon canexplain rotation curves since they would be attractive. Thereare several advantages in this model compared with the NSSmodel. First, we need not require the nonminimal coupling,which are constrained astrophysically @8#. Second, theywould also be model black holes in the central galaxies.Third, the core mass can be chosen to be astronomicallylarge, contrary to the NSS model, where the core mass isnecessarily microscopic. Therefore it is important to studythe properties of such global monopoles, and discuss whetheror not they can be a realistic candidate as galactic halos,taking astrophysical bounds into account @9#.This paper is written as follows. In Sec. II, we explain ourmodel and basic equations. In Sec. III, we investigate globalmonopoles with an event horizon in two situations sepa-rately. In Sec. III A, we consider the case where the size ofevent horizon is comparable to the core radius of the mono-pole to compare with regular monopoles. In Sec. III B, weconsider the case where the size of an event horizon is as-trophysically large. In Sec. IV, we denote concluding re-marks and discuss problems concerning the restriction fromobservation.II. BASIC EQUATIONS FOR NUMERICAL ANALYSISWe begin with the actionS5E d4xA2gF R16pG 2 ~„Fa!22 2l4 ~FaFa2v2!2G ,~1!where G and Fa are the gravitational constant and the realtriplet Higgs field, respectively. The theoretical parameters vand l are the vacuum expectation value and the self-coupling constant of the Higgs field, respectively.We assume that the space time is static and sphericallysymmetric, in which the metric is written asds252 f ~r !e22d(r)dt21 f ~r !21dr21r2dV2, ~2!where f (r)“122Gm(r)/r . We adopt the hedgehog ansatzgiven byFa5vrah~r !, ~3!where ra is a unit radial vector.Under the above assumptions, the basic equations are*Electronic address: tamaki@tap.scphys.kyoto-u.ac.jp†Electronic address: sakai@ke-sci.kj.yamagata-u.ac.jpPHYSICAL REVIEW D 69, 044018 ~2004!0556-2821/2004/69~4!/044018~5!/$22.50 ©2004 The American Physical Society69 044018-1m¯ 85r¯2v¯ 2F f2 ~h8!21UG , ~4!d852v¯ 2r¯~h8!2, ~5!h952h8r¯11f Fh~h221 !1 2hr¯212r¯v¯ 2h8U2h8r¯G , ~6!where 85d/dr¯ andU5~h221 !24 1h2r¯2. ~7!We have introduced the following dimensionless variables:r¯5vAlr , m¯ 5GvAlm , v¯5vA4pG . ~8!We assume the regular event horizon at r5rH :m¯ H5r¯H2 , dH,‘ , ~9!hH8 5hH@21r¯H2 ~hH2 21 !#r¯H~122r¯H2 v¯ 2UH!. ~10!The variables with subscript H are evaluated at the horizon.We introduce the variablem¯ core~r¯ !“m¯ ~r¯ !2v¯ 2r¯ ~11!and assume the boundary conditions at spatial infinity asm¯ core~‘!5:M¯ 5const, d~‘!50, h~‘!51, ~12!which means that the space time is asymptotically ‘‘flat’’with deficit angle. Here M¯ corresponds to a core mass of themonopole, which determines a particle motion we will seebelow. We will obtain the black hole solutions numericallyby solving Eqs. ~4!–~6! iteratively with the boundary condi-tions ~9!, ~10!, and ~12!. For our numerical calculation, weuse the double-precision Bulirsch-Store Method based onRef. @10#.III. PROPERTIESThe space-time structure of global monopole black holesdepends on the expectation value of the Higgs field @1#. Forv¯ 2,1/2, there is a solution with asymptotically ‘‘flat’’ spacetime. We concentrate on this realistic case.A. Small horizonTypically, it is supposed that a global monopole has anontrivial structure in the core r&rcore“2/vAl @5#, whilethe field is almost constant, h>1, outside the core r*rcore . Therefore, we expect that a monopole black holewith a small horizon rH>rcore ~i.e., r¯H>1) may have newproperties.First, we show field distributions of black hole solutionsin Figs. 1~a! r¯-h and 1~b! r¯-m¯ core . We choose v¯50.6,0.2 andr¯H50.3. In Fig. 1~a!, we find that the Higgs field has anontrivial structure extending to ;10 and it does not dependon the expectation value of the Higgs field, which is impor-tant in the later analysis. In this solution, h increases mono-FIG. 1. The behavior of ~a! h, ~b! m¯ core for r¯H50.3 and v¯50.6, 0.2.T. TAMAKI AND N. SAKAI PHYSICAL REVIEW D 69, 044018 ~2004!044018-2tonically with r¯ . Although solutions where h is not mono-tonic exist @3#, here we only consider monotonic solutionsfor simplicity. Figure 1~b! shows that m¯ core for v¯50.6 de-creases with r¯ much faster than that for v¯50.2. This is natu-ral because of the factor v¯ 2r¯ in Eq. ~11!. The important pointis that m¯ core becomes negative for v¯50.6 and positive forv¯50.2 in the asymptotic region. As we will discuss below,the sign of m¯ core at large r ~i.e., M¯ ) determines the qualitativebehavior of particle motions around the monopole.Let us consider the geodesic equation of a test particle inthe equatorial plane. In our coordinate system ~2!, this isexpressed asE22 5r˙ 22 e22d1Veff , ~13!where ˙5d/dt and E is the energy of the particle per unitrest mass. Veff is defined asVeff“e22dS 12 2Gmr D S 11 L2r2 D , ~14!where L is the angular momentum of the particle per unit restmass.We show the effective potential Veff for r¯H50.3 in Figs.2~a! v¯50.2 and 2~b! v¯50.6. The angular momentum is cho-sen as (L/rH)250 and 5. For (L/rH)255, there is no poten-tial minimum for v¯50.6, while there is for v¯50.2. For(L/rH)250, there is a potential maximum for v¯50.6, whilethere is not for v¯50.2. While the properties of the monopoleblack hole with v¯50.2 are essentially the same as those ofthe Schwarzschild black hole, larger v¯ changes propertiesqualitatively. These different properties are determined bythe sign of M¯ .To evaluate the potential minimum, we substitute theasymptotic form d→0 and m¯ →v¯ 2r¯1M¯ into Eq. ~14!. Then,dVeff /dr¯50 is satisfied at the positionsr¯65L¯ 2~122v¯ 2!6AL¯ 4~122v¯ 2!2212L¯ 2M¯ 22M¯, ~15!where L¯ is defined as L¯“vAlL . We find r¯6,0 whenL¯ 2(122v¯ 2)2.12M¯ 2 and M¯ ,0. Therefore, there is no po-tential minimum even if a test particle has large angular mo-mentum.On the other hand, the potential maximum cannot beevaluated from the asymptotic form of the solution because itis near the horizon if it exists, as shown by Fig. 2. However,we can discuss its existence as follows. If M¯ ,0,Veff de-creases with r¯ and approaches 122v¯ 2 Asymptotically, sinceVeff50 at the horizon, there is at least one local maximum atsome r¯ . If M¯ .0, on the other hand, Veff increases with r¯and approaches 122v¯ 2 asymptotically; whether or not thepotential maximum appear depends on L¯ as in the Schwarzs-child black hole. Thus, the sign of M¯ is important to deter-mine the particle motion around the black hole.Such a convex form of the potential for a L50 particlemotion is characteristic of a global monopole black hole, anddoes not appear neither in a Schwarzschild black hole nor ina regular global monopole. In the case of a regular globalmonopole, the whole space time is repulsive @5#, whichmeans that Veff decreases monotonically.Figure 3 shows the relation between r¯H and M¯ for variousv¯ . M¯ is negative in the limit r¯H→0 as it is expected from theregular solutions. As v¯ increases, the region of M¯ ,0 ex-tends to larger r¯H . As it was pointed out in Ref. @2#, there areFIG. 2. The behavior of Veff for r¯H50.3, and (L/rH)250, 5. ~a!v¯50.2 and ~b! v¯50.6.PROPERTIES OF GLOBAL MONOPOLES WITH AN . . . PHYSICAL REVIEW D 69, 044018 ~2004!044018-3solutions with M¯ 50. In this case, a potential minimum r¯2goes to infinity and a test particle cannot feel a black hole.B. Large horizonNSS argued that global monopoles with the nonminimallycoupled gravity may explain rotation curves in spiral galax-ies @4#. However, the nonminimally coupled gravity has beenconstrained astrophysically. Moreover, in the NSS model thebound orbits exist only in the microscopic region r&rcore .Thus, it is desirable to seek for other possibilities.On the other hand, Wetterich discussed the possibility thata massless scalar field may explain rotation curves of galac-tic halos @11#. In his solutions, however, physical boundaryconditions were not taken into account. If we assume regu-larity at the center or an event horizon of a black hole, onlya trivial solution remains. Nonexistence of nontrivial blackhole solutions are guaranteed by no hair theorem @12#. In thissense, his model is also unrealistic.Then, we turn to a global monopole with an event hori-zon. This model is free from the above difficulties existing inthe NSS model. Furthermore, the existence of an event hori-zon is realistic because massive black holes are observed inthe central regions of galaxies.Let us consider astrophysical bounds from the mass den-sity of monopoles in the universe at first. If we demand thatmass density of monopoles should be less than 10 times ofcritical density @9#, we haven,1023S 1016 GeVv D3Mpc23, ~16!where n is the number density of monopoles. Thus, v!1016 GeV is required to explain rotational curves of galac-tic halos. For definiteness, we choose v¯50.231024 and dis-cuss the case where the event horizon is cosmological size.This corresponds to the case r¯H@1 unless the self-couplingconstant of the Higgs field l takes extremely smaller valuethan 1.To see the structure of the Higgs field for r¯H@1, we plotrH-(12hH) in Fig. 4. To maintain numerical accuracy, wechange the variable from h(r) to s(r)“12h(r). In this dia-gram, the error is below a percent. We find hH>1 and (12hH)>1/rH2 . We can check this relation analytically as fol-lows. In the asymptotic region, supposing the asymptoticformh511 (n51‘Cnr2n, ~17!we have C150 and C2521, which are consistent with ournumerical results above.Because of this asymptotic behavior, M¯ can be estimatedby substituting h51 into Eq. ~4!. Then, we have theasymptotic relationM¯ 5r¯H2 ~122v¯2!. ~18!We show the relation r¯H-M¯ in Fig. 5 for v¯50.231024 andv¯50.2, which confirms the relation ~18!. Actually, Fig. 3shows that this approximation is fairly good even for r¯H;1 when v¯ is small.Let us consider particle motions. Setting h51, we haveVeff;S 122v¯ 21 2M¯r¯D S 11 L¯ 2r¯2D . ~19!FIG. 3. The relation between r¯H and M¯ for various v¯ .FIG. 4. The relation between r¯H and (12hH) for v¯50.231024.T. TAMAKI AND N. SAKAI PHYSICAL REVIEW D 69, 044018 ~2004!044018-4Since v¯ is small as constrained by Eq. ~16!, we find that theparticle motion is practically the same as that in Schwarzs-child black hole. We can regard this kind of behavior asnatural since the massive field has its structure comparable tothe Compton wavelength. Therefore, as long as we consideran astronomical-sized event horizon, the effect of scalarfields on particle motions is negligible.IV. CONCLUSION AND DISCUSSIONWe investigated properties of global monopoles with anevent horizon and revealed interesting features which hadnot been known so far. The main features of test particlemotions are determined by the sign of the core mass; if it isnegative and if the event horizon is as small as the coreradius, there is an unstable circular orbit even for a particlewith zero angular momentum. We also found the asymptoticform of the solutions when the event horizon is much largerthan the core radius; the qualitative features of the monopoleblack hole is the same as that of the Schwarzschild blackhole.Although our model does not explain observed rotationcurves very well, we obtain some lessons here. Our resultsindicates that massive scalar fields would encounter with thesame difficulty as in our model. A typical mass scale of par-ticle physics is so large that it generally contributes only tomicroscopic structure, whose size is of order of the inverseof the mass. In this sense, a massless scalar field consideredby Wetterich @11# might be useful. Although his model itselfdoes not satisfy physical boundary conditions neither of ablack hole nor of a regular solution, interaction with mattermay be a key ingredient to solve this problem. Including thispossibility, we also want to consider other solitonlike modelssuch as boson-fermion stars in the future @13#.ACKNOWLEDGMENTSN.S. thanks Luca Amendola for discussions and his hos-pitality at Osservatorio Astronomico di Roma. This workwas supported in part by Grant-in-Aid for Scientific Re-search Fund of the Ministry of Education, Science, Cultureand Technology of Japan, No. 154568 ~T.T.! and No.15740132 ~N.S.!. This work was also supported in part by aGrant-in-Aid for the 21st Century COE ‘‘Center for Diver-sity and Universality in Physics,’’ and by the exchanging-researcher project between Japan Society for the Promotionof Science and National Research Council of Italy.@1# S.L. Liebling, Phys. Rev. D 60, 061502 ~1999!; 61, 024030~2000!; D. Maison and S.L. Liebling, Phys. Rev. Lett. 83, 5218~1999!.@2# U. Nucamendi and D. Sudarsky, Class. Quantum Grav. 17,4051 ~2000!.@3# D. Maison, gr-qc/9912100.@4# U. Nucamendi, M. Salgado, and D. Sudarsky, Phys. Rev. Lett.84, 3037 ~2000!; Phys. Rev. D 63, 125016 ~2001!.@5# D. Harari and C. Lousto´, Phys. Rev. D 42, 2626 ~1990!.@6# H. Watabe and T. Torii, gr-qc/0307074; Phys. Rev. D 66,085019 ~2002!.@7# M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63, 341 ~1989!.@8# C. Will, Theory and Experiment in Gravitational Physics~Cambridge University Press, Cambridge, 1981!.@9# W.A. Hiscock, Phys. Rev. Lett. 64, 344 ~1990!.@10# W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flan-nery, Numerical Recipes in FORTRAN ~Cambridge UniversityPress, Cambridge, 1986!.@11# C. Wetterrich, Phys. Lett. B 522, 5 ~2001!.@12# J.D. Bekenstein, Phys. Rev. D 5, 1239 ~1972!; S.W. Hawking,Commun. Math. Phys. 25, 167 ~1972!.@13# A.B. Henriques and L.E. Mendes, astro-ph/0301015.FIG. 5. The relation between r¯H and M¯ for v¯50.231024 andv¯50.2 by a solid line and by a dotted line, respectively.PROPERTIES OF GLOBAL MONOPOLES WITH AN . . . PHYSICAL REVIEW D 69, 044018 ~2004!044018-5

Properties of global monopoles with an event horizon

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Properties of global monopoles with an event horizon

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