Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
arXiv:1309.4671v1 [hep-th] 18 Sep 2013
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BIPARTITE-FINSLER SPACES AND THE BUMBLEBEE MODEL J. EUCLIDES G. SILVA∗ and C.A.S. ALMEIDA Physics Department, Cear´ a Federal University , Fortaleza, Cear´ a ZIP 6030, 60455-760, Brazil ∗ E-mail:
[email protected] We present a proposal to include Lorentz-violating effects in gravitational field by means of the Finsler geometry. In the Finsler set up, the length of an event depends both on the point and the direction in the space-time. We briefly review the bumblebee model, where the Lorentz violation is induced by a spontaneous symmetry breaking due to the bumblebee vector field.The main geometrical concepts of the Finsler geometry are outlined. Using a Finslerian Einstein-Hilbert action we derive the bumblebee action from the bipartite Finsler function with a correction to the gravitational constant.
1. Bumblebee model A model to include gravity in the Standart model extension (SME)1 is provided by a vector field Bµ , the so-called bumblebee field, which couples with the usual geometrical tensors through the action2 Z √ (1) SLV = K (uR + sµν Rµν ) −gd4 x M
where sµν = ξ B µ B ν − 41 B 2 g µν and u = ξB µ Bµ . Inspite this model introduces the Lorentz violation into the geometry,3 the geometrical tensors remain Lorentz-invariants. This drawback can be overcame by means of the Finsler geometry2 . 2. Finsler geometry A Finsler geometry is an extension of the Riemannian geometry where given a curve γ : [0, 1] → M , its arc length is given by4 s=
Z
0
1
F (x, y)dt,
(2)
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
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where, x ∈ M, y ∈ Tx M . The function F (x, y) is called the Finsler principal function. Note that the interval depends both on the position x as on the direction y. As in the Riemannian case, it is possible to define a Finsler metric by4 F gµν (x, y) =
1 ∂2F 2 . 2 ∂y µ y ν
(3)
The Finsler metric (3) is a symmetric and an anisotropic quadratic form on T T M . Differenting the metric yields the so-called Cartan ten∂g from which it is possible to define a nonlinear sor Aαβγ (x, y) = F4 ∂yαβ γ Aδ
β ǫ ξ δ y y which decouples T T M into connection by Nδα = γαβ y β − Fαβ γǫξ L T T M = hT T M vT T M , where δxδα = ∂x∂α − Nαβ ∂y∂ β is the basis for the horizontal section hT T M and F ∂y∂α is a base for the vertical section vT T M . Aδ
The compatible Cartan connection is given by ωαδ = Γδαβ dxβ + Fαβ δy β , F δǫ (δα gǫβ + δβ gǫα − δǫ gαβ ).4 where Γδαβ = 21 gαβ Following the approach proposed by Pfeifer and Wohlfarth,5 here we are concerned with only the Lorentz violation effects on tensor fields defined on hT T M . The horizontal Ricci curvature is defined as4 Rαβ = δγ Γγαβ − δβ Γγαγ + Γγǫβ Γǫαγ − Γγǫγ Γǫαβ ,
(4)
F and the scalar curvature RF = g F αβ Rαβ .
3. The Bipartite space Based in a previous work on the classical point particles Lagrangians,6 Kostelecky proposed a new Finsler function of form7 F (x, y) =
q q gµν (x)y µ y ν + lP (aµ (x)y µ ± sµν (x)y µ y ν ),
(5)
where sµν (x) = b2 (x)gµν (x) − bµ (x)bν (x). The Planck length scale lP provides a scale of length where the anisotropic effects have to be taken in account. Kostelecky, Russel and Tso supposed aµ = 0 and enhanced the sµν tensor to be any symmetrical one.8 This geometry is so-called bipartite. The bipartite-Finsler function yields the Finsler metric8 F gµν =
F gµν + lP2 α
F sµν − ασkµ kν σ
,
(6)
Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)
3 ∂α 1 ∂σ where, kµ = α1 ∂y µ − σ ∂y µ . Furthermore, for dimM = 4, the relation between the Finslerian and Lorentzian volume element is given by 5 2 p √ F S F −g = −g. (7) α σ
4. Finslerian Einstein-Hilbert action Assuming the dynamics of the space-time is governed by an Einstein-Hilbert action then, Z p SF = κ RF −g F d4 xd4 y Z Z √ √ 4 = κ R −gd x + κF 8b2 R −gd4 x Z + κ (4 + b2 lP2 )sµν Rµν d4 x + .... (8) it is possible to regain some interaction terms of the bumblebee model action (1). As a perspective we expect to obtain the dynamical terms of the bumblebee model and some other interaction terms as well. 5. Acknowledgements We are grateful to Alan Kosteleck´ y and Yuri Bonder for useful discussions and to the Graduate Physics Program of the Cear´ a Federal University and CNPq (National Council for Scientific and Technologic Developing) for financial supporting. References 1. D. Colladay and V. A. Kostelecky, Phys. Rev. D 55, 6760 (1997). 2. V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004). 3. R. V. Maluf, V. Santos, W. T. Cruz and C. A. S. Almeida, Phys. Rev. D 88, 025005 (2013) 4. D. Bao, S. Chern, Z. Shen, An introduction to Riemann-Finsler geometry, Springer, 1991. 5. C. Pfeifer and M. N. R. Wohlfarth, Phys. Rev. D 85, 064009 (2012) 6. A. V. Kostelecky and N. Russell, Phys. Lett. B 693, 443 (2010). 7. A. Kostelecky, Phys. Lett. B 701, 137 (2011). 8. V. A. Kostelecky, N. Russell and R. Tso, Phys. Lett. B 716, 470 (2012).
