A nonlinear dynamics for the scalar field in Randers spacetime JID PLB AID 32556 /SCO Doctopic Theory [m5Gv1 3; v1 195; Prn 19/01/2017; 7 32] P 1 (1 5) Physics Letters B ••• (••••) •••–••• 1 66 2 67 3[.]
JID:PLB AID:32556 /SCO Doctopic: Theory [m5Gv1.3; v1.195; Prn:19/01/2017; 7:32] P.1 (1-5) Physics Letters B ••• (••••) •••–••• Contents lists available at ScienceDirect 66 67 Physics Letters B 68 69 70 71 www.elsevier.com/locate/physletb 72 73 74 10 75 11 12 13 14 15 16 A nonlinear dynamics for the scalar field in Randers spacetime b 77 78 J.E.G Silva a , R.V Maluf b , C.A.S Almeida b a 76 79 Universidade Federal Cariri (UFCA), Instituto de formaỗóo de professores, Rua Olegário Emídio de Arẳjo, Brejo Santo, CE, 63.260.000, Brazil Universidade Federal Ceará (UFC), Departamento de Física, Campus Pici, Fortaleza, CE, C.P 6030, 60455-760, Brazil 80 81 17 82 18 83 19 a r t i c l e i n f o 84 a b s t r a c t 20 21 22 23 24 25 85 Article history: Received 29 October 2016 Received in revised form January 2017 Accepted 13 January 2017 Available online xxxx Editor: M Cvetiˇc 26 27 28 29 Keywords: Local Lorentz violating gravity Finsler gravity Randers spacetime 30 We investigate the properties of a real scalar field in the Finslerian Randers spacetime, where the local Lorentz violation is driven by a geometrical background vector We propose a dynamics for the scalar field by a minimal coupling of the scalar field and the Finsler metric The coupling is intrinsically defined on the Randers spacetime, and it leads to a non-canonical kinetic term for the scalar field The nonlinear dynamics can be split into a linear and nonlinear regimes, which depend perturbatively on the even and odd powers of the Lorentz-violating parameter, respectively We analyze the plane-waves solutions and the modified dispersion relations, and it turns out that the spectrum is free of tachyons up to secondorder © 2017 Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 86 87 88 89 90 91 92 93 94 95 31 96 32 97 33 98 34 99 35 36 100 Introduction 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Despite the current lack of a complete theory of quantum gravity, several candidate theories assume that some symmetries present at low energy regimes might no longer be valid at Planck scale For instance, the string theory [1], spacetime noncommutativity [2], Horava–Lifshitz gravity [3], loop quantum gravity (LQG) [4], Doubly Special Relativity (DSR) [5] and the Very Special Relativity (VSR) [6] admit the possibility of absence of Lorentz symmetry for the spacetime The violation of Lorentz symmetry may be the result of a spontaneous breaking of tensor fields which acquire nonvanishing vacuum expectation values [7] or the condensation of a ghost (scalar) field leading to modifications of the dispersion relations [8] A field theoretical framework to test the Lorentz symmetry is provided by the Standard Model Extension (SME) [9] For a comprehensive review of tests on Lorentz and CPT violation, we indicate the Ref [10] The violation of the local Lorentz symmetry can be extended to curved spacetimes by means of the so-called Finsler geometry [11–13] In this anisotropic geometry, the intervals are evaluated by a non-quadratic function, called the Finsler function [14,15] The lack of quadratic restriction provides modified dispersion relation for the fields, a hallmark of Lorentz violation [16–18] Applications 60 61 62 63 64 65 E-mail address: carlos@fisica.ufc.br (C.A.S Almeida) of Finsler geometry can also be found in optics [19] and condensed matter physics [20] One of the most important Finsler spacetimes is the Randers spacetime where the anisotropy is driven by a background vector field aμ which changes the length of intervals of the spacetime [21,22] The cosmological and astrophysical effects of the Randers spacetime were analyzed in Refs [23,24] In the context of the SME, the Randers spacetime arises as a kind of bipartite-Finsler space, in the classical point particle Lagrangian for the CPT-Odd fermionic sector [25,26] Other SME-based Finsler spaces can be found in Refs [27] In this work, we propose a minimal coupling of a real scalar field and the Finsler metric in the Randers spacetime Unlike the tangent bundle theories [28,29], whose dynamics lies on T T M, we propose a position dependent field and Lagrangian Our dynamics also differs from the osculating method, where the directiondependent is worked out as a constraint [30] For a Finslerian action, we employ an extension of the known as the Shen functional, where the square of the components of the gradient of the field is evaluated with the Finsler metric As a Finsler volume, we choose the Busemann–Hausdorff volume which provides an anisotropic factor, such as in Bogoslovsky space [31] The work is organized as follows In section we review the basic definitions and properties of the Randers spacetime In section we propose the Finsler action, obtain the Finslerian equation of motion and analyze the important regimes The modified dispersion relation and stability are studied in section and section 5, http://dx.doi.org/10.1016/j.physletb.2017.01.025 0370-2693/© 2017 Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 JID:PLB AID:32556 /SCO Doctopic: Theory [m5Gv1.3; v1.195; Prn:19/01/2017; 7:32] P.2 (1-5) J.E.G Silva et al / Physics Letters B ••• (••••) •••–••• 2 for linear and nonlinear regimes, respectively Final remarks, conclusions and perspectives are outlined in section The Finsler metric also provides a deformation of the mass shell, given by [17,31,13,32] 66 R g ∗ R μν (x, P μ ) P μR P νR = −m2 68 Randers spacetime In Randers spacetime, given a four-velocity x˙ μ = dxμ /dt, the infinitesimal interval of a worldline is defined by ds = F R (x, x˙ )dt where [23] 10 11 F R (x, x˙ ) := α (x, x˙ ) + β(x, x˙ ) = 12 − g μν (x)˙xμ x˙ ν + ζ aμ (x)˙xμ , (1) 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 and ζ is a real parameter controlling the local Lorentz-violation The norm of the background Randers covector aμ is evaluated with the Lorentzian metric g, a2 (x) := g μν (x)aμ aν In this work we adopt the mostly-plus metric convention (−, +, +, +) The perturbative character of the Lorentz violation is encoded in the small value of the linear term, constrained to ≤ ζ a < We assume that ζ is a constant, bigger than Planck length and with a dimension of length L The Randers background vector is assumed to have mass dimension one, as expected for a background vector field arising from reminiscent quantum gravity effects in four dimensions [32] The Randers function can be written as F R = R where the anisotropic timelike Randers metric g μν (x, x˙ ) is defined as [17,12,29] 29 30 31 R g μν (x, x˙ ) := − 36 37 α dt F R (x, x˙ ) = −m S R = −m 40 I R (x, x ˙ )˙xμ x˙ ν dt − g μν 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 (3) I The free particle action (3) is analogous to the Lagrangian of a charged particle in a Lorentzian spacetime with an electromagnetic background vector aμ The canonical momentum is R Pμ = P μ − mζ aμ , where P μ := mg μν (x)U ν is the Lorentzian conjugate momentum and U μ := x˙ μ /α (˙x) is the Lorentzian unitary R 4-velocity The Finsler metric g μν (x, x˙ ) provides a nonlinear dualR ity between the covariant P μ and the contravariant P R μ , given by R R Pμ = g μν (x, P R ) P R ν [17] Thus, the contravariant components of the momentum are given by P R μ = m x˙ μ / F (x, x˙ ) [17] The Finsler R function for the covariant vector P μ is 65 inside the Lorentz-invariant lightcone, similar to the dispersion relations analyzed in the Ref [33] Furthermore, the asymptote of the deformed mass-shell corresponds to the Randers lightcone Then, though the particle reaches Lorentzian superluminal velocities, its speed does not exceed the speed of light in the Randers spacetime 71 72 73 74 75 76 77 78 79 80 81 82 83 84 87 After analyzing the dynamics of a point particle in the anisotropic Randers spacetime, let us now consider the real scalar field dynamics in this local Lorentz violating spacetime We assume that the real scalar field is a function of the position x only, i.e., = (x) Consider the action functional defined by a minimal coupling between the scalar field and the dual Finsler metric, namely 5 d4 x − g (x) ∓ ζ a2 (x) 88 89 90 91 92 93 94 95 96 × g F ∗μν (x, d)∂μ ∂ν + V () , 97 (5) 98 99 where V () = m2 2 for the free scalar field and the signs −, + stand for a timelike and spacelike background vector a respec√ tively The anisotropic volume form d4 x − g (x)(1 ∓ ζ a2 (x))5/2 , is an extension of the Busemann–Hausdorff volume for the Randers spacetime [34,15] It is worthwhile to mention that the Finslerian action in Eq (5) bears some resemblance to the so-called k-essence models [35] The nonquadratic Lagrangian density defined by Eq (5) can be split into two terms 100 101 102 103 104 105 106 107 108 109 NQ L = Lb + L , (6) where Lb is the bilinear Lagrangian density constructed with the quadratic terms field derivatives ∂ and with the potential term, given by b L = − 2 2 + V ()(1 ∓ ζ a ) 112 113 114 116 117 −(1 ∓ ζ 2a2 ) g , 111 115 (1 ∓ ζ a ) g μν (x) − 2ζ 2aμ (x)aν (x) ∂μ ∂ν + 2 110 118 (7) 119 120 121 (1 ∓ ζ a2 ) g μν − ζ 2aμ aν ∗μν a∗μ (x) = −ζ g ( x ) = , (1 ∓ ζ 2a2 ) (1 ∓ ζ a2 )2 aμ The signs −, + stand for a timelike and spacelike background vecR ) is defined by [17, tor aμ [15,22] The dual Finsler metric g ∗ F (x, P μ 15] g 70 86 where ∗ F μν 69 85 Scalar field dynamics R F R∗ (x, P μ ) = − g ∗μν (x) P μR P νR + a∗μ (x) P μR , 63 64 −m2 yields to ( g μν + ζ aμ aν ) P R μ P R ν − 2ζ maμ P R μ = −m2 In a flat spacetime g μν = ημν and for a constant Randers background vector aμ = (−a0 , a), the modified mass-shell is (1 − ζ a20 ) E + )) E −|p |2 +ζ (a · p )[2m −ζ (a · p )] = −m2 For a time2ζ a0 (m −ζ ( a· p ), the modified mass-shell lies like background vector aμ = (−a0 , (2) 41 42 In Randers spacetime, the modified dispersion relation (MDR) is R R g μν P μ P ν − 2mζ (a · P R ) = −(1 ∓ ζ a2 )m2 , which is an elliptical hyperboloid of two sheets [26] Considering the momentum R ), the dual mass-shell g μν 4-vector P R μ = ( E , p (x, P R ) P R μ P R ν = S = − where ˜lμ := ∂ α /∂ x˙ μ = − g μν (x)˙xν /α , such that ˜lμ x˙ μ = α We define the action for a free massive particle in the Randers– Finsler spacetime as [21,17] 38 39 α 33 35 ∂ F R2 (x, y ) ∂ x˙ μ ∂ x˙ ν β FR = g μν + ˜lμ˜lν − ζ (˜lμ aν + ˜lν aμ ) − ζ aμ aν , 32 34 R (x, x ˙ )˙xμ x˙ ν , − g μν (4) 67 R (x, P μ ) = − ∂ F ∗2 (x, P μ R ) ∂PR∂PR μ μ NQ and L corresponds to the nonquadratic terms in the Lagrangian L , whose expression is given by NQ L = −ζ (a · ∂) [−(1 ∓ ζ 2a2 )(∂)2 + ζ (a · ∂)2 ] × −(1 ∓ ζ 2a2 ) g , 122 123 124 125 126 (8) where (∂)2 := g μν (x)∂μ ∂ν and a · ∂ := aμ ∂μ Note that for a Lorentzian spacetime, i.e ζ = 0, the nonquadratic Lagrangian 127 128 129 130 JID:PLB AID:32556 /SCO Doctopic: Theory [m5Gv1.3; v1.195; Prn:19/01/2017; 7:32] P.3 (1-5) J.E.G Silva et al / Physics Letters B ••• (••••) •••–••• NQ density L vanishes and the bilinear Lagrangian reduces to the √ Lorentz-invariant L = − 12 g μν (x)∂μ ∂ν + 2V () − g where := λ/ L and the phase function ψ(x) is called the eikonal The differential of the field (15) is given by The bilinear Lagrangian Lb in (7) can be expanded in powers of ζ as d = b L = − 10 − 11 ζ2 2 13 14 18 19 20 21 22 23 24 2 (9) Therefore, the local Lorentz violating effects arise in second order in ζ parameter The LV terms have a similar form of those proposed in the context of the Standard Model Extension (SME) [9] μν Indeed, defining the symmetric and dimensionless tensor k as μν k := ζ aμ aν , the bilinear term can be regarded as a CPT even Lorentz-violating Lagrangian of the Higgs sector in the minimal SME [9] Extremizing the Finslerian action (5), the Euler–Lagrange equation yields the equation of motion for the field 25 27 28 29 30 31 32 √ ∂μ (1 ∓ ζ 2a2 ) − g g F ∗μν (x, d)∂ν 26 (1 ∓ ζ 2a2 ) √ −g ∂ V () = ∂ (10) The Eq (10) is a nonlinear Finslerian extension of the Klein– Gordon equation By defining the nonlinear D’Alembertian operator 33 34 35 F := (1 ∓ ζ 2a2 ) √ −g 5√ × ∂μ (1 ∓ ζ 2a2 ) − g g F ∗μν (x, d)∂ν , 36 37 (11) 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 the nonlinear Finslerian Klein–Gordon equation can be rewritten as ∂ V () 2F = ∂ (12) The nonlinear Finslerian D’Alembertian operator F is an extension of the so-called Shen Laplacian [15], defined on Riemann– Finsler spaces, to Pseudo-Finsler Spacetimes The free nonlinear Finslerian Klein–Gordon equation, where V () = m2 2 , can be rewritten as g F ∗μν (x, d)∂μ ∂ν 2 52 √ F ∗μν (x, d) ∂ν = m2 + ∂μ log (1 ∓ ζ a ) − g g (13) 54 55 56 57 58 59 60 61 62 63 64 65 For a flat spacetime with a constant background field a, the anisotropic volume factor (1 ∓ ζ a2 )5/2 can be absorbed in a change of coordinates and the Klein–Gordon equation (13) yields to g F ∗μν (d)∂μ ∂ν + ∂μ ( g F ∗μν (d))∂ν = m (14) Consider the ray approximation, where the wave length λ is much smaller than the geometrical characteristic length L, i.e., λ