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# Isometric group of ( α, β ) -type Finsler space and the symmetry of Very Special Relativity 

$^{1}$Institute of Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, China 

Xin Li 1 , $^{4}$, ∗ Zhe Chang 2 , $^{4}$, † and Xiaohuan Mo 3 ‡ 

$^{2}$Institute of High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China 

$^{3}$Key Laboratory of Pure and Applied Mathematics School of Mathematical Sciences, Peking University, Beijing 100871, China 

$^{4}$Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences 

The Killing equation for a general Finsler space is set up. It is showed that the Killing equation of ( α, β ) space can be divided into two parts. One is the same with Killing equation of a Riemannian metric, another equation can be regarded as a constraint. The solutions of Killing equations present explicitly the isometric symmetry of Finsler space. We find that the isometric group of a special case of ( α, β ) space is the same with the symmetry of Very Special Relativity (VSR). The Killing vectors of Finsler-Funk space are given. Unlike Riemannian constant curvature space, the 4 dimensional Funk space with constant curvature just have 6 independent Killing vectors. 

In the past few years, two interesting theories of in- vestigating the violation of Lorentz Invariance (LI) are proposed. One is the so called Doubly Special Relativity (DSR) [1–5]. This theory takes Planck-scale effects into account by introducing an invariant Planckian parameter in the theory of special relativity. Another is the so called Very Special Relativity (VSR) developed by Cohen and Glashow [6]. This theory suggested that the exact sym- metry group of nature may be isomorphic to a subgroup SIM(2) of the Poincare group. And the SIM(2) group semi-direct product with the spacetime translation group gives an 8-dimensional subgroup of the Poincare group called ISIM(2) [7]. Under the symmetry of ISIM(2), the CPT symmetry is preserved and many empirical suc- cesses of special relativity are still functioned. 

Recently, Physicists found that the two theories men- tioned above are related with Finsler geometry. Girelli, Liberati and Sindoni [8] showed that the Modified dis- persion relation (MDR) in DSR can be incorporated into the framework of Finsler geometry. The symmetry of the MDR was described in the Hamiltonian formalism. Also, Gibbons, Gomis and Pope [9] showed that the Finslerian line element ds = ( η$_{µν}$dx $^{µ}$dx$^{ν}$) (1 − b ) / $^{2}$( n$_{ρ}$dx $^{ρ}$)b is invari- ant under the transformations of the group DISIM$_{b}$(2) (1-parameter family of deformations of ISIM(2)). 

Finsler geometry as a natural generation of Riemann geometry could provide new sight on modern physics. the model based on Finsler geometry could explain the recent astronomical observations which Einstein’s gravity could not. An incomplete list includes: the flat rotation curves of spiral galaxies can be deduced naturally without invoking dark matter [10]; the anomalous acceleration[11] 

PACS numbers: 02.40.-k,11.30.-j 

## I. INTRODUCTION 

in solar system observed by Pioneer 10 and 11 spacecrafts could account for a special Finsler space-Randers space [12]; the secular trend in the astronomical unit[13, 14] and the anomalous secular eccentricity variation of the Moon’s orbit[15] could account for effect of the length change of unit circle in Finsler geometry[16]. 

Thus, the symmetry of Finslerian spacetime is worth investigating. The way of describing spacetime symme- try in a covariant language (the symmetry should not depend on any particular choice of coordinate system) involves the concept of isometric transformation. In fact, the symmetry of spacetime is described by the so called isometric group. The generators of isometric group is directly connected with the Killing vectors[17]. In this paper, we use solutions of the Killing equation to estab- lish the symmetry of a class of Finslerian spacetime. In particular, we show that the isometric group of a special kind of ( α, β ) space is equivalent to the symmetry of the VSR. 

## II. KILLING VECTOR IN RIEMANN SPACE 

In this section, we give a brief review of the Killing vec- tors in Riemann space (further material can be found, for example, in [18]). Under a given coordinate transforma- tion x → ¯ x , the Riemannian metric g$_{µν}$ ( x ) transforms as 

Any transformation x → ¯ x is called isometry if and only if the transformation of the metric g$_{µν}$ ( x ) satisfies 

One can check that the isometric transformations do form a group. It is convenient to investigate the isometric 

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