General Relativity and Gravitation manuscript No. (will be inserted by the editor) Brane Cosmology And Motion Of Test Particles In Five-Dimensional Warped Product Spacetimes Sarbari Guha1, Subenoy Chakraborty2 1 Department of Physics, St. Xavier’s College (Autonomous), Kolkata 700 016, India 0 e-mail: [email protected] 1 2 Department of Mathematics, Jadavpur University, Kolkata 700032, India 0 e-mail: [email protected] 2 n Received: January 5, 2010/ Accepted: date / Published online: date a (cid:13)c The Author 2009 J 5 Abstract In the ”braneworld scenario” ordinary standard model matter ] c andnon-gravitationalfieldsareconfinedbysometrappingmechanismtothe q 4-dimensionaluniverseconstitutingtheD3-braneswhichareembeddedina - (4 + n)-dimensional manifold referred to as the ’bulk’ (n being the number r g of extra dimensions). The notion of particle confinement is necessary for [ theorieswithnon-compactextradimensions,otherwise,theparticleswould 4 escapefromour4-dimensionalworldalongunseendirections.Inthispaper, v we have considered a five-dimensional warped product space-time having 2 an exponential warping function which depends both on time as well as on 7 the extra coordinates and a non-compact fifth dimension. Assuming that 0 the lapse function may either be a constant or a function of both time and 5 . oftheextracoordinates,wehavestudiedthenatureofthegeodesicsoftest 2 particles and photons and have analyzed the conditions of stability in this 1 8 geometrical framework. We have also discussed the possible cosmology of 0 the corresponding (3 + 1)-dimensional hypersurfaces. : v i Key words Braneworlds · Geodesic motion · Cosmology. X r a 1 Introduction According to the string theory postulate, standard model matter and non- gravitational fields are confined by some trapping mechanism to the 4- dimensionaluniverseconstitutingtheD3-branes(4-dimensionaltimelikehy- persurfaces) that are embedded in a (4+n)-dimensional manifold referred 2 Sarbari Guha, Subenoy Chakraborty to as the ’bulk’ (n being the number of extra dimensions). This has led to a renewed interest in extra-dimensional theories of gravity. The success of the Kaluza-Klein [1] theory in particle physics led several workers to construct numerous models of non-compact higher-dimensional theories of physics [2]-[6], and eventually to the so-called ”braneworld scenario” [7]- [9]. In the braneworld models with non-compact extra dimension, particles and non-gravitational fields are assumed to be confined to the branes. At low energies, gravity is localized at the brane along with the particles but at high energies gravity ”leaks” into the higher-dimensional bulk and is propagated therein. These ”braneworld” models have been used to address several issues, one of which is to explain why the observable universe is found to be 4-dimensional [10],[11]. Consequently, it is necessary to deter- mine how closely the corresponding apparent 4-dimensional world resemble the observed world [12], and thus we need to verify the extent to which these theories satisfy the geodesic postulates concerning the motion of test particles and light rays. The notion of particle confinement is necessary for theories with non- compact extra dimensions, as otherwise, the particles would escape from our 4-dimensional world along unseen directions. In the classical context, confinementofatestparticletothebraneeliminatestheeffectsofextradi- mensions,therebyrenderingthemundetectable.Inthebraneworldscenario, thestabilityoftheconfinementofmatterfieldsatthequantumlevelismade possiblebyassuminganinteractionofmatterwithascalarfield.Itappears thatnon-gravitationalforcesactinginthebulkandorthogonaltothespace- timearenecessarytokeepthetestparticlesmovingonthebrane,thesource of these confining forces being interpreted in different ways [13]-[15]. The alternative explanation in case of higher-dimensional theories involve the invocation of geometrical mechanisms to support lower-dimensional con- finement. In this case, confinement is due purely to the classical gravi- tational effects, without requiring the presence of brane-type confinement mechanisms.Thishasledtotheinvestigationofwarpedproductspacesand their geometrical properties [16]. Randall and Sundrum [10],[11], achieved itthroughtheconstructionoffive-dimensionalwarpedproductspacesusing an exponential warp factor in a non-factorizable metric furnished with mir- rorsymmetry,evenwhenthefifthdimensionwasinfinite.Incaseofwarped product spaces, it has been observed that a general qualitative analysis of the behavior of massive particles and photons in the fifth dimension can be made from the knowledge of the warping function. A general picture of these geodesic motions has been obtained by using the natural decoupling occurring between motions in the brane and motion in the fifth dimension in case of such spaces [17],[18]. For a thin asymmetric braneworld, bulk and brane geodesics does not coincide in general. However, if appropriate energy conditions are satisfied by the matter confined to the brane, then in presence of mirror-symmetry, test particles can be confined by gravity to a small region about the brane [19],[20]. The Z -symmetric braneworlds 2 have their apparent and bulk geodesics coinciding on the brane. Although Brane Cosmology And Motion Of Test Particles 3 null bulk geodesic motion in the RS2-type braneworlds, as well as in the static universe in the bulk of a charged topological AdS black hole have been studied [21], the same is not found for timelike geodesics. Another important program in the study of higher dimensional models is the cosmological interpretation of the corresponding 4-dimensional ge- ometry. By use of the Einstein equations, it has been shown that [22] the embedding of a surface in a flat space of co-dimension one imposes the re- strictionthatthesurfacehasaconstantcurvature,ifitsdimensionisn>2. TheeffectiveequationsforgravityinfourdimensionswereobtainedbyShi- romizuetal[23]andsubsequentlycosmologicalsolutionshavebeenstudied by some authors [24],[25]. In this paper, we have considered RS-type braneworlds with the bulk in the form of a five-dimensional warped product space-time, having an ex- ponential warping function which depends both on time as well as on the extra coordinates and a non-compact fifth dimension. We know that the exponential warp factor reflects the confining role of the bulk cosmological constant [26] to localize gravity at the brane through the curvature of the bulk.Itis,therefore,possiblethatsuchlocalizationmayincludesometime- dependence and hence the choice. Employing the technique used by Dahia [17], we have obtained a mechanism for the confinement of geodesics on co-dimension one hypersurfaces, considering that the confinement is purely due to the classical gravitational effects. We have been able to obtain the description of the geodesic motions by using the natural splitting occurring between the motion in the extra dimension and the motion in the four- dimensional hypersurfaces. Such splitting helps us to use the phase space analysis to determine the nature of the geodesic motions in the neighbour- hood of the hypersurfaces and analyze the conditions of stability. Further, we have assumed the lapse function to be either a constant or a function of both time and of the extra coordinates. This is possible since, in our case, the square of the lapse function is identical to the metric coefficient for the fifth dimension, which may therefore be of such a type. Finally, the cosmological interpretations of the corresponding (3 + 1)-dimensional hypersurfaces have also been discussed on the basis of the field equations. 2 Geometric Construction of Five-Dimensional Warped Product Spaces A warped product space [27],[28], is constructed as follows: Let us consider two manifolds (Riemannian or semi-Riemannian) (Mm,h) and (Mn,h¯) of dimensions m and n, with metrics h and h¯ respectively. Given a smooth function f : Mn → ℜ (henceforth called the warping function), we can build a new Riemannian (or semi-Riemannian) manifold (M,g) by setting M = Mm ×Mn, which is defined by the metric g = e2fh h¯. Here (M, g) is called a warped product manifold. The case dim M = 4 corresponds L to (M, g) being a spacetime, and is called a warped product spacetime (or simply warped spacetime). 4 Sarbari Guha, Subenoy Chakraborty InlocalcoordinateszA,thelineelementcorrespondingtothebulkmetric is denoted by dS2 =g dzAdzB. (1) AB Theclassofwarpedgeometriesconsideredbyusisrepresentedingeneral by the line element of the form dS2 =e2fh dxαdxβ +h¯ dyadyb, (2) αβ ab where the extra coordinates are represented by ya, the coordinates on the m-dimensional submanifold by xµ, f is a scalar function, h = h (x) is αβ αβ the warp metric on the submanifold of dimension m and h¯ is the metric ab representing the extra dimensional part. In this paper, we consider m = 4 and n = 1, so that M = M4 ×M1, where M4 is a Lorentz manifold with signature (+ - - -). The canonical metric due to Mashhoon et. al. [29],[30] is more general than the Randall- Sundrum metric and encompasses all the metrics usually considered in the braneworld and Induced Matter Theory (IMT) approaches [6],[31]-[34]. For this,thephysicalmetricin4-dimensionisassumedtobeconformallyrelated to the induced metric [35], i.e. dS2 =Ωds2+εΦ2dy2, (3) where Ω is the ”warp” factor that satisfies the condition Ω > 0 and the scalar Φ which normalizes the vector normal to the hypersurfaces y = constant is known as the lapse function. We assume that the brane is de- fined by the y = constant hypersurface, where y is a Gaussian normal coordinate orthogonal to the brane, representing the fifth dimension, which is non-compact and curved (warped) [26],[11]. In general, the warp factor is assumed to be a function of extra coordi- nates only, but here we assume that it is a function of both time, as well as of the extra coordinates. As in the RS models, gravity is localized on the branethroughthecurvatureofthebulk.Weknowthatthebulkcosmologi- cal constant acts to squeeze the gravitational field closer to the brane, with the exponential warp factor reflecting the confining role of the bulk cos- mological constant [26]. Mathematically, the time dependence of the warp factor does not affect its smooth nature. Physically, it takes into account thepossibilitythattheconfiningroleofthebulkcosmologicalconstantand the curvature of the bulk may have some dependence on time, in addition to their dependence on the extra dimensional coordinate. When the extra dimension is spacelike, we have ε = −1. In our case, Φ2 = p = g (which yy means that the square of the lapse function is identical to the metric coef- ficient for the fifth dimension) and Ω =e2f(t,y). The expression for the line element therefore takes the form dS2 =e2f(t,y)h dxαdxβ −pdy2. (4) αβ Brane Cosmology And Motion Of Test Particles 5 Here, p may either be a constant or a function of coordinates. We shall consider two cases, viz. p=1 and p=p(t,y) (for example, the second type canbefoundin[36]).Totakeintoaccountthetime-dependenceofthewarp factor, we assume the scalar function f(t,y) to be of the form f(t,y)=at+lln(cosh(cy)) (5) where a, l and c are constants. We will not make any assumption on the topology of the extra dimension, so that, in principle −∞<y <+∞. 3 Geodesic Motion Dahia [17] demonstrated oscillatory confinement of massive particles and light rays for branes of finite thickness in the context of 4 + 1-dimensional warped product spaces, whereby, the particles in the four-dimensional hy- persurface can oscillate about the hypersurface, while remaining close to it. They have shown that such behaviour can occur for large classes of bulks which possess warped product geometries. Writing the geodesic equations forwarpedproductspaces,theyshowedthattheequationthatdescribesthe motion in 5D decouples from the rest. They made a qualitative analysis of the motions by rewriting the geodesic equation in the fifth dimension as an autonomous planar dynamical system and then employed the phase plane analysis to study the motion of particles with nonzero rest mass and pho- tons respectively. Consequently they could draw very general conclusions about the possible existence of confined motions and their stability in the neighbourhood of the hypersurfaces. In our analysis, which now follows, we have used this method to arrive at the results. The equations of geodesics in the 5-dimensional space M is given by d2zA dzB dzC +(5)ΓA =0, (6) dλ2 BC dλ dλ whereλisanaffineparameterand(5)ΓA arethe5-dimensionalChristoffel BC symbols of the second kind defined by (5)ΓA = 1gAD(g +g − BC 2 DB,C DC,B g ).Ifwedenotethefifthcoordinatez4 byy andtheremaining”space- BC,D time” coordinates zµ by xµ i.e. zA = (xµ,y), it can be shown that the 4-dimensional part of the geodesic equations (6) can be rewritten in the form [17] d2xµ dxαdxβ +(4)Γµ =ξµ, (7) dλ2 αβ dλ dλ where dy 2 dxα dy 1 dxαdxβ ξµ =−(5)Γµ −2(5)Γµ − gµ4(g +g −g ) , 44 dλ α4 dλ dλ 2 4α,β 4β,α αβ,4 dλ dλ (cid:18) (cid:19) (8) with (4)Γµ = 1gµν(g +g −g ). αβ 2 να,β νβ,α αβ,ν 6 Sarbari Guha, Subenoy Chakraborty We consider that the bulk spacetime is a 5-dimensional warped product spacetime in the braneworld scenario, represented by the line element (4) with the warping function given by (5). Let us now assume that this bulk space-time is foliated by a family of hypersurfaces defined by the equation y =constant. The geometry of each such leaves of foliation denoted by, say y =y will be determined by the induced metric 0 ds2 =gαβ(x,y0)dxαdxβ =e2f(t,y0)hαβ(x)dxαdxβ =ηαβdxαdxβ. (9) The extrinsic curvature of such a hypersurface is given by 2K = αβ −(∂ηαβ). In that case, the quantities (4)Γµ which appear on the left side ∂y αβ of Eq. (7), are the Christoffel symbols associated with the induced metric in the leaves of foliation defined above. 3.1 Case 1: p=1 For the class of warped geometries given by Eq. (4) and (5), we can easily see that for p=1 we have (5)Γµ =0 and (5)Γµ = 1gµν ∂gνα . 44 α4 2 ∂y In this case, the 4-dimensional part of the geodesic eq(cid:16)uation(cid:17)s reduces to the form d2xµ dxαdxβ dxµ dy +(4)Γµ =−2f′ . (10) dλ2 αβ dλ dλ dλ dλ whereaprimedenotesdifferentiationwithrespecttoy.Similarly,thegeodesic equation for the fifth coordinate y in this warped product space becomes d2y dxαdxβ +f′e2fh =0. (11) dλ2 αβ dλ dλ The above two geodesic equations (10) and (11) are identical to those inRef.[17].Thishappensduetothespecialnatureofthewarpingfunction chosen by us. We like to point out that the type of warping function in our caseisdifferentfromthatusedin[17],astheyconsideredawarpingfunction which depends on the extra coordinates only. So, we can conclude that the geodesic equations do not depend on the explicit form of the warping function for this particular choice when p = 1. The result can be stated in the form of the following proposition: Proposition1: Thegeodesic equations forthe 4-dimensional spacetime and for the fifth dimension remains the same as in the case with a warp factor which depends only on the extra coordinate, even after the inclusion of the additional linear time-dependence of the warping function, provided g =1. yy Astudyofthe5-dimensionalmotionofparticleswithnon-zerorestmass nearthehypersurfacecanbedonebyconsideringthe5-dimensionaltimelike Brane Cosmology And Motion Of Test Particles 7 geodesics (g dzAdzB = 1), for which the above equation can be easily AB dλ dλ decoupled fromthe4-dimensional spacetime coordinates asin [17], yielding d2y dy 2 ′ +f 1+ =0. (12) dλ2 dλ (cid:18) (cid:19) ! The motion of photons in 5-dimension can similarly be analyzed by consideringthe null geodesics, for which (g dzAdzB =0) and we get from AB dλ dλ Eq. (11) d2y dy 2 ′ +f =0. (13) dλ2 dλ (cid:18) (cid:19) Equations(12)and(13)aresecondorderordinarydifferentialequations, which can be solved if we know the nature of the warping function f. 3.1.1 Motion of particles in four-dimensional spacetime To determine the nature of motion of particles in the four-dimensional world, we need to find the constraints on the warping function that will determine whether the geodesics on the bulk manifold may coincide with those on the hypersur- face or not. Considering the expression for the extrinsic curvature of the hypersurfaceinthiscase,wefindthattheresultisidenticaltothoseinRef. [18]. 3.1.2 Motion of particles in the fifth dimension A qualitative analysis of themotioninfifthdimensioncanbedonewithoutactuallysolvingEqs.(12) and (13) by defining q = dy and investigating the autonomous dynamical dλ systems [37] dy =q (14) dλ dq =P(q,y) (15) dλ with P(q,y)=−f′(ǫ+q2), where ǫ=1 for timelike geodesics and ǫ=0 for null geodesics. The equilibrium points of the system of equations (14) and (15) are given by dy =0 and dq =0. Knowledge of these points along with dλ dλ theirstabilitypropertiescanprovidealotofinformationaboutthebehavior allowed by this dynamical system. For the warping function considered by us, the analysis is very similar to the one done by Dahia et. al. [17]. Therefore,wefindthattheresultsobtainedin[17]and[18]aretrueeven if we include a linear time-dependence of the warping function, provided g =1. yy 8 Sarbari Guha, Subenoy Chakraborty 3.2 Case 2: p=p(t,y) The geodesic motion of the test particle in the bulk spacetime is still de- scribed by the equations (6). Let us consider a bulk metric of the form: dS2 =e2f(t,y) dt2−btdr2−btr2dθ2−btr2sin(θ)2dφ2 −p(t,y)dy2 (16) where b is anoth(cid:0)er constant and f is given by Eqn. (5(cid:1)) as f(t,y) = at+ lln(cosh(cy)). On substituting the explicit expression for the bulk metric, the geodesic equations reduce to the following five second order differential equations: d2t dt 2 (2at+1)b dr 2 dθ 2 dφ 2 +a + +r2 +sin2θ =0, dλ2 dλ 2 dλ dλ dλ (cid:18) (cid:19) "(cid:18) (cid:19) ((cid:18) (cid:19) (cid:18) (cid:19) )# (17) d2r (2at+1) dt dr dθ 2 dφ 2 + −r +sin2θ =0, (18) dλ2 2t dλdλ dλ dλ ((cid:18) (cid:19) (cid:18) (cid:19) ) d2θ dθ (2at+1) dt 1dr dφ 2 + + −sinθcosθ =0, (19) dλ2 dλ 2t dλ rdλ dλ (cid:26) (cid:27) (cid:18) (cid:19) d2φ dφ (2at+1) dt 1dr cosθ dθ + + + =0, (20) dλ2 dλ 2t dλ rdλ sinθ dλ (cid:26) (cid:27) d2y 1 ∂p dt dy ∂p dy 2 + 2 + =0, (21) dλ2 2p ∂t dλdλ ∂y dλ ( (cid:18) (cid:19) ) Let us now analyse the equations of 5-dimensional motion of both pho- tonsaswellastheparticlesofnon-zerorestmass,nearthehypersurface.For that we consider the equations of the 5-dimensional geodesics to be given by dzAdzB g =ǫ, (22) AB dλ dλ where, ǫ = 1 for timelike geodesics and ǫ = 0 for null geodesics. Using the expression for the bulk metric, we get from (22), 2 ǫ+p dy dr 2 dθ 2 dφ 2 1 dt 2 dλ b +r2 +sin2θ =  − (cid:26) (cid:16) (cid:17) (cid:27), dλ dλ dλ t dλ e2f "(cid:18) (cid:19) ((cid:18) (cid:19) (cid:18) (cid:19) )# (cid:18) (cid:19)      (23)  Substituting from (23) in (17), we get Brane Cosmology And Motion Of Test Particles 9 d2t dt 2 (2at+1) dt 2 1 dy 2 +a + − ǫ+p =0, (24) dλ2 dλ 2t dλ e2f dλ (cid:18) (cid:19) "(cid:18) (cid:19) ( (cid:18) (cid:19) )# Onceagain,aqualitativeanalysisofthemotionofparticlescanbedone without actually solving the Eqs. (17) to (21) by defining a dynamical sys- tem in the following way: dt U = , (25) dλ and dy V = , (26) dλ which, when substituted in (24) and (21) yields the equations dU 1 1 + U2 2a+ − ǫ+pV2 =0, (27) dλ 2t e2f (cid:20) (cid:18) (cid:19) (cid:21) (cid:8) (cid:9) and dV 1 ∂p ∂p + 2UV +V2 =0. (28) dλ 2p ∂t ∂y (cid:26) (cid:27) Thesetofequations(25)to(28)areusedtoanalysetheevolutionofthe above dynamical system. The trajectories of this dynamical system can be describedina2-dimensionalphasespaceintermsofthevariablesU andV. The system of equations (25) to (28) represent a real non-linear dynamical system [38],[39] of the type dt =U, dλ dy =V, dλ dU =P(U,V,t,y), dλ dV =Q(U,V,t,y), (29) dλ where 1 1 P(U,V,t,y) =−U2 2a+ + ǫ+pV2 2t e2f (cid:18) (cid:19) 1 ∂p ∂p(cid:8) (cid:9) Q(U,V,t,y) =− 2UV +V2 2p ∂t ∂y (cid:26) (cid:27) withǫ=1,0fortimelikeandnullgeodesicsrespectively.BothPandQhave continuousfirstpartialderivativesforall(U,V).Theevolutionofthesystem inthe(U,V)phaseplaneisgivenbythesolutionoftheset(29).Itisevident 10 Sarbari Guha, Subenoy Chakraborty that the solution of the system for a given value of the parameter λ is no longer unique, thereby, giving rise to very complicated trajectories. In such cases, the critical point of the system must be a zero of the equations for all times for which the system is defined. Since no linear terms are present in the set (29), there will generally be several critical points of the system. We know that the zeros of the system correspond to the fixed points of the phase trajectories. For a nonautonomous system, generally each point of the phase space is intersected by many distinct trajectories. Therefore the collection of phase trajectories as well as the analysis for the identification ofthefixedpointsbecomesextremelycomplicated[40].Thedynamicsofthe phase trajectories is then strongly controlled by the underlying topology of the manifold under consideration. Such an analysis is beyond the scope of thepresentpaper.Onesimplemethodistomakesuitabletransformationsto reducethenon-autonomoussystemstoautonomousonesbyappendingtto the depending variables. However, such a transformation has the drawback that the non-autonomous problem frequently looses its original algebraic structure. Hence it is not considered here. 4 Phase trajectories and Critical Points of the system for the case p=p(t,y) In spite of the fact that we are dealing with a nonautonomous system, we can still interpret the nature of the trajectories with the help of a simple analysis. The equation dV Q = , (30) dU P at some given value of y specifies the phase trajectory of the system in the (U,V) phase plane, provided P 6= 0 at this point. Generally P 6= 0 except at the point U = V = 0 for null geodesics (which, as we shall find later, represents the critical point of the system in such a case). Substituting the explicit forms of P and Q in (30) and integrating, we arrive at the result 1∂p 1 1 ∂p p ǫ U2V −(2a+ ) +UV2 +V3 +V +K =0, (31) p ∂t 2t 2p∂y 3e2f e2f (cid:20) (cid:21) where K is the arbitrary constant of integration. Setting this constant to zero and simplifying, we can recast eqn. (31) into the form AU2+BUV +CV2+D =0, (32) whereA,B,C,andDarefunctionsofcoordinates,specifically,tandy.The above equation is useful to understand the phase trajectories even when no fixed points are found. For null geodesics, D =0 and (32) reduces to AU2+BUV +CV2 =0. (33)