Bipartite Finsler Spaces and the Bumblebee Model

Bipartite Finsler Spaces and the Bumblebee Model
of 3
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
    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:  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
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...

Sign Now!

We are very appreciated for your Prompt Action!