Bipartite-Finsler symmetries
J. E. G. Silva R. V. Maluf C. A. S. Almeida
gr-qc hep-th










ufca]J. E. G. Silva



pici]R. V. Maluf




pici]C. A. S. Almeida



[ufca]Universidade Federal do Cariri(UFCA), Av. Tenente Raimundo Rocha, 
 Cidade Universitária, Juazeiro do Norte, Ceará, CEP 63048-080, Brasil

[pici]Universidade Federal do Ceará (UFC), Departamento de Física, Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil








In this work, we study the symmetries of a Lorentz violating bipartite-Finsler spacetime. By using the Finslerian Killing equation for the bipartite-Finsler metric with Lorentzian signature, we analyze how the anisotropy of the bipartite spacetime modifies the local Lorentz symmetries. The symmetries of the pseudo-Finsler metric allow us to obtain the bipartite algebra and interpreted it as a deformed Lorentz algebra. Causality effects driven by the particle modified dispersion relations are investigated upon Minkowski and the Schwarzschild spacetimes. The background bipartite tensor yields to superluminal velocities and modifies the effective gravitational potential.



Lorentz violation, Finsler metric, Randers space, dispersion relations












§ INTRODUCTION


A complete description of the structure of space-time and fields dynamics near the Planck scale is still a matter of debate and investigation. Some theories suggest that the cornerstones symmetries of the standard model (SM), the CPT and Lorentz symmetries may be violated in the UV regime. In string theories, the vacuum expectation values (VEV) of tensorial fields at some specific vacua might couple to the SM fields driving Lorentz symmetry effects <cit.>. The so-called Very Special Relativity (VSR) <cit.> assumes a sub-group of the Lorentz group, the DSIM(2) whereas for the Doubly Special Relativity a larger group leaving invariant both the speed of the light c and the Planck length l_P is assumed <cit.>. Noncommutative geometries <cit.>, Horava theories <cit.> and Loop Quantum Gravity theory <cit.> also contain Lorentz symmetry breaking effects.

A broad framework to probe CPT-Lorentz symmetry violation is provided by the so-called Standard Model Extension (SME) <cit.>. The Lorentz-violating effects were investigated on several different systems involving neutrinos <cit.>, photons <cit.>, mesons <cit.>. In the gravitational sector, the Lorentz-violating terms might couple to the metric and the connection yielding to an Einstein-Cartan theory <cit.>. In the Riemannian limit, assuming a spontaneous symmetry breaking driven by a vector (bumblebee) field <cit.> preserves the Bianchi identities and the influence of LV was investigated in cosmology <cit.>.

A conspicuous feature of Lorentz symmetry violation is the modified dispersion relation (MDR) <cit.>. The MDR effects can be encompassed into the geometric structure of the space-time by means of the so-called Finsler geometry <cit.>. Unlike Riemann geometry, in Finsler geometry the norm of tangent vector v is computed with a general function F(x,v) called Finsler function <cit.>. Applications of Riemann-Finsler geometry can be found in anisotropic optics <cit.> and graphene <cit.>.

In SME-based Finsler geometry a Riemann-Finsler function is derived from a point-particle Lagrangian with SME coefficients <cit.>. By means of a Wick rotation, the spin-dependent point-particle Lagrangians provide Riemann-Finsler functions whose departure from the usual Riemannian structure depends on the spin particle <cit.>. In the photon <cit.> and in the scalar sector <cit.> new Riemann-Finsler structures were obtained. A CPT-odd fermion has a Finsler structure given by the well-known Randers space <cit.>, whereas a chiral CPT-odd fermion point-particle describes a geodesic motion in a novel Finsler space called b-space <cit.>. Both Randers and b-space have Finsler functions (and hence the local Lorentz violation) driven by background vectors. The effects of the background vector field were investigated in cosmology <cit.> and astrophysics <cit.>.

The Randers and b-space are special cases of a general Finsler geometry called bipartite space, where the Finsler function is defined from the Riemannian metric and a symmetric background tensor s_μν <cit.>. This bipartite-Finsler structure provides quartic MDR from CPT-even SME interactions <cit.>. By considering the geometric features of this Finsler spacetime, we can find new symmetries. Although not Lorentz invariant, the Bipartite symmetries reveal what quantities are left invariant by the presence of the bipartite background tensor.

In this work, we seek for the symmetry transformations that preserve the bipartite-Finsler pseudo-metric with Lorentzian signature. Using the Finsler Killing equation, we find the particle transformations leaving unchanged the particle Lagrangian. We define bipartite generators of translations, boosts and rotations. The generators are modified by the inclusion of the bipartite tensor. The corresponding bipartite algebra is defined showing a resemblance with the κ-algebra <cit.>. A similar analysis has been carried out for the Randers <cit.> and Bogoslovsky <cit.> Finsler spaces. Moreover, the effects of the bipartite tensor on the momentum and energy conservation, as well as the causality analysis is performed. It turns out that the bipartite geometry yields to superluminal signals, an issue exhibited by other Lorentz violating models <cit.>. We discuss the modifications of the bipartite Lorentz symmetry breaking on the bound orbit stability and compare to other Finslerian models <cit.>.




§ BIPARTITE-FINSLER SPACE


The Bipartite-Finsler space is an anisotropic spacetime whose interval for a point timelike particle has the form <cit.>

    ds_B =F_B(x,u)dt = (√(g_μνu^μu^ν)+√(s_μνu^μu^ν))dt,

where u^μ=dx^μ/dt 
is the 4-velocity, s_μν=s_νμ is the background Lorentz violating Bipartite tensor and we are assuming the mostly minus metric signature (+,-,-,-). The Finsler function F_B can be written as F_B(x,u)=α(x,u)+σ(x,u) where α(x,u):=√(g_μνu^μu^ν) is the local Lorentz invariant interval and σ(x,u):=√(s_μνu^μu^ν) is the Lorentz violating term. For s_μν=b_μb_ν, the Bipartite tensor provides two copies of the Randers space F_R(x,u)=√(g_μνu^μu^ν)± b_μ u^μ. Another example of Bipartite space is the so-called b-space, whose Bipartite tensor has the form s_μν=b^2 g_μν - b_μb_ν <cit.>. These both Finsler spacetimes stem from the classical Lagrangian for fermion under chiral CPT-odd Lorentz violating interactions <cit.>. The indexes for the Bipartite tensor are raised and lowed with the local Lorentz invariant pseudo-metric g_μν.

Let us assume the particle action as an integral over the particle worldline <cit.>

    S_B = m∫(√(g_μνu^μu^ν)+√(s_μνu^μu^ν))dt.

The action (<ref>) inherits the Lorentz violating from the spacetime geometry.



Consider the basis B'={e_0, e_1,e_2, e_3 }, formed from the eigenvectors of s_μν i.e., s^μ_ν(e_ρ)^ν=λ_ρδ_ν^μ(e_ρ)^ν. Then, 𝐬=λ_0 e^0⊗ e^0 + ∑_i=1^3λ_i e^i⊗ e^i, and therefore σ=√(λ_0 + ∑_i=1^3λ_i (v^i)^2).  In order to guarantee the condition s_μνu^μu^ν≥ 0, we assume that λ_0 ≥ 0 and λ_i ≥ 0, a condition similar to the weak energy condition for the stress-energy tensor. For the Randers space, s^μ_ν = b^μb_ν and the background vector b^μ is an eigenvector of s^μ_ν with eigenvalue λ=b^2. Thus, the Randers vector b^μ defines a privileged direction in spacetime. In Randers spacetime the bipartite tensor is idempotent, i.e.,

    s^μ_ρs^ρ_ν = ζ s^μ_ν,

where ζ=b^2=g_μνb^μb^ν. We assume the condition Eq. (<ref>) is valid for any bipartite space.


We can define an anisotropic bipartite-Finsler metric g^B_μν(x,u) such that ds^2_B = g^B_μν(x,u)u^μu^ν by <cit.>

    g^B_μν(x,u)     =    1/2∂^2F_B^2/∂ u^μ∂ u^ν
        =    F/αg_μν +  (F/σs_μν-ασ k_μk_ν),

where, α:=√(g_μνu^μu^ν), β:=√(s_μνu^μu^ν) and k_μ=(g_μν/α-s_μν/σ)u^ν. The bipartite-Finsler metric encompasses the bipartite background tensor and its Lorentz violating effects.
The inverse bipartite-Finsler metric which satisfies the condition g^Bμρg^B_ρν=δ^μ_ν have the form <cit.>

    g^Bμν(x,u)     =    α/F[g^μν+(σ^⊥/σ)^2 α/Sλ^μλ^ν-α/Ss^μν],

where λ_μ :=(1/σ^⊥)(s_μν-(σ S/F)g_μν)u^ν, σ^⊥ :=√((ζ g_μν-s_μν)u^μ u^ν) and S:=ζα +σ <cit.>.

The directional dependence of the bipartite metric can be measured by the so-called Cartan tensor C_μνρ, defined as <cit.>

    C_μνρ    =    F_B (x,u)/2∂ g^B_μν(x,u)/∂ u^ρ
        =    1/ασg_μνs_ρλu^λ - σ/α^3g_μνu_ρ+1/ασs_μνu_ρ -(σ/αu_ρ + α/σs_ρλu^λ)k_μ k_ν +
        -    ασ(∂ k_μ/∂ u^ρk_ν + ∂ k_ν/∂ u^ρk_μ).

In fact, the Cartan tensor vanishes for the usual Riemannian geometry.

Since the bipartite metric depends on both the position x and the velocity u, a complete description of this Finsler spacetime demands we interpreted it as a configuration space (x,u) and consider both position and direction variation, respectively ∂/∂ x^μ and ∂/∂ u^μ <cit.>.

Consider the so-called horizontal derivative

    δ_μ:=∂/∂ x^μ - N_μ^ν∂/∂ u^ν

and the vertical derivative ∂̅_μ:=F∂/∂ u^μ, where N^μ_ν:=γ^μ_νλu^λ/F-C^μ_νλγ^λ_δϵ(u^δu^ϵ/F^2) and the usual Christoffel symbols are γ^μ_νλ:=g^Bμρ/2(∂_νg^B_ρλ+∂_λ g^B_ρν-∂_ρ g^B_νλ) <cit.>. The basis (δ_μ , ∂̅_μ) is orthogonal and splits the configuration space TM_4/0={(x,u), x∈ M_4,  u∈ T_xM_4} into a horizontal space spanned by (δ_μ) and a vertical space spanned by (∂̅_μ) <cit.>. Using the dual basis (dx^μ, δ u^μ), where δ u^μ := (1/F)(du^μ +N^μ_ν dx^ν), the configuration space has a metric ds^2= g_μν^B dx^μ dx^ν +(g^B_μν/F^2)δ u^μδ u^ν <cit.>. Although the tangent bundle TM/0 contains the whole metric directional-dependence of the spacetime, it is worthwhile to stress that, for a tangent vector v^μ∈ T_xM_4 only the horizontal metric defines its norm.

In order to study how the spacetime geometry varies from point to point and explore the effects of these changes upon the particle symmetries and dynamics we have to define a connection ω^μ_ν=Γ^μ_νρdx^ρ + Γ'^μ_νρδ y^ρ <cit.>. The metric changes as ∇ g= g^B_μν|ρdx^μ⊗ dx^ν⊗ dx^ρ+g^B_μν;ρdx^μ⊗ dx^ν⊗ (δ u^ρ/F), where g^B_μν|ρ=δ_ρ g_μν^B -Γ_ρμ^λg_λν^B-Γ_ρν^λg_λμ^B
and g^B_μν;ρ=∂̅_ρg_μν^B -Γ_ρμ^'λ g_λν^B -Γ_ρν^'λg_λμ^B. Assuming a metric compatible connection ∇ g=0, we obtain the so-called Cartan connection whose horizontal connection has a familiar form 

    Γ^Bμ_νρ(x,u):=g^Bμσ(x,u)/2[δ_ν g^B_σρ(x,u)+δ_ρ g^B_σν(x,u)-δ_σg^B_νρ(x,u)],

whereas the vertical connection is given by the Cartan tensor Γ_ρν^'λ=C_ρν^'λ <cit.>. Therefore, the connection Γ^Bμ_νρ(x,u) preserves the Finsler metric g^F_μν(x,u) as we move along the Bipartite space. This fact will be used in the next section to find the local isometries (symmetries) of the bipartite spacetime.




§ BIPARTITE TRANSFORMATIONS


In this section we study the coordinate transformation which preserves the bipartite particle interval (action) Eq. (<ref>). 

The action is clearly invariant upon usual observer Lorentz transformations x'^μ:=(Λ)^μ_νx^ν. For particle transformations, we consider an actual displacement of the particle along a trajectory preserving the action. To do so, let us consider the Finsler Killing equation which provides the local symmetries of the bipartite Finsler metric g^B_μν(x,u).


The general way to find the local symmetries ξ^ρ of the action is through the Lie derivative of the Finsler metric, given by <cit.>

    ℒ_ξg^F_μν(x,u)   :=   ξ^ρg^F_μν|ρ(x,u) + g^F_μρ(x,u)ξ^ρ_|ν+g^F_νρ(x,u)ξ^ρ_|μ+
       +    2C_μνρ(x,u)ξ^ρ_|σu^σ,

where the Finsler covariant derivative is defined by the horizontal connection (<ref>).
A Finslerian Killing vector satisfies ℒ_ξg^F_μν(x,u)≡ 0 and then, the Killing equation turns into

    g^F_μρ(x,u)ξ^ρ_|ν+g^F_νρ(x,u)ξ^ρ_|μ+2C_μνρ(x,u)ξ^ρ_|σu^σ=0.



It is worthwhile to mention that we are interested in the particle symmetries as the particle moves in M_4 and not on TTM. Hence, the Killing vector ξ∈ T_xM_4, i.e., ξ=ξ(x) has no u-dependence. In ref. <cit.> the authors study the symmetry effects on the Randers spacetime considering variations on the direction as well.

For active particle transformations, let us consider the displacement vector in the direction of the particle motion, i.e., ξ^ρ=u^ρ. The compatible connection (<ref>) guarantees that the 4-acceleration is orthogonal to the 4-velocity, i.e., u_|σ^ρu^σ=0. Thus, the Killing equation turns to

    ξ_μ|ν+ξ_ν|μ=0.

The Killing equation for the particle symmetries (<ref>) is rather similar to the usual Lorentz invariant Killing equation. The difference lies only on the horizontal connection (<ref>).

Let us first consider the case where g_μν=η_μν and s_μν are constants.
The covariant horizontal derivative turns into ξ_μ|ν=δ_νξ_μ and a constant 4-vector ϵ^μ satisfies the equation (<ref>). Thus, we can use the Killing vector ϵ^μ to define a translation operation by

    T(λ):=e^i(ϵ^μP^F_μ)λ,

where λ is a real parameter and the translation generator is defined as

    P_μ^F:=-iδ_μ.

The infinitesimal translation is ξ^ρ=ϵ^μδ_μx^ρ=ϵ^ρ.


In order to define the bipartite transformations and probe the directional symmetry, we define ω_μν:=δ_νξ_μ, which yields Eq. (<ref>) to

    ω_μν+ω_νμ=0.

Eq. (<ref>) is also satisfied by the Lorentz generators. The velocity dependence can be introduced by

    ω^Fμ_ν    :=     g^Fμρ(u)ω_ρν
        =    1/1+σ[η^ρν+(σ^⊥/√(S)σ)^2 λ^ρλ^ν - 1/Ss^ρν]ω_νσ

where S:=ζ +σ, σ^⊥:=±√(ζ - s_μνu^μu^ν) and λ^ρ:=1/σ^⊥(s^ρ_ϵ-σ S/Fδ^ρ_ϵ)u^ϵ.






Therefore, the bipartite transformations are the result of the usual Lorentz transformations and the action of the background Lorentz-violating tensor s_μν on the 4-velocity u^μ. Accordingly, a Lorentz violating particle motion under a bipartite background  tensor can be regarded as a bipartite-symmetric movement.

There is still another important class of symmetry transformations in bipartite spacetime. Indeed, the Finslerian vielbein E^Fa_μ allow us to rewrite g^B_μν(x,u)u^μu^ν=η_abu'^au'^b, where g^B_μν(x,u)=E^Fa_μ(x,u)E^Fb_ν(x,u)η_ab and u'^a=E^Fa_μu^μ. In the bipartite spacetime, the first-order vielbein has the form

    E^Ba_μ(x,u)=√(F/α)E^a_μ + √(F/ζσ)s^a_μ +√(ασ/-k^2)k^ak_μ,

where E^a_μ is the local Lorentzian vielbein and k_μ is assumed as timelike vector. The bipartite vielbein represents a local anisotropic transformation that transforms the coordinate basis {∂_μ} into a Finslerian 
orthogonal basis {e_a=E^μ_a∂_μ}. The vielbein changes the 4-velocity u^μ into the anisotropic vector u'^a=E^Fa_μu^μ, such that the Lorentz violation is moved from the spacetime (Finsler metric) into the
u'^a. This is analogous to the light propagation in anisotropic media where the electric and displacement fields are related by the anisotropic polarization tensor, D^i=ϵ^i_jE^j. 

Once we found the infinitesimal bipartite transformations, let us study the proper bipartite generators



where

    J^F_μν   :=    x_μP_Fν-x_νP_Fμ
        =     J_μν + m/σ(x_μ s_νρ - x_ν s_μρ)u^ρ.

In Randers space, the generators take the form J^F_μν=J_μν + m(x_μ b_ν - x_ν b_μ), which is similar to one found in Ref. <cit.>. Defining the bipartite generators operators

    J_μν^B= -i(x_μδ_ν-x_νδ_μ),

the proper bipartite particle transformation takes the form

    Λ_BP(λ):=e^i1/2(J^F_μνω^μν)λ.












§ BIPARTITE ALGEBRA


The Finsler algebra is strongly modified by the anisotropy of the
action. In fact, the momentum algebra is given by

    [P^F_μ , P^F_ν]	   	=	   	-[δ_μ , δ_ν]=2 δ_[μ N^ρ_ν]∂/∂ u^ρ
       	=	   	 R^Fσ_ρμν(x,P^F)P^Fρ∂/∂ P^Fσ.



As a matter of fact, the anisotropy causes non-inertial effects measures with the Finslerian horizontal curvature R^Fσ_ρμν(x,u). Indeed, this results in the nonvanishing commutator in Eq. (<ref>). The commutator (<ref>) differs from one found in Ref. <cit.> due to the definition of the momentum generator in function of the horizontal derivative δ_μ in Eq. (<ref>).

The angular momentum generators are even more sensible to the anisotropic structure. In fact, for a constant s_μν,  the  angular momentum generators satisfy

    [J^F_μν , J^F_λρ]	   	=	   	-g^F_μλ(P^F)J^F_νρ-g^F_νρ(P^F)J^F_μλ
       	+	   g^F_μρ(P^F)J^F_νλ + g^F_νλ(P^F)J^F_μρ.

Note the presence of the Finsler metric g^F_μλ(P^F) which enhances the momentum dependence when compared to the 
Ref. <cit.>.

We can rewrite Eq. (<ref>) as 

    [J^F_μν , J^F_λρ]   =   α/F(-η_μλJ^F_νρ-η_νρJ^F_μλ +	η_μρJ^F_νλ + η_νλJ^F_μρ) -D^F_μλ(P^F)J^F_νρ
       -   D^F_νρ(P^F)J^F_μλ D^F_μρ(P^F)J^F_νλ + D^F_νλ(P^F)J^F_μρ,







where D_μν^F:=g_μν^F-α/Fη_μν.
Assuming the background vector is still constant, the commutator between P^F_μ and J^F_νλ takes the form

    [P^F_μ, J^F_νλ]	   	=	   	g^F_μν(P^F)P^F_λ - g^F_μλ(P^F)P^F_ν.


It is worthwhile to say that the commutation relations in equations (<ref>), (<ref>) and (<ref>) have an analogous form of that of the Lorentz invariant transformations, changing only J^F_μν→ J^F_μν and η_μν→ g^F_μν(P^F).

The Finsler metric deforms the algebra between momentum, Randers transformations and position. Indeed, for a constant s_μν,

    [P^F_μ , x_ν]=-i(g^F_μν(x,P^F)+δ_μ(g^F_νρ(x,P^F)x^ρ),


    [J^F_μν , x_λ ]=-i{x_μ g^F_νλ(x,P^F)-x_ν g^F_μλ(x,P^F)}.

Since for ζ=0, g^F_μν(x,P^F)=η_μν, the Randers algebra resembles the deformed Heisenberg-Weyl algebra of the κ-Minkowski spacetime <cit.>.

From the bipartite generators, we define the boost generators

    K_i^F	   	:=	   	J^F_0i=K_i + m/σ(x_0 s_iρ - x_i s_0ρ)u^ρ,

and the bipartite angular momentum generator

    J^F_i	   	:=	   	-1/2ϵ^jk_iJ^F_jk =J_i - ζ m ϵ^jk_i a_j x_k.

Remarkably, the background vector a^μ produces not only an expected background boost generator but an angular momentum generator as well.


At the particle rest frame where g^F_00(u)=1, the boosts and angular momentum generators satisfy










    [K^F_i , K^F_j]=ϵ^k_ijJ^F_k + 2 g^F_0[iK^F_j],


    [K^F_i , J^F_j]=ϵ^k_ijK^F_k + 2 g^F_0[iJ^F_j],


    [J^F_i , J^F_j]=ϵ^k_ijJ^F_k.

For a timelike background vector a^μ, where we can set g^F_0i(u)=0, the Randers algebra takes a form of an extension of the Lorentz algebra by K_μ→ K^F_μ and J_μν→ J^F_μν. For a spacelike and lightlike a^μ, the presence of the terms g^F_0i provides mixtures between boosts and Randers generators.





§ SYMMETRY EFFECTS


Once we found the symmetries of bipartite spacetime let us consider the motion of a massive particle in the bipartite space and the effects of this motion on conserved quantities.

The covariant canonical momentum is P_μ ^B=(m u_μ + 1/σs_μνu^ν). Using the local Lorentz-invariant metric g^μν, the canonical momentum 4-vector is P̂^μ=g^μνP_ν ^B=(δ^μ_ν +1/σs^μ_ν )P^ν. However, P̂^μ is not preserved as the particle moves. Instead, using the bipartite-Finsler metric, the Finslerian momentum P^Bμ:=g^Bμν(x,u)P^B_ν = m/Fu^μ is a covariant constant 4-vector. 
If we parametrize the particle worldline to have a constant Finsler norm, i.e., assuming F(x,u)=1, then P^Bμ=P^μ. 
That is equivalent to define a bipartite-Finsler proper time dτ_B= β_Bdt, where the bipartite factor is β_B := √(1-v^2)+√(λ_0 + ∑_i=1^3λ_i (v^i)^2).

The Finslerian momentum P^Bμ satisfies the Finsler geodesic equation

    DP^Bμ/dt=0	   ⇒   dP^Bμ/dt+Γ^Bμ_νρ(x,u)u^ν P^Bρ=0.

Consider the connection coefficients in the first-order of s_μν and up to the quadratic terms in u. The equation of motion (EoM) is given by

    dP^μ/dt+mα/σ(u^ν∂_ν s^μ_ρ)u^ρ-mα/2σ(∂ ^μs_νρ)u^νu^ρ=0,

and in a static background s_μν, it is equivalent to the pair of equations

    dE/dt   =   -mα/σv⃗·∇λ_0,
    dP⃗/dt   =   mα/σ(∇λ_0 - v⃗·∇(λ_i)v^i+ ∇(s_ij)v^iv^j).

In the low velocities regime, the EoM looks rather similar to the Newtonian limit whose potential is given by mα/σ∇λ_0. However, note the presence of the terms proportional to the 4-velocity, which can yield to unstable motions.


For the point particle 4-momentum P^μ=m u^μ, defining the effective metric tensor g̃_μν:=g_μν-s_μν, the modified dispersion relation (MDR), provided by the coupling F(x,P)=√(g_μνP^μP^ν)+ √(s_μνP^μP^ν)=m, yields to a quartic MDR






    (g̃_μνP^μ P^ν - m^2)^2=4m^2 s_μν P^μ P^ν.


In the Randers space, the Eq. (<ref>) leads to the quadratic MDR P^2 + 2 λ (b· P) - λ^2 (b· P)^2=m^2. 

On the other hand, the modified dispersion relation in a flat Minkowsky space g_μν=η_μν (<ref>) leads to

    [(1-λ_0)E^2 -∑_i=1^3(1-λ_i) (P^i)^2-m^2]^2 = 4m^2 [λ_0 E^2 + ∑_i=1^3λ_i (P^i)^2].

The group velocity  v_g^j = dE/dP_j takes the form

    v^j={δ (1-λ_j) + 2m^2 λ_j/δ (1-λ_0) - 2m^2 λ_0}P^j/E,

where δ:=(1-λ_0)E^2 -∑_i=1^3(1-λ_i) (P^i)^2-m^2.
For a pure timelike bipartite tensor s_00=λ_0 >0 and s_ij=0, 




we plot the deformed mass-shell in figure (<ref>) and the group velocity in figure (<ref>).


The bipartite geometry exhibits causal issues for high momenta, such as superluminal velocities, a common feature of Lorentz violating models <cit.>. 

Now let us consider the effects of the bipartite background tensor upon a particle moving near a massive and spherically symmetric body. Assuming a sufficient weak Lorentz violation such that the geometry is still static and spherically symmetric, the existence of a timelike Killing vector ζ^μ=∂/∂ t provides a conserved energy per unit of mass given by E=g_μν(x)ζ^μu^ν whereas the angular symmetry ψ^μ=∂/∂ϕ yields to a conserved particle angular momentum L=g_μνϕ^μζ^ν. By considering that the Schwarzschild geometry is not altered by the bipartite tensor, the geodesic condition for a timelike particle g_μν^B(x,u)u^μu^ν=1 yields to the modified particle dispersion relation

    [E^2/f - ṙ^2/f - (1+L^2/r^2+s_μνu^μu^ν)]^2 =4s_μνu^μu^ν,

where ds^2 = f(r)dt^2 -f^-1dr^2-r^2 dθ^2 -r^2sin^2θ dϕ^2, u^μ =dx^μ/dτ=(ṫ,ṙ,θ̇,ϕ̇) and f(r)=(1-2GM/r).

In Randers spacetime for  b^μ=(b^0 , b^r , 0, b^ϕ) the dispersion relation takes the form

    E^2/2   =   ṙ^2/2+f{1+1/r^2L^2 +(b^0 E/f - f^-1b^r ṙ-b^ϕ L)^2 
       +    (b^0 f^-1E - f^-1b^r ṙ -L b^ϕ)}.

The radial component of the Lorentz-violating vector b^r provides velocity-dependent potential term. A background vector with angular component b^μ=(0,0,0,b^ϕ) yields to an effective potential whose asymptotic value depends on the test particle angular momentum. For a timelike background vector b^μ=(b^0,0⃗), the first-order perturbed MDR is

    E^2=ṙ^2/2+f(L^2/r^2+1)± 2b^0 f √(ṙ^2 + f(L^2/r^2+1)),

and hence, the effective gravitational potential also exhibits a velocity-dependent term. These examples show that the MDR caused by the bipartite tensor leads to instabilities of particle orbits and asymptotic exotic effects even in the Randers spacetime.




















§ FINAL REMARKS AND PERSPECTIVES


We analyzed the symmetries of the Lorentz violating bipartite-Finsler spacetime. Since the Finsler metric contains corrections due to the background Lorentz violating bipartite tensor, the Finslerian Killing equation enabled us to 
obtain the particle transformations leaving invariant the particle Lagrangian. It is worthwhile to mention that we assumed
particle displacements on the bipartite spacetime M_4, not on its tangent bundle TTM_4.

We obtained bipartite translations, rotations and boosts, provided the bipartite tensor be constant. The presence of the bipartite tensor in the particle transformations indicates that the Lorentz invariant conserved quantities must be modified by the bipartite tensor in order to be conserved. Moreover, the direction-dependence of the bipartite metric yields to a deformation of the bipartite vielbeins and the Lorentz algebra resembling the κ-algebra. The quartic dispersion relation yields to causality issues at the UV regime and modifies the gravitational potential by introducing dissipative-like terms. These features suggest the bipartite-Finsler structure might exhibit interesting properties in early cosmology effects.



§ ACKNOWLEDGMENTS

The authors thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), grants no 312356/2017-0 (JEGS), no 305678/2015-9 (RVM) and no 308638/2015-8 (CASA) for financial support.





§ REFERENCES

99

KS  V. A. Kostelecký and S. Samuel, Phys. Rev. D 39,
683 (1989);  Phys. Rev. Lett. 63, 224 (1989). Phys. Rev. Lett. 66, 1811
(1991).

vsr
   A. G. Cohen and S. L. Glashow,

   Phys. Rev. Lett.  97, 021601 (2006).
R. V. Maluf, J. E. G. Silva, W. T. Cruz and C. A. S. Almeida,
  
  Phys. Lett. B 738, 341 (2014).


dsr
   J. Magueijo and L. Smolin,

  Phys. Rev. Lett.  88, 190403 (2002).


noncommutative

S. M. Carroll, J. A. Harvey, V. A. Kostelecký, C. D. Lane, and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001).

horava P. Hořava, Phys. Rev. D 79, 084008 (2009).




lqg
J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia,
  
  Phys. Rev. Lett.  84, 2318 (2000).
  
  
  Phys. Rev. D 65, 103509 (2002).



cptviolation
  D. Colladay and V. A. Kostelecký,
  
  Phys. Rev. D 55, 6760 (1997).  Phys. Rev. D 58, 116002 (1998).




  
 

neutrinos
V. A. Kostelecký and M. Mewes,
  
  Phys. Rev. D 69, 016005 (2004).
  
photon
R. Casana, M. M. Ferreira, Jr, A. R. Gomes and P. R. D. Pinheiro,
  
  Phys. Rev. D 80, 125040 (2009).  
  
neutralmesons  
 V. A. Kostelecký,
  
  Phys. Rev. Lett.  80, 1818 (1998).




  
 



kosteleckygravity
V. A. Kostelecký,
  
  Phys. Rev. D 69, 105009 (2004).

bumblebee
R. Bluhm and V. A. Kostelecký, Phys. Rev. D 
71, 065008 (2005).

  R. V. Maluf, J. E. G. Silva and C. A. S. Almeida,
  
  Phys. Lett. B 749, 304 (2015).

bumblebeecosmology 
  D. Capelo and J. Páramos,
  
  Phys. Rev. D 91, 104007 (2015).

 
 
kosteleckymdr
   V. A. Kostelecký and N. Russell,

   Phys. Lett. B 693, 443 (2010).
   

 

  
   
  
  

Gibbons
   G. W. Gibbons, J. Gomis and C. N. Pope,

   Phys. Rev. D 76, 081701 (2007). 

kouretsis   
A. P. Kouretsis, M. Stathakopoulos and P. C. Stavrinos,
  
  Phys. Rev. D 79, 104011 (2009).
  
   
girelli
  F. Girelli, S. Liberati and L. Sindoni,
  
  Phys. Rev. D 75, 064015 (2007).
  
vacaru
   S. I. Vacaru,

   Class. Quant. Grav.  28, 215001 (2011).
  
  JHEP 9809, 011 (1998).
  
pfeifer
   C. Pfeifer and M. N. R. Wohlfarth,

   Phys. Rev. D 85, 064009 (2012).  Phys. Rev. D 84, 044039 (2011).
  
  
rund
H. Rund, The differential geometry of Finsler spaces, Springer (1959).

   
 chern
D. Bao, S. Chern, Z. Shen, An introduction to Riemann-Finsler geometry,
Springer, 1991.












 
  

   
optics A. Joets and R. Ribotta, Opt. Commun., 107, (1994).
G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick and M. C. Werner,
   
Phys. Rev. D 79, 044022 (2009).

graphene M. Cvetic and G. W. Gibbons,
  
 Annals Phys.  327, 2617 (2012)
 
kosteleckyfinsler1
   V. A. Kostelecký,
   
   Phys. Lett. B 701, 137 (2011).




ColladayMc
  D. Colladay and P. McDonald,
  
  Phys. Rev. D 85, 044042 (2012). Phys. Rev. D 92,  085031 (2015).
  

russell
  N. Russell,
  
  Phys. Rev. D 91, 045008 (2015).
  


marco M. Schreck,
  
  Phys. Rev. D 91, 105001 (2015).

  
 


marcofinsler1 M. Schreck,
 
  Phys. Rev. D 92, 125032 (2015).

edwards B. R. Edwards and V. A. Kostelecky,
  
  Phys. Lett. B 786, 319 (2018).
  
  



randers
 G. Randers,
 Phys. Rev. 59, 195 (1941).

euclidesranders 
  J. E. G. Silva, R. V. Maluf and C. A. S. Almeida,
  
  Phys. Lett. B 766, 263 (2017)
 
  

foster
  J. Foster and R. Lehnert,
  
  Phys. Lett. B 746, 164 (2015).


randerscosmology
  Z. Chang and X. Li,
  
  Phys. Lett. B 676, 173 (2009). 




  
 
  
  
  


randersastrophysics
 Z. Chang and X. Li,
  
  Phys. Lett. B 668, 453 (2008). X. Li and Z. Chang,
  
  Phys. Lett. B 692, 1 (2010).
 
  



Kosteleckybipartite
    V. A. Kostelecký, N. Russell and R. Tso,

   Phys. Lett. B 716, 470 (2012).






euclides
  J. E. G. Silva and C. A. S. Almeida,
  
  Phys. Lett. B 731, 74 (2014).


kalgebra
J. Lukierski, A. Nowicki and H. Ruegg,
  
  Phys. Lett. B 293, 344 (1992).



 
  
  








 











 























  
  


changmdr
   Z. Chang and X. Li,

   Phys. Lett. B 663, 103 (2008).

changkilling
X. Li and Z. Chang,
  
  Differ. Geom. Appl.  30, 737 (2012).

 

bogoslovsky
G. Y. Bogoslovsky and H. F. Goenner,
  
  
  

   Phys. Lett. A 323, 40 (2004).


  
  

 
  
  
  
  



  
 





 
  
  
  

stability
 V. A. Kostelecký and R. Lehnert,
  
  Phys. Rev. D 63, 065008 (2001). 
  
  
  

Lammerzahl
  C. Lämmerzahl, V. Perlick and W. Hasse,
  
  Phys. Rev. D 86, 104042 (2012).
  
  
  


 
  
  
  
  
  
  


 
  
  
  
  
  
  
  

 




  
 


  
  
  


 
  









  
 
  
  
 
  
  
 
  
 
  
  
  
  


 
  
  


 
  
  
  
  
  
 
  
  








