Electric Polarization Induced by Gravity in Fat Branes
aa r X i v : . [ h e p - t h ] N ov Electric Polarization Induced by Gravity in Fat Branes
F. Dahia
Dep. of Physics, Univ. Fed. da Para´ıba, Jo˜ao Pessoa, Para´ıba, Brazil and Dep. of Physics,Univ. Fed. de Campina Grande, Campina Grande, Para´ıba, Brazil.
Alex de Albuquerque Silva
Dep. of Physics, Univ. Fed. da Para´ıba, Jo˜ao Pessoa, Para´ıba,Brazil and Dep. of Physics, Univ. Fed. de Campina Grande, Sum´e, Para´ıba, Brazil.
C. Romero
Dep. of Physics, Univ. Fed. da Para´ıba, Jo˜ao Pessoa, Para´ıba, Brazil.
In the fat brane model, also known as the split fermion model, it is assumed that leptons andbaryons live in different hypersurfaces of a thick brane in order to explain the proton stability withoutinvoking any symmetry. It turns out that, in the presence of a gravity source M , particles willsee different four-dimensional (4D) geometries and hence, from the point of view of 4D-observers,the equivalence principle will be violated. As a consequence, we show that a hydrogen atom inthe gravitational field of M will acquire a radial electric dipole. This effect is regulated by theHamiltonian H d = − µ A · δ r , which is the gravitational analog of the Stark Hamiltonian, where theelectric field is replaced by the tidal acceleration A due to the split of fermions in the brane and theatomic reduced mass µ substitutes the electric charge. PACS numbers: 11.10.Kk, 04.50.-h, 04.40.-b, 31.15.aj
I. INTRODUCTION
In the braneworld models, our four-dimensional spacetime is viewed as a submanifold isometrically embedded inan ambient space with higher dimensions. The basic feature of this scenario is the confinement of matter and fieldsin the brane (when they have an energy lower than a certain level which is expected to be of the order of 1TeV, atleast), while gravity has access to all dimensions [1–3]. In this framework, the extra dimensions might be much largerthan Planck length. As a matter of fact, in the RSII model it was shown that the extra dimension might even havean infinity length without any phenomenological conflict [3].A modified version of the RSII model, known as fat brane, assumes that the brane has a thickness and that leptonsand baryons live in different hypersurfaces of the thick brane [4]. The original motivation of this model is to explainthe stability of protons without using any symmetry. The conservation of the baryonic number, that protects theproton from decaying, is just a consequence of the split of fermions in the thick brane, since this separation producesa strong suppression in couplings between quarks and leptons. On the other hand, gauge fields have access to all thebrane. Thus, if the thickness of the brane is of TeV order then we might expect that traces of the extra dimensionscould be detected in experiments at LHC [4, 5].By virtue of the confinement, particles do not see the geometry of the whole bulk but they feel the induced metricon the hypersurface where they live. If there is no gravity source then all fermions see the same Minkowski spacetime.However, under the gravitational influence of a mass M in the brane, the induced metric will be different for distinctslices. This means that leptons and baryons will feel different geometries. From the point of view of 4D-observers(not aware of the extra dimensions), this will be seen as a violation of the equivalence principle, since particles inthe same four-dimensional brane coordinates will feel different gravitational accelerations. This tidal acceleration A ,due to the split of fermions in the extra dimension, produces an internal force in a hydrogen atom, inducing, in thisway, an electric dipole in a tangential direction of the brane. As we shall see, the Hamiltonian associated with theinteraction between the atom and the gravitational field of M contains a dipole term which has exactly the same formof the Stark Hamiltonian, H d = − µ A · δ r (where δ r denotes the internal relative coordinates), in the first order of GM , where G is Newton’s gravitational constant. II. GRAVITY IN THICK BRANES
In the context of thick brane models, the brane is usually described as a domain wall generated by a certain scalarfield φ [1, 6]. The presence of a mass M (a star or a black hole) localized in the brane will affect the domain wallsolution. An exact solution for this system is not known so far. Solutions for lower dimension (2-brane) are known[7] and some numeric solutions for a black hole confined to a thin 3-brane were found recently [8]. However, in thecase of a mass M trapped in a thick 3-brane, no exact solution of the Einstein equations coupled to the scalar fieldequation has been found yet. However, based on the symmetry of the problem, it is expected that a non-rotatingmass should give rise to an axisymmetric, static spacetime in five dimensions. In such spaces, as is well known, thereare coordinates in which the metric assumes the Weyl canonical form[9], which can be put, by means of a convenientcoordinate transformation, in a Gaussian form adapted to the brane: ds = − e A ( r,z ) dt + e B ( r,z ) dr + e C ( r,z ) d Ω + dz , (1)where z = 0 gives the localization of the center of the brane. Due to symmmetry the scalar field will depend only onthe coordinates r and z , i.e., φ = φ ( r, z ).Even when M = 0, the Einstein equations (5) G µν = κT ( φ ) µν coupled to the scalar field equation (cid:3) φ − V ′ ( φ ) = 0in five dimensions are not easy to solve. Here κ is the five-dimensional gravitational constant, T ( φ ) µν is the usualenergy-momentum tensor of the scalar field subjected to the potential V ( φ ) and c = 1. In some special situations,when the potential is conveniently choosen, an exact solution of a self-gravitating domain wall can be obtained. Forexample, taking V ( φ ) = λ/ × (cid:0) φ − η (cid:1) − βλ/ η × φ (cid:0) φ − η (cid:1) , the solution is [10] : ds = e a ( z ) (cid:0) − dt + dr + r d Ω (cid:1) + dz , (2)2 a ( z ) = − β ln cosh zε − β tanh zε , (3) φ = η tanh zε , (4)where ε = 2 /λη , β = κη /
9. This solution can be interpreted as a regularized version of the RSII brane model.Indeed, taking the parameter ε (the thickness of the wall) equal to zero, while keeping the condition ε/ β = const. ≡ ℓ ( ℓ defines the curvature radius of AdS space) the RSII solution is recovered[10].Let us now consider a mass M describing a body or a black hole confined in the core of the domain wall. Thepresence of this gravity source certainly will modify both the original metric and the scalar field. Considering theamount of technical difficulty to solve this problem exactly, let us try to employ approximation methods[11]. At largedistances from M , where the weak field regime is valid, the modification can be treated as a small perturbation ofthe original solution. In this case, we can write ds = e a (cid:2) − (1 + f ) dt + (1 + m ) dr + r (1 + h ) d Ω (cid:3) + dz , (5) φ = η (cid:16) tanh zε + k (cid:17) , (6)where f, m, h and k , which are functions of r and z , give the small corrections of the unperturbed metric andthe scalar field. It may happen that, due to the perturbation, the coordinate z will not be adapted to the levelsurface of the scalar field anymore. For instance, the center of the domain wall ( φ = 0), which originally coincideswith z = 0, is now, in the first order approximation, given by the equation z = − εk ( r, z = 0. The great advantage ofworking in these coordinates is the fact that “initial conditions”, i.e., the value that the correction functions assumein the center of wall, can be easily established. For example, as the center corresponds to z = 0, then, we shouldhave k ( r,
0) = 0. Another important condition can be immediately deduced based on the expectation that the metricshould be symmetric with respect to the center of the wall. As ∂ z is the normal vector of the hypersurface z = 0, then,it follows that in the center of the domain the first derivative with respect to z should be zero: f z = m z = h z = 0 . Theremaining set of the initial conditions, i.e, f ( r, , m ( r, , h ( r,
0) and k z ( r, M localized in the thin brane in the first approximation order of GM . With the purpose to obtain aconnection with this thin brane solution, we are going to impose that in z = 0 our solution reproduces Garriga andTanaka’s result. This condition allows us to determine f ( r, , m ( r, , h ( r, k z ( r,
0) = 0 follows from the ( zz )-component of the Einstein equations, which corresponds to a constraint equation.Now by using the dynamical components of the Einstein equations (those that involve second derivatives with respectto z ) to propagate the initial conditions into the bulk, we find the metric around the center of the brane as a powerseries in z . In the first correction order, the line element for z << ε and r >> GM is given by ds = − e a (cid:18) − GMr (cid:18) ℓ r − ℓ r z (cid:19)(cid:19) dt + e a (cid:18) GMr (cid:18) ℓ r + ℓ r z (cid:19)(cid:19) dr + e a (cid:18) r − GM ℓ r z (cid:19) d Ω + dz , (7)and the scalar field solution is simply φ = η tanh z/ε in the first approximation order of GM . III. ELECTRIC DIPOLE INDUCED BY GRAVITY
It is well known that the confinement of matter in a thick brane can be obtained by means of a Yukawa-typeinteraction between the Dirac field and the scalar field [1]. Under this interaction, the wave packet of a massless Diracfield has a peak at the center of the domain wall and decay exponentially in the extra dimension. When a non-nullmass is taken into account, the peak is shifted by a certain amount that depends on the particle mass [4]. Therefore,electrons and quarks will be localized in different slices of the brane.In this scenario, particles are in a bound state with respect to the transversal direction, but they might be free in theparallel direction. If we want to study the motion of a particle along the brane it is convenient to consider a classicalapproach for this problem. In order to achieve this, it is necessary first to provide a mechanism of confinement of testparticles to the brane, which may simulate classically the confinement of the matter field. In Ref. [13], a particularLagrangian based on the Yukawa interaction was proposed to describe the particle’s motion in this context. It wasthen shown that the new Lagrangian has the effect of increasing the effective mass of the particle due to the interactionwith the scalar field, and this modification is sufficient to ensure the localization of the particle. This new Lagrangianwas defined as L = − (cid:0) m + h ϕ (cid:1) ˜ g AB ˙ x A ˙ x B , where m is the rest mass of the particle in a free state and h is thecoupling constant of the interaction. The split of fermions in different slices can be added in our model by redefiningthe Lagrangian as L = − (cid:16) m + h ( ϕ + αm ) (cid:17) ˜ g AB ˙ x A ˙ x B , (8)where α is a new parameter related to the interaction. Calculating the 5 D -momentum P A = ∂L/∂ ˙ x A of the particle,we find P A P A = − m − h ( ϕ + αm ) ≡ − m ef , for massive test particles (cid:0) ˜ g AB ˙ x A ˙ x B = − (cid:1) . Then, we can verifydirectly that the effective mass m ef is now affected by the presence of the scalar field. Of course, the usual relationis recovered by turning off the interaction, i.e., by setting h = 0. It is worthy of mention that a similar kind ofLagrangian was also employed, in a different context, to describe the interaction between test particles and dilatonicfields [14].Due to the interaction with the scalar field, particles will move with a proper acceleration A A = − Π AC ˜ ∇ C ln M ,which is the gradient of the mass potential M ≡ e a m ef /m projected by the tensor Π AC ≡ ˜ g AC + ˙ x A ˙ x C into thefour-space orthogonal to the particles’ proper velocity ˙ x A .When there is no additional gravity source ( M = 0) the metric in the bulk is given by (2) and the transversalmotion decouples from the motion in the tangential direction. In this case, the first integral of the equation of motionin the z direction can be obtained directly, also implying that the transversal motion is bounded by the mass potential M according to the equation M ˙ z = E − M , where E is a constant related to the initial condition of the motion.The function M plays the role of a confining potential and has a stable equilibrium point when the parameters h and α satisfy appropriate conditions. It is important to stress here that the equilibrium position z depends on the massof the particle. If we admit that z is small in comparison with the brane thickness ε , then we can show that theparticle with mass m is confined to a slice approximately specified by z = √ αm √ εℓκ/ . As we can see, electronsare localized closer to the center than quarks. Of course we can manipulate the Lagrangian in order to get the inverseresult, namely, quarks stuck in the center of the brane and electrons in an upper slice. For our purpose, what isimportant here is that we can formulate a simple classical model which has the essential characteristic of the fat branemodel, namely, the split of leptons and baryons in different slices of the brane. On the other hand, in the tangentdirection both fermions move freely in the same induced four-dimensional Minkowski spacetime.This situation changes when we consider the presence of a mass M in the brane. For the sake of simplicity, hereafterwe are going to admit that quarks are confined in the center of the brane, while electrons will be stuck in some slice z . With this choice, we can admit that almost all the mass M is localized in the center of the brane, since the baryonsare stuck in that hypersurface. It follows then that the metric will given by (7) and therefore leptons and baryonswill see different four-dimensional geometries, since the induced metric depends on the value of z. As a consequence,the equivalence principle will be violated from the 4D perspective and this fact can produce interesting phenomenain the brane as, for example, the induction of an electric dipole in a hydrogen atom by gravitational effects. In orderto investigate this, let us consider the motion of a particle in the spacetime with metric (7). Due to the symmetry ofthis spacetime, the energy E and the axial angular momentum L will be conserved and the particle’s motion shouldobey the following equations − m ef ˜ g tt ˙ t = E (9) m ef ˜ g φφ ˙ φ = L (10)We can also verify that θ = π is a solution of the equations. On the other hand, the motion in the z directionis not decoupled from the radial motion. However, if the particle is in a circular motion or static ( r = const ) thenthe effect of the mass M is just to modify the equilibrium position of the particle by an amount of the order of GM .Analyzing the radial motion we can see that there are stable circular orbits for appropriate values of energy andangular momentum. Considering this, it follows from equations (9) and (10) that the angular frequency of a particlein a circular motion of radius r is given by: ω = ˙ φ ˙ t ! = GMr (cid:18) ℓ r − ℓ r z (cid:19) . (11)This clearly shows that the angular frequency depends on the mass of the particle through z . From the perspectiveof 4D observers, this mass dependence will be seen as a violation of the equivalence principle. In fact, in a circularorbit with the same radius r , a proton (stuck in the center of the brane) will move faster than an electron.Now if we consider a hydrogen atom orbiting the mass M with a certain angular frequency or static, then basedon the previous reasoning we are led to expect that the radius of the electron’s orbit and the radius of the proton’sorbit should be different. As matter of fact, the center of the negative charge tends to circulate in an outer orbit incomparison with the proton’s orbit, as we shall see next.In the weak field regime, we can consider that the gravitational potential of the mass M is given by ϕ = − GMr (cid:18) ℓ r − ℓ r z (cid:19) . (12)Here it is important to note that, based on the equation (9), the warping factor can be incorporated in a redefinition ofthe mass of the particle and for this reason we might define the potential ϕ without using it. Thus, the non-relativisticHamiltonian of the hydrogen atom is H = P p m p + P e m e + m p ϕ p + m e ϕ e + U, (13)where U is the potential energy of the electric and gravitational interaction between the proton and the electron.Introducing the coordinates of the center of mass R and the relative coordinates δ r = r p − r e , we can verify that H contains an unusual dipole term due to the fact that electrons and protons live in different slices of the brane. Thisnew term H d has the same form of the Stark Hamiltonian H d = − µ A · δ r , (14)where the tidal acceleration A between the electron and the proton replaces the electric field. We must emphasizethat A is the relative gravitational acceleration between the electron and the proton when they have the same four-dimensional brane coordinates. It is clear that the origin of A is the split of fermions in the extra dimensions. It canalso be shown that A = − GM (cid:18) ℓ R z (cid:19) R . (15)We should mention that, considering the procedure usually adopted to deal with atomic systems in gravitational field[15] (which is based on the expansion of metric around the center of the mass of the system in Fermi coordinates),this tidal acceleration comes from the term R αβγλ ; σ u β u γ s λ s σ , where u β is the proper velocity of the center of massand s σ = (0 , , , , z ) is the separation vector (or the relative coordinates). The semi-colon indicates the covariantderivative of the five-dimensional Riemmann tensor R αβγλ .The Hamiltonian H d will induce an electric dipole p in the hydrogen atom, whose direction tends to be alignedwith the tidal field. In order to make a rough estimate of the dipole magnitude we are going to describe the atomfollowing a semi-classical approach. First, we admit that the electron charge is uniformly distributed in a cloudaround the proton. The tidal acceleration inside the hydrogen atom will produce a radial separation between theproton and the center of the negative charge, giving rise to an electric force F between them. To calculate F , we aregoing to make some considerations. We begin by recalling that, according to the fat brane model, the particle stateis described by a very narrow wave packet along the extra dimension and thus the wave function can be consideredas a delta-distribution in z direction. So we can assume that the electronic cloud is spread in a spherically symmetric3-volume of the slice z . A sphere attached to a hypersurface constitutes a kind a 3D-disk from the bulk perspective.We are going to admit that the radius of this disk is equal to the Bohr radius a . Additionally, it is reasonable toexpect that the radial separation δr is not greater than z , and therefore, in our calculations, we have to take intoaccount that in such domain the electric force has a 5D behavior. Considering all these assumptions, and also that z << a , we can show that F = (cid:0) ke ε/π a (cid:1) δ r , where k is the known electrostatic constant in 4D spacetime.Since at the equilibrium state the internal electric force and the tidal force are balanced, it follows that theelectric dipole induced by gravity is proportional to the tidal acceleration p = (cid:0) π a µ/ keε (cid:1) A . The proportionalityfactor defines the electric polarizability of the hydrogen atom induced by gravity and it can be written as α G = (cid:0) π aµ/ εe (cid:1) α E , where α E is the usual electric polarizability of the atom induced by electric fields. In order to makea comparison between the polarization induced by gravity and that induced by an electric field, let us consider theequivalent electric field E eqv ≡ µ/e A , which is capable to produce the same acceleration A in a particle with charge e and mass µ. In a TeV-brane, the gravity-induced dipole will be approximately 10 ( ≃ a/ε ) greater than the dipoleinduced by the equivalent electric field. Despite this impressive magnification, as the tidal acceleration A producedby astronomical bodies will be very tiny, we should expect that gravity-induced dipole will be more significant in thepresence of microscopic black holes. A. Final remarks
We would like to mention that in Ref. [16], it was shown that an effective Hamiltonian of a hydrogen atom, deducedfrom the covariant Dirac equation in a curved spacetime up to order v /c , contains terms that can mix oppositive-parity state. However, these terms arise as a result of a relativistic effect in post-Newtonian approximation, while theelectric polarization induced by gravity discussed here is a consequence of the split of fermions in the extra dimension.Finally, we should also emphasize that in the fat brane, as electrons and protons are stuck in different slices of thebrane, then every atom should have an electric dipole with a non-null component in the z direction. However, wehave shown here that the gravitational field of a mass M will induce a tangential component in the atomic dipole andhence the atom will produce an electric field in the brane whose pattern can be recognized by 4D-observers. [1] V. Rubakov and M. Shaposhnikov, Phys. Lett. B , 136 (1983).[2] P. Horava and E. Witten, Nucl. Phys. B , 506 (1996); N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B , 263 (1998); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B , 257 (1998); L. Randall.and R. Sundrum, Phys. Rev. Lett. , 3370 (1999).[3] L. Randall and R. Sundrum, Phys. Rev. Lett. , 4690 (1999).[4] N. Arkani-Hamed and M. Schmaltz, Phys. Rev. D , 033005 (2000); N. Arkani-Hamed, Y. Grossman and M. Schmaltz,Phys. Rev. D , 115004 (2000).[5] T. Han, G. D. Kribs, and B. McElrath, Phys. Rev. Lett. , 031601 (2003); G. Barenboim, G. C. Branco, A. de Gouvea,and M. N. Rebelo, Phys.Rev. D , 073005 (2001); G. C. Branco, A. de Gouvea, M. N. Rebelo Phys.Lett. B , 115(2001); H. Georgi, A. K. Grant, G. Hailu, Phys. Rev. D , 064027 (2001); A. De Rujula, A. Donini, M. B. Gavela, S.Rigolin, Phys. Lett. B , 195 (2000); P. Q. Hung, Ngoc-Khanh Tran, Phys. Rev. D , 064003 (2004).[6] O. De Wolfe , D.Z. Freedman, S.S. Gubser, A. Karch Phys. Rev. D , 046008 (2000); A. Chamblin and G.W. Gibbons,Phys. Rev. Lett. , 1090 (2000); M. Gremm, Phys. Lett. B , 434 (2000); C. Cs´aki, J. Erlich, T. J. Hollowood, andY. Shirman, Nucl. Phys. B , 309 (2000); R. Guerrero, A. Melfo and N. Pantoja, Phys. Rev. D , 125010 (2002); F.Brito, M. Cvetic and S. C. Yoon, Phys. Rev. D , 064021 (2001).[7] R. Emparan, R. Gregory, and C. Santos. Phys. Rev. D , 104022 (2001).[8] P. Figueras and T. Wiseman, Phys. Rev. Lett. , 081101 (2011); S. Abdolrahimi, C. Catto¨en, Don N. Page and S.Yaghoobpour-Tar, Phys. Lett. B , 405 (2013).[9] C. Charmousis and R. Gregory. Class.Quant.Grav. , 527 (2004). [10] A. Kehagias and K. Tamvakis, Phys.Lett. B, , 38 (2001).[11] M. Giovannini, Phys. Rev. D , 064023 (2001).[12] J. Garriga and T. Tanaka, Phys. Rev. Lett. , 2778 (2000).[13] F. Dahia and C. Romero, Phys. Lett. B , 232 (2007).[14] W. T. Kim and E. J. Son JHEP
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