Magnetic dilaton strings in anti-de Sitter spaces

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Physics Letters B 672 (2009) 101105leet/loM pAa Db RaArReReAcAvEdation spacnde ptond.1.The construction and analysis of black hole solutions in thebarespcotimofasiccelimcaststfelosisttaeantimcotiopo*BoSisha03dockground of anti-de Sitter (AdS) spaces is a subject of muchcent interest. This interest is primarily motivated by the corre-ondence between the gravitating elds in an AdS spacetime andnformal eld theory living on the boundary of the AdS space-e [1]. This equivalence enables one to remove the divergencesthe action and conserved quantities of gravity in the same wayone does in eld theory. It was argued that the thermodynam-s of black holes in AdS spaces can be identied with that of artain dual conformal eld theory (CFT) in the high temperatureit [2]. Having the AdS/CFT correspondence idea at hand, onen gain some insights into thermodynamic properties and phaseructures of strong t Hooft coupling conformal eld theories byudying the thermodynamics of asymptotically AdS black holes.On another front, scalar coupled black hole solutions with dif-rent asymptotic spacetime structure is a subject of interest for ang time. There has been a renewed interest in such studies evernce new black hole solutions have been found in the context ofring theory. The low energy effective action of string theory con-ins two massless scalars namely dilaton and axion. The dilatonld couples in a nontrivial way to other elds such as gauge eldsd results into interesting solutions for the background space-e. It was argued that with the exception of a pure cosmologicalnstant, no dilaton-de Sitter or anti-de Sitter black hole solu-n exists with the presence of only one Liouville-type dilatontential [3]. Recently, the dilaton potential leading to (anti)-deCorrespondence address: Department of Physics, Shahid Bahonar University, POx 76175, Kerman, Iran.E-mail address: sheykhi@mail.uk.ac.ir.dilaton potentials, a class of static dilaton black hole solutions in(A)dS spaces has been obtained by using a coordinates transfor-mation which recast the solution in the Schwarzschild coordinatessystem [4]. More recently, a class of charged rotating dilaton blackstring solutions in four-dimensional anti-de Sitter spacetime hasbeen found in [5]. Other studies on the dilaton black hole solu-tions in (A)dS spaces have been carried out in [6,7].In this Letter, we turn to the investigation of asymptoticallyAdS spacetimes generated by static and spinning string sources infour-dimensional EinsteinMaxwell-dilaton theory which are hori-zonless and have nontrivial external solutions. The motivation forstudying such kinds of solutions is that they may be interpreted ascosmic strings. Cosmic strings are topological structure that arisefrom the possible phase transitions to which the universe mighthave been subjected to and may play an important role in theformation of primordial structures. A short review of papers treat-ing this subject follows. The four-dimensional horizonless solutionsof Einstein gravity have been explored in [8,9]. These horizonlesssolutions [8,9] have a conical geometry; they are everywhere atexcept at the location of the line source. The spacetime can beobtained from the at spacetime by cutting out a wedge and iden-tifying its edges. The wedge has an opening angle which turnsto be proportional to the source mass. The extension to includethe Maxwell eld has also been done [10]. Static and spinningmagnetic sources in three and four-dimensional EinsteinMaxwellgravity with negative cosmological constant have been explored in[11,12]. The generalization of these asymptotically AdS magneticrotating solutions to higher dimensions has also been done [13].In the context of electromagnetic cosmic string, it has been shownthat there are cosmic strings, known as superconducting cosmicContents lists availabPhysics Lwww.elsevier.comagnetic dilaton strings in anti-de Sitter shmad Sheykhi a,b,epartment of Physics, Shahid Bahonar University, PO Box 76175, Kerman, Iranesearch Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iranr t i c l e i n f o a b s t r a c tticle history:ceived 22 October 2008ceived in revised form 9 December 2008cepted 19 December 2008ailable online 10 January 2009itor: M. CveticWith an appropriate combinof spinning magnetic dilatobackground of anti-de Sitter shave a conic geometry. We to the rotation parameter. Waction in the presence of dilausing the counterterm methoIntroduction70-2693/$ see front matter 2009 Elsevier B.V. All rights reserved.i:10.1016/j.physletb.2008.12.070at ScienceDirectters Bcate/physletbacesn of three Liouville-type dilaton potentials, we construct a new classtring solutions which produces a longitudinal magnetic eld in theetime. These solutions have no curvature singularity and no horizon, butthat the spinning string has a net electric charge which is proportionalresent the suitable counterterm which removes the divergences of thepotential. We also calculate the conserved quantities of the solutions by 2009 Elsevier B.V. All rights reserved.tter-like solutions of dilaton gravity has been found [4]. It wasown that the cosmological constant is coupled to the dilaton invery nontrivial way. With the combination of three Liouville-type10 s B 6stteofindisoofdibeneLedieprofco2.diIGwelsteteThtharthapbiVVHcaThbytrorbaexsusobefoabEiaceRThththcaredithrezeerWofIcOVasbocastfodiTThwthcosuinawinVasebeQwspthtamMJThmth2 A. Sheykhi / Physics Letterrings, that behave as superconductors and have interesting in-ractions with astrophysical magnetic elds [14]. The propertiesthese superconducting cosmic strings have been investigated[15]. It is also of great interest to generalize the study to thelaton gravity theory [16]. While exact magnetic rotating dilatonlution in three dimensions has been obtained in [17], two classesmagnetic rotating solutions in four [18] and higher dimensionallaton gravity in the presence of one Liouville-type potential haveen constructed [19]. Unfortunately, these solutions [18,19] areither asymptotically at nor (A)dS. The purpose of the presenttter is to construct a new class of static and spinning magneticlaton string solutions which produces a longitudinal magneticld in the background of anti-de Sitter spacetime. We will alsoesent the suitable counterterm which removes the divergencesthe action, and calculate the conserved quantities by using theunterterm method.Basic equationsOur starting point is the four-dimensional EinsteinMaxwell-laton action= 116Md4xg(R 2 V () e2 F F) 18Md3x( ), (1)here R is the scalar curvature, is the dilaton eld, F =A A is the electromagnetic eld tensor, and A is theectromagnetic potential. is an arbitrary constant governing therength of the coupling between the dilaton and the Maxwellld. The last term in Eq. (1) is the GibbonsHawking surfacerm. It is required for the variational principle to be well-dened.e factor represents the trace of the extrinsic curvature fore boundary M and is the induced metric on the bound-y. While = 0 corresponds to the usual EinsteinMaxwell-scalareory, = 1 indicates the dilaton-electromagnetic coupling thatpears in the low energy string action in Einsteins frame. For ar-trary value of in AdS space the form of the dilaton potential() is chosen as [4]() = 23(2 + 1)2[2(32 1)e2/ + (3 2)e2+ 82e(1/)]. (2)ere is the cosmological constant. It is clear that the cosmologi-l constant is coupled to the dilaton eld in a very nontrivial way.is type of the dilaton potential was introduced for the rst timeGao and Zhang [4]. They derived, by applying a coordinatesansformation which recast the solution in the Schwarzschild co-dinates system, the static dilaton black hole solutions in theckground of (A)dS universe. For this purpose, they required theistence of the (A)dS dilaton black hole solutions and extractedccessfully the form of the dilaton potential leading to (A)dS-likelutions. They also argued that this type of derived potential canobtained when a higher dimensional theory is compactied tour dimensions, including various supergravity models [20]. In thesence of the dilaton eld the action (1) reduces to the action ofnsteinMaxwell gravity with cosmological constant. Varying thetion (1) with respect to the gravitational eld g , the dilatonld and the gauge eld A , yields = 2 + 12gV ()+ 2e2(F F 14g F F), (3)spof72 (2009) 1011052 = 14V2e2 F F , (4)(ge2 F)= 0. (5)e conserved mass and angular momentum of the solutions ofe above eld equations can be calculated through the use ofe substraction method of Brown and York [21]. Such a procedureuses the resulting physical quantities to depend on the choice ofference background. A well-known method of dealing with thisvergence for asymptotically AdS solutions of Einstein gravity isrough the use of counterterm method inspired by AdS/CFT cor-spondence [22]. In this Letter, we deal with the spacetimes withro curvature boundary, Rabcd( ) = 0, and therefore the countert-m for the stress energy tensor should be proportional to ab .e nd the suitable counterterm which removes the divergencesthe action in the form (see also [23])t = 18Md3x(1l+6V ()2). (6)ne may note that in the absence of a dilaton eld where we have() = 2 = 6/l2, the above counterterm has the same formin the case of asymptotically AdS solutions with zero-curvatureundary. Having the total nite action I = IG + Ict at hand, onen use the quasilocal denition to construct a divergence freeress-energy tensor [21]. Thus the nite stress-energy tensor inur-dimensional Einstein-dilaton gravity with three Liouville-typelaton potentials (2) can be written asab = 18[ab ab +(1l+6V ()2) ab]. (7)e rst two terms in Eq. (7) are the variation of the action (1)ith respect to ab , and the last two terms are the variation ofe boundary counterterm (6) with respect to ab . To compute thenserved charges of the spacetime, one should choose a spacelikerface B in M with metric i j , and write the boundary metricADM (ArnowittDeserMisner) form:b dxa dxb = N2 dt2 + i j(d i + V i dt)(d j + V j dt),here the coordinates i are the angular variables parameteriz-g the hypersurface of constant r around the origin, and N andi are the lapse and shift functions, respectively. When there isKilling vector eld on the boundary, then the quasilocal con-rved quantities associated with the stress tensors of Eq. (7) canwritten as() =Bd2x Tabnab, (8)here is the determinant of the metric i j , and na are, re-ectively, the Killing vector eld and the unit normal vector one boundary B. For boundaries with timelike ( = /t) and ro-tional ( = /) Killing vector elds, one obtains the quasilocalass and angular momentum=Bd2x Tabnab, (9)=Bd2x Tabnab. (10)ese quantities are, respectively, the conserved mass and angularomenta of the system enclosed by the boundary B. Note thatey will both depend on the location of the boundary B in theacetime, although each is independent of the particular choicefoliation B within the surface M.s B 63.EqthdsThdistdila2[ggaariscoasonrirotobeFwisshinas222wtr2RThRSuwwstdishfwrethtrAdshanloneancasizepostisf(wchthrer2WdswRfROinetlirThcuA. Sheykhi / Physics LetterStatic magnetic dilaton stringHere we want to obtain the four-dimensional solution ofs. (3)(5) which produces a longitudinal magnetic elds alonge z direction. We assume the following form for the metric [11]2 = 2l2R2()dt2 + d2f ()+ l2 f ()d2 + 2l2R2()dz2. (11)e functions f () and R() should be determined and l has themension of length which is related to the cosmological con-ant by the relation l2 = 3/. The coordinate z has themension of length and ranges < z < , while the angu-r coordinate is dimensionless as usual and ranges 0

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