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Bipartite Finsler Spaces and the Bumblebee Model

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Bipartite Finsler Spaces and the Bumblebee Model
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    a  r   X   i  v  :   1   3   0   9 .   4   6   7   1  v   1   [   h  e  p  -   t   h   ]   1   8   S  e  p   2   0   1   3 Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) 1 BIPARTITE-FINSLER SPACES AND THE BUMBLEBEEMODEL 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: euclides@fisica.ufc.br  We present a proposal to include Lorentz-violating effects in gravitational fieldby means of the Finsler geometry. In the Finsler set up, the length of anevent depends both on the point and the direction in the space-time. Webriefly review the bumblebee model, where the Lorentz violation is inducedby a spontaneous symmetry breaking due to the bumblebee vector field.Themain geometrical concepts of the Finsler geometry are outlined. Using a Finsle-rian Einstein-Hilbert action we derive the bumblebee action from the bipartiteFinsler function with a correction to the gravitational constant. 1. Bumblebee model A model to include gravity in the Standart model extension (SME) 1 isprovided by a vector field  B µ , the so-called bumblebee field, which coupleswith the usual geometrical tensors through the action 2 S  LV    =  K    M  ( uR  +  s µν  R µν  ) √ − gd 4 x  (1)where  s µν  =  ξ   B µ B ν  −  14 B 2 g µν    and  u  =  ξB µ B µ .  Inspite this model in-troduces the Lorentz violation into the geometry, 3 the geometrical tensorsremain Lorentz-invariants. This drawback can be overcame by means of theFinsler geometry 2 . 2. Finsler geometry A Finsler geometry is an extension of the Riemannian geometry where givena curve  γ   : [0 , 1] → M  , its arc length is given by 4 s  =    10 F  ( x,y ) dt,  (2)  Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) 2 where,  x ∈ M,y  ∈ T  x M  . The function  F  ( x,y ) is called the Finsler principalfunction. Note that the interval depends both on the position  x  as on thedirection  y .As in the Riemannian case, it is possible to define a Finsler metric by 4 g F µν  ( x,y ) = 12 ∂  2 F  2 ∂y µ y ν   .  (3)The Finsler metric (3) is a symmetric and an anisotropic quadraticform on  TTM  . Differenting the metric yields the so-called Cartan ten-sor  A αβγ  ( x,y ) =  F  4 ∂g αβ ∂y γ  from which it is possible to define a nonlinearconnection by  N  αδ  =  γ  δαβ y β −  A δαβ F   γ  βǫξ y ǫ y ξ which decouples  TTM   into TTM   =  hTTM   vTTM  , where  δδx α  =  ∂ ∂x α  − N  βα∂ ∂y β  is the basis for thehorizontal section  hTTM   and  F   ∂ ∂y α  is a base for the vertical section  vTTM  .The compatible Cartan connection is given by  ω δα  = Γ δαβ dx β +  A δαβ F   δy β , where Γ δαβ  =  12 g Fδǫαβ  ( δ  α g ǫβ  +  δ  β g ǫα − δ  ǫ g αβ ). 4 Following the approach proposed by Pfeifer and Wohlfarth, 5 here we areconcerned with only the Lorentz violation effects on tensor fields definedon  hTTM  . The horizontal Ricci curvature is defined as 4 R αβ  =  δ  γ  Γ γ αβ  − δ  β Γ γ αγ   + Γ γ ǫβ Γ ǫαγ   − Γ γ ǫγ  Γ ǫαβ ,  (4)and the scalar curvature  R F  =  g Fαβ R F αβ . 3. The Bipartite space Based in a previous work on the classical point particles Lagrangians, 6 Kostelecky proposed a new Finsler function of form 7 F  ( x,y ) =   g µν  ( x ) y µ y ν  +  l P  ( a µ ( x ) y µ ±   s µν  ( x ) y µ y ν  ) ,  (5)where  s µν  ( x ) =  b 2 ( x ) g µν  ( x ) − b µ ( x ) b ν  ( x ) .  The Planck length scale  l P   pro-vides a scale of length where the anisotropic effects have to be taken inaccount.Kostelecky, Russel and Tso supposed  a µ  = 0 and enhanced the  s µν  tensor to be any symmetrical one. 8 This geometry is so-called bipartite.The bipartite-Finsler function yields the Finsler metric 8 g F µν   =  F αg µν   +  l 2 P   F σ s µν   − ασk µ k ν   ,  (6)  Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) 3 where,  k µ  =  1 α∂α∂y µ  − 1 σ∂σ∂y µ . Furthermore, for  dimM   = 4, the relation be-tween the Finslerian and Lorentzian volume element is given by   − g F  =  F α  5  S σ  2 √ − g.  (7) 4. Finslerian Einstein-Hilbert action Assuming the dynamics of the space-time isgovernedby an Einstein-Hilbertaction then, S  F   =  κ    R F    − g F  d 4 xd 4 y =  κ    R √ − gd 4 x  +  κ F     8 b 2 R √ − gd 4 x +  κ    (4 +  b 2 l 2 P  ) s µν  R µν  d 4 x  +  ....  (8)it is possible to regain some interaction terms of the bumblebee modelaction (1). As a perspective we expect to obtain the dynamical terms of thebumblebee model and some other interaction terms as well. 5. Acknowledgements We are grateful to Alan Kosteleck´y and Yuri Bonder for useful discussionsand to the Graduate Physics Program of the Cear´a Federal University andCNPq (National Council for Scientific and Technologic Developing) for fi-nancial 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). View publication statsView publication stats
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