###### Abstract

Lorentz invariance is one of the foundations of modern physics; however,
Lorentz violation may happen from the perspective of quantum gravity, and
plenty of studies on Lorentz violation have arisen in recent years. As a good
tool to explore Lorentz violation, Finsler geometry is a natural and
fundamental generalization of Riemann geometry. The Finsler structure depends
on both coordinates and velocities. Here, we simply introduce the mathematics
of Finsler geometry. We review the connection between modified dispersion
relations and Finsler geometries and discuss the physical influence from
Finsler geometry. We review the connection between Finsler geometries and
theories of Lorentz violation, such as the doubly special relativity, the
standard-model extension, and the very special relativity.

###### keywords:

Finsler geometry; Lorentz violation; modified dispersion relation

15 5 978 Academic Editor: Giovanni Modanese 26 March 2023 20 April 2023 21
April 2023 25 April 2023 https://doi.org/10.3390/sym15050978 Lorentz Violation
in Finsler Geometry Lorentz Violation in Finsler Geometry Jie Zhu 1 and Bo-
Qiang Ma 1,2,3,∗ Jie Zhu and Bo-Qiang Ma Zhu, J.; Ma, B.-Q. Correspondence:
<EMAIL_ADDRESS>
##  1 Introduction

As a basic symmetry of spacetime, Lorentz invariance is a basic assumption of
Einstein’s relativity, and now it is the foundation of general relativity and
quantum field theory in modern physics. However, in quantum gravity (QG)
Amelino (2005), Lorentz violation (LV) may happen at the Planck scale ($E_{\rm
Pl}\simeq 1.22\times 10^{19}~{}\rm{GeV}$). There are many kinds of theoretical
models including LV effects, including the QG theories such as string theory
Danielsson (2001); Li (2021, 2022, 2023) and loop quantum gravity Rovelli
(2008); Ashtekar (2021); Li (2023), spacetime structure theories such as
doubly special relativity (DSR) Amelino (2001, 2002); Smolin (2002, 2003) and
very special relativity (VSR) Glashow (2006), and the effective theory with
extra terms, such as the standard-model extension (SME) Colladay (1997, 1998);
Kostelecký (2004). For a recent review on Lorentz invariance violation, see He
(2022).

Among many LV theories, modified dispersion relations (MDRs) are introduced,
and for energy $E\ll E_{\rm Pl}$, the MDR can be expressed at the leading
order as

$E^{2}=m^{2}+p^{2}+\alpha p^{n+2},$ (1)

where $n$ is the broken order here, and $\alpha$ is a parameter with mass
dimension and $[\alpha]=-n$. In 2007, Girelli et al. found the relations
between MDRs and Finsler geometries, and in the same year, Gibbons et al.
found that the general VSR is Finsler geometry. Since then, Finsler geometry
has drawn more attention to LV studies, for example, the application of
Randers–Finsler geometry on LV ChangZ (2008, 2008).

Finsler geometry is the generalization of Riemann geometry and is named after
Paul Finsler who studied it in his doctoral thesis in 1917. Finsler geometries
have many applications in different physics domains. Since Riemann geometry
describes gravity, it is natural to study gravitation via Finsler geometry to
pursue new physics. The gravitation theory in Berwald–Finsler spacetime is
built up in Ref. LiX (2010), and the vacuum field equation in Finsler
spacetime is present in Ref. Pfeifer (2012). The modification of Newtonian
dynamics is also considered in Finsler geometry LiX (2013). In addition,
Finsler geometries are applied to cosmology problems, such as dark matter
ChangZ (2008); Saridakis (2013), dark energy ChangZ (2008), bounce cosmology
Saridakis (2019), and anisotropy of the universe ChangZ (2014, 2014), and
Finsler-like theories Saridakis (2019, 2021) are also applied to cosmology.
Gravitational waves are also studied in Finsler spacetime LiX (2013). Even
neutrino oscillations have a realization in Finsler spacetime Antonelli
(2018). One can conclude from the above examples that Finsler geometry has an
infinite number of degrees of freedom to choose from, which can be modified at
will to meet different requirements in different physics domains. It is not
reasonable to introduce all of the applications of different kinds of Finsler
geometries in a short review. In this review, we focus on the applications of
Finsler geometry on Lorentz invariance violation.

This review is organized as follows: Section 2 briefly introduces the
mathematics of Finsler geometry. Section 3 introduces the connection between
MDRs and Finsler geometries. Section 4 introduces the connection between LV
theories and Finsler geometries, including DSR, SME, and VSR. Section 5
reviews and discusses the topic of this review.

##  2 Introduction to Finsler Geometry

###  2.1 A First Glance at Finsler Geometry

Essentially, a Finsler manifold is a manifold $M$ where each tangent space is
equipped with a Minkowski norm, that is, a norm that is not necessarily
induced by an inner product. The metric of a Finsler manifold depends on not
only the points of $M$, but also the directions in the tangent bundle of $M$.

For a manifold $M$, denote by $T_{x}M$ the tangent space at $x\in M$, and by
$TM$ the tangent bundle of $M$. Each element of $TM$ has the form $(x,y)$,
where $x\in M$ and $y\in T_{x}M$. The natural projection $\pi:TM\rightarrow M$
is given by $\pi(x,y)\equiv x$. A Finsler structure or Finsler norm of $M$ is
a function

$F:TM\rightarrow[0,\infty)$ (2)

with the following properties:

  1. 1.

Regularity: $\mathrm{F}$ is $C^{\infty}$ on the entire slit tangent bundle
$TM_{0}:=TM\backslash\\{0\\}$.

  2. 2.

Positive homogeneity : $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$.

  3. 3.

Strong convexity: The $n\times n$ Hessian matrix

$g_{ij}:=\left(\frac{1}{2}F^{2}\right)_{y^{i}y^{j}}$ (3)

is positive-definite at every point of $TM_{0}$, where we use the notation
()${}_{y^{i}}=$ $\frac{\partial}{\partial y^{i}}()$.

$g_{ij}$ in Equation (3) is called the Finsler metric. From Euler’s
homogeneous function theorem, the Finsler norm $F$ has attributions such that

$y^{i}F_{y^{i}}=F,\quad y^{i}F_{y^{i}y^{j}}=0,\quad
F^{2}=g_{ij}(x,y)y^{i}y^{j}.$ (4)

Finsler geometry has its genesis in integrals of the form

$L_{F}(C)=\int_{C}F\left(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau}\right)d\tau,$
(5)

and its geometric meaning is the distance between two points in the Finsler
manifold through a certain path $C$. Given a manifold $M$ and a Finsler
structure $F$ on $TM$, the pair $(M,F)$ is called a Finsler manifold. The
Finsler structure $F$ is a function of $\left(x^{i},y^{i}\right)$. In the case
of $g_{ij}$, depending on $x^{i}$ only, the Finsler manifold reduces to
Riemannian manifold.

To make things easier understood by physicists, here are some examples. As
mentioned above, Riemann geometry is a special Finsler geometry. Riemann
geometry can be described by the Riemann metric

$ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu},$ (6)

and, here, $g_{\mu\nu}(x)$ depends only on the point $x$. In the language of
Finsler geometry, the Finsler structure of the above Riemann metric is

$F=\sqrt{g_{\mu\nu}(x)y^{\mu}y^{\nu}},$ (7)

where $y^{\mu}=dx^{\mu}/d\tau$. A kind of special Finsler geometry is of
Randers type, and its Finsler norm is modified from Riemann metric as

$F=\sqrt{a_{\mu\nu}(x)y^{\mu}y^{\nu}}+b_{\mu}(x)y^{\mu}=\alpha(x,y)+\beta(x,y),$
(8)

where $\alpha(x,y)=\sqrt{a_{\mu\nu}(x)y^{\mu}y^{\nu}}$ and
$\beta(x,y)=b_{\mu}(x)y^{\mu}$. Generally, any form of

$F=\alpha(x,y)\phi(s),\quad s=\frac{\beta(x,y)}{\alpha(x,y)},$ (9)

is a Finsler structure. Another example is that in a universe where the
Pythagorean theorem is modified from $c^{2}=a^{2}+b^{2}$ to
$c^{4}=a^{4}+b^{4}$, the length element in a two-dimensional plane is
$ds^{4}=dx_{1}^{4}+dx_{2}^{4}$; thus, the Finsler norm in this plane is

$F=(y_{1}^{4}+y_{2}^{4})^{\frac{1}{4}}.$ (10)

We can calculate the Finsler metric from Equation (3) as

$g_{ij}(x,y)=\begin{pmatrix}\frac{y_{1}^{6}+3y_{1}^{2}y_{2}^{4}}{\left(y_{1}^{4}+y_{2}^{4}\right)^{3/2}}&-\frac{2y_{1}^{3}y_{2}^{3}}{\left(y_{1}^{4}+y_{2}^{4}\right)^{3/2}}\\\
-\frac{2y_{1}^{3}y_{2}^{3}}{\left(y_{1}^{4}+y_{2}^{4}\right)^{3/2}}&\frac{3y_{1}^{4}y_{2}^{2}+y_{2}^{6}}{\left(y_{1}^{4}+y_{2}^{4}\right)^{3/2}}\end{pmatrix}.$
(11)

In this example, the Finsler metric depends only on $y$. We can identify the
differences between Finsler metrics and Riemann metrics in two ways: whether
the square of the length element is a quadratic form of $y$, or whether the
metric $g_{ij}(x,y)$ contains $y$.

To describe the “1 + 3” spacetime, instead of Finsler geometry we turn to
pseudo-Finsler geometry, similar to the relation between Riemann geometry and
pseudo-Riemann geometry that describes general relativity. To simplify the
description, in the following context, we do not distinguish between Finsler
geometry and pseudo-Finsler geometry; we call them Finsler geometry.

###  2.2 Mathematical Concepts of Finsler Geometry

Finsler geometry contains analogs for many of the natural objects in
Riemannian geometry, but their formulas are very different. For example, the
Christoffel symbol in Riemann geometry plays the role in the connection and
can be expressed as

$\gamma_{jk}^{i}=\frac{g^{il}}{2}\left(\frac{\partial g_{lj}}{\partial
x^{k}}+\frac{\partial g_{lk}}{\partial x^{j}}-\frac{\partial g_{jk}}{\partial
x^{l}}\right),$ (12)

where $g^{il}$ is the inverse matrix of $g_{il}$. In Finsler geometry,
$\gamma_{jk}^{i}$ in Equation (12) is less important and is called the _formal
Christoffel symbol._ Since this review is for physicists to quickly understand
Finsler geometry, here we just review some major concepts in Finsler geometry
quickly. For detailed discussions see Ref. Bao (2000).

The _Hilbert form_ is a 1-form defined as

$\omega:=F_{y^{i}}dx^{i}=\frac{y^{j}}{F}g_{ji}dx^{i},$ (13)

and its dual vector field is

$l=l^{i}\frac{\partial}{\partial
x^{i}}:=\frac{y^{i}}{F}\frac{\partial}{\partial x^{i}}.$ (14)

The Hilbert form is invariant under coordinate transformations. Assume that
$(\tilde{x},\tilde{y})$ is another local coordination system, thus

$\tilde{y}^{i}=\frac{\partial\tilde{x}^{i}}{\partial
x^{j}}y^{j},F_{y^{i}}=\frac{\partial\tilde{x}^{j}}{\partial
x^{i}}F_{\tilde{y}^{j}},$ (15)

so

$\omega=F_{y^{i}}dx^{i}=F_{\tilde{y}^{i}}d\tilde{x}^{i}.$ (16)

From Equation (4), the distance integration in Equation (5) can also be
expressed as

$L_{F}(C)=\int_{C}\omega.$ (17)

The _Cartan tensor_ is defined as

$\begin{split}&A_{ijk}:=\frac{F}{2}\frac{\partial g_{ij}}{\partial
y^{k}}=\frac{F}{4}(F^{2})_{y^{i}y^{j}y^{k}},\\\ &A:=A_{ijk}dx^{i}\otimes
dx^{j}\otimes dx^{k}.\end{split}$ (18)

Obviously, a Finsler manifold is a Riemann manifold if and only if $A\equiv
0$. The Cartan tensor is a symmetric covariant tensor with an attribute

$y^{i}A_{ijk}=y^{j}A_{ijk}=y^{k}A_{ijk}=0.$ (19)

As a vector field on $TM$, the transformation formula of
$\frac{\partial}{\partial x^{i}}$ is

$\frac{\partial}{\partial\tilde{x}^{i}}=\frac{\partial
x^{k}}{\partial\tilde{x}^{i}}\frac{\partial}{\partial
x^{k}}+\frac{\partial^{2}x^{k}}{\partial\tilde{x}^{i}\partial\tilde{x}^{j}}\tilde{y}^{j}\frac{\partial}{\partial
y^{k}}.$ (20)

To obtain the same transformation law as a tensor, we need to replace
$\frac{\partial}{\partial x^{i}}$ by $\frac{\delta}{\delta x^{i}}$ as

$\frac{\delta}{\delta x^{i}}:=\frac{\partial}{\partial
x^{i}}-N^{j}_{i}\frac{\partial}{\partial y^{j}},$ (21)

where

$N^{j}_{i}:=\gamma^{j}_{ik}y^{k}-g^{jl}\frac{A_{ikl}}{F}\gamma^{k}_{pq}y^{p}y^{q}$
(22)

is connection, and $\gamma^{i}_{jk}$ is the formal Christoffel symbol in
Equation (12). One can easily check

$\frac{\delta}{\delta\tilde{x}^{i}}=\frac{\partial
x^{k}}{\partial\tilde{x}^{i}}\frac{\delta}{\delta x^{k}}.$ (23)

Denote by $T(TM_{0})$ the tangent bundle of $TM_{0}$, and $T(TM_{0})$ can be
divided into the horizontal part $HTM$ spanned by $\\{\frac{\delta}{\delta
x^{i}}\\}$ and the vertical part $VTM$ spanned by $\\{\frac{\partial}{\partial
y^{i}}\\}$. Their dual bases are $\\{dx^{i}\\}$ and $\\{\delta y^{i}\\}$,
where

$\delta y^{i}:=dy^{i}+N^{i}_{j}dx^{j}.$ (24)

In other words,

$\begin{split}&T(TM_{0})=HTM\oplus VTM=\text{span}\\{\frac{\delta}{\delta
x^{i}}\\}\oplus\text{span}\\{\frac{\partial}{\partial y^{i}}\\},\\\
&T^{*}(TM_{0})=H^{*}TM\oplus
V^{*}TM=\text{span}\\{dx^{i}\\}\oplus\text{span}\\{\delta
y^{i}\\}.\end{split}$ (25)

Since Finsler geometry is discussed on $TM$, the division above is important
and shows the differences with Riemann geometry.

_Chern connections_ are connection 1-forms $\omega^{i}_{j}$ with attributes
torsion freeness

$dx^{j}\wedge\omega_{j}^{i}=0$ (26)

and almost $g$-compatibility

$dg_{ij}-g_{kj}\omega_{i}^{k}-g_{ik}\omega_{j}^{k}=2\frac{A_{ijs}}{F}\delta
y^{s}.$ (27)

The Chern 1-form can be expressed as

$\omega^{i}_{j}=\Gamma^{i}_{jk}dx^{k},$ (28)

where $\Gamma^{i}_{jk}$ is the _Christoffel symbol (or the connection
coefficients) of the Chern connection_ with an attribute
$\Gamma^{i}_{jk}=\Gamma^{i}_{kj}$. The Christoffel symbol of the Chern
connection can be expressed as

$\begin{split}\Gamma^{l}_{ij}&=\frac{1}{2}g^{lk}\left(\frac{\delta
g_{ik}}{\delta x^{j}}+\frac{\delta g_{jk}}{\delta x^{i}}-\frac{\delta
g_{ij}}{\delta x^{k}}\right)\\\
&=\gamma^{l}_{ij}-\frac{g^{lk}}{F}(A_{jkp}N^{p}_{i}+A_{kip}N^{p}_{j}-A_{ijp}N^{p}_{k}).\end{split}$
(29)

For any tensor on $TM_{0}$, we can define its covariant derivative via Chern
connections. Since $T^{*}(TM_{0})$ can be divided into two parts, the
covariant derivatives can be also divided into horizontal parts and vertical
parts. For example, for a $(1,1)$-tensor
$T=\tensor{T}{{}^{i}_{j}}\frac{\partial}{\partial x^{i}}\otimes dx^{j}$, its
covariant derivative can be defined as

$\nabla\tensor{T}{{}^{i}_{j}}:=d\tensor{T}{{}^{i}_{j}}+\tensor{T}{{}^{k}_{j}}\omega^{i}_{k}-\tensor{T}{{}^{i}_{k}}\omega^{k}_{j}=\tensor{T}{{}^{i}_{j|k}}dx^{k}+\tensor{T}{{}^{i}_{j;k}}\delta
y^{k},$ (30)

where $\tensor{T}{{}^{i}_{j|k}}$ and $\tensor{T}{{}^{i}_{j;k}}$ are called
_horizontal covariant derivative_ and _vertical covariant derivative_ ,
respectively. Their expressions are shown as follows:

$\begin{split}\tensor{T}{{}^{i}_{j|k}}&=\frac{\partial\tensor{T}{{}^{i}_{j}}}{\partial
x^{k}}+\tensor{T}{{}^{s}_{j}}\Gamma^{i}_{sk}-\tensor{T}{{}^{i}_{s}}\Gamma^{s}_{jk}-\tensor{T}{{}^{i}_{j;s}}N^{s}_{k}\\\
&=\frac{\delta\tensor{T}{{}^{i}_{j}}}{\delta
x^{k}}+\tensor{T}{{}^{s}_{j}}\Gamma^{i}_{sk}-\tensor{T}{{}^{i}_{s}}\Gamma^{s}_{jk},\\\
\tensor{T}{{}^{i}_{j;k}}&=\frac{\partial\tensor{T}{{}^{i}_{j}}}{\partial
y^{k}}.\end{split}$ (31)

The _geodesic spray coefficient_ is $G^{i}$, given by

$G^{i}=\frac{1}{4}g^{ij}\left((F^{2})_{y^{j}x^{k}}y^{k}-(F^{2})_{x^{j}}\right).$
(32)

The geodesic equation for the Finsler manifold is given as

$\frac{d^{2}x^{i}}{d\tau^{2}}+2G^{i}=0.$ (33)

The connections between geodesic spray coefficients and Christoffel symbols
are

$G^{i}=\frac{1}{2}\gamma^{i}_{jk}y^{j}y^{k}=\frac{1}{2}\Gamma^{i}_{jk}y^{j}y^{k},$
(34)

so the geodesic equations in Finsler manifolds are the same as in Riemann
manifolds. The connections between $\Gamma^{l}_{ij}$, $N^{i}_{j}$ and $G^{i}$
are

$N^{i}_{j}=\Gamma^{i}_{jk}y^{k}=\frac{\partial G^{i}}{\partial y^{j}},$ (35)

thus $N^{i}_{j}$ only depends on $G^{i}$.

The curvature 2-forms of the Chern connection are

$\begin{split}\Omega^{i}_{j}:&=d\omega^{i}_{j}-\omega^{k}_{j}\wedge\omega^{i}_{k}\\\
&=\frac{1}{2}\tensor{R}{{}_{j}^{i}{}_{k}{}_{l}}dx^{k}\wedge
dx^{l}+\tensor{P}{{}_{j}^{i}{}_{k}{}_{l}}dx^{k}\wedge\delta y^{l},\end{split}$
(36)

where $R$ and $P$ are the $hh-$ and $hv-$_curvature tensors of the Chern
connection_ , respectively. The expression of the $hh-$curvature tensor is

$\tensor{R}{{}_{j}^{i}{}_{k}{}_{l}}=\frac{\delta\Gamma^{i}_{jl}}{\delta
x^{k}}-\frac{\delta\Gamma^{i}_{jk}}{\delta
x^{l}}+\Gamma^{i}_{km}\Gamma^{m}_{jl}-\Gamma^{i}_{lm}\Gamma^{m}_{jk},$ (37)

and the $hh-$curvature tensor has attributes

$\begin{split}&\tensor{R}{{}_{j}^{i}{}_{k}{}_{l}}+\tensor{R}{{}_{j}^{i}{}_{l}{}_{k}}=0,\\\
&\tensor{R}{{}_{j}^{i}{}_{k}{}_{l}}+\tensor{R}{{}_{k}^{i}{}_{l}{}_{j}}+\tensor{R}{{}_{l}^{i}{}_{j}{}_{k}}=0.\end{split}$
(38)

The expression of the $hv-$curvature tensor is

$\tensor{P}{{}_{j}^{i}{}_{k}{}_{l}}=-\frac{\partial\Gamma^{i}_{jk}}{\partial
y^{l}}.$ (39)

Here, we can see another difference between Riemann manifolds and Finsler
manifolds: the curvature tensors have extra $hv$ parts in Finsler manifolds.
The $hv-$curvature tensors vanish if and only if the Christoffel symbol of the
Chern connection $\Gamma^{i}_{jk}=\Gamma^{i}_{jk}(x)$ is only the function of
the point $x$ of the manifold $M$. In this case, we call $M$ a
_Berwald–Finsler manifold_. A Riemann manifold is a Berwald–Finsler manifold,
and not vice versa. For Berwald–Finsler manifolds, the $hh-$curvature tensors
reduce to

$\tensor{R}{{}_{j}^{i}{}_{k}{}_{l}}=\frac{\partial\Gamma^{i}_{jl}}{\partial
x^{k}}-\frac{\partial\Gamma^{i}_{jk}}{\partial
x^{l}}+\Gamma^{i}_{km}\Gamma^{m}_{jl}-\Gamma^{i}_{lm}\Gamma^{m}_{jk},$ (40)

similar to Riemann manifolds.

The _flag curvature tensor_ is defined as

$R_{ij}:=g_{ki}\tensor{R}{{}^{k}_{j}}=g_{ki}y^{p}\tensor{R}{{}_{p}^{k}{}_{j}{}_{q}}y^{q}=y^{p}\tensor{R}{{}_{p}{}_{i}{}_{j}{}_{q}}y^{q}=-\tensor{R}{{}_{p}{}_{i}{}_{q}{}_{j}}y^{p}y^{q}.$
(41)

The _Ricci scalar_ is defined as

$Ric:=g^{ij}R_{ij}=\tensor{R}{{}^{i}_{i}}.$ (42)

The _Ricci tensor_ is defined as

$Ric_{ij}:=\frac{1}{2}(Ric)_{y^{i}y^{j}}.$ (43)

If $F$ is Riemann metric and $F^{2}=g_{ij}(x)y^{i}y^{j}$, there are relations
such that

$\tensor{R}{{}^{i}_{k}}=\tensor{R}{{}_{j}^{i}{}_{k}{}_{l}}(x)y^{j}y^{l},Ric=Ric_{ij}(x)y^{i}y^{j},$
(44)

and $Ric_{ij}(x)$ is the Ricci tensor in Riemann manifolds.

##  3 Modified Dispersion Relations and Finsler Geometry

A common feature among Lorentz violating theories is that most of them admit
modified versions of dispersion relations for single particles. When the
energies of particles are far below the Planck scale, the MDRs for elementary
particles can be written in the form of Taylor series as

$\displaystyle E^{2}$ $\displaystyle=m^{2}+p^{2}+D(p,\mu,M)$ (45)
$\displaystyle=m^{2}+p^{2}+\sum_{n=1}^{\infty}\alpha_{n}(\mu,M)p^{n},$

where $\alpha_{n}$ are dimensional coefficients, $\mu$ is some particle
physics mass scale, and $M$ is the scale associated with the new physics
responsible for the correction of the dispersion relation, and usually taken
to be the Planck scale $M_{P}\approx 10^{19}~{}\mathrm{GeV}.$ The spacetime
structure corresponding to an MDR has been studied in the existent research
(see, e.g., Kimberly (2004)). The key feature of the corresponding spacetime
is described by a rainbow metric Magueijo (2004), which depends on the energy
of the particle. This is a natural concept from the QG point of view and arose
in different contexts such as spacetime foam Magueijo (2004), the
renormalization group applied to gravity Girelli (2007), or as a consequence
of averaging over QG fluctuations Aloisio (2006). However, the rainbow metric
still lacks a rigorous formulation since it involves a metric defined on the
tangent bundle while depending on a quantity associated with the cotangent
bundle (the 4-momentum of the particle). On the road to geometrical structures
leading to MDRs, Riemannian geometry is abandoned, and the generalization of
Riemannian geometry is adopted Girelli (2007). This is Finsler geometry.

###  3.1 The Bridge between Modified Dispersion Relations and Finsler Geometry

The connection between MDRs and Finsler geometry was first proposed by Girelli
et al. Girelli (2007). Here, we review the procedure in Ref. Girelli (2007).

Starting from an MDR as $\mathcal{M}(p)=m^{2}$, one can suppose the action of
a particle to be

$I=\int\left[\dot{x}^{\mu}p_{\mu}-\lambda\left(\mathcal{M}(p)-m^{2}\right)\right]d\tau,$
(46)

where $\lambda$ is a Lagrange multiplier that transforms appropriately under
an arbitrary change of time parameter to ensure reparametrization invariance
of the action. Applying Hamilton equations on Equation (46), one obtains

$p_{\mu}=\lambda\frac{\partial\mathcal{M}}{\partial\dot{x}^{\mu}}.$ (47)

If the relation above is invertible, one can rewrite the action in terms of
velocities and the multiplier, hence obtaining

$I=\int{L}(x,\dot{x},\lambda)d\tau.$ (48)

The multiplier can be eliminated by varying the action concerning it, and thus
one obtains an action depending on $x$ and $\dot{x}$ as

$I=\int{L}(x,\dot{x},\lambda(x,\dot{x}))d\tau.$ (49)

For a particle in Finsler spacetime, its action can be expressed as

$I=m\int F(x,\dot{x})d\tau,$ (50)

so the Finsler norm is finally identified by

$mF(x,\dot{x})={L}(x,\dot{x},\lambda(x,\dot{x})).$ (51)

To make the procedure clear, here we present the example Amelino (2014) of the
MDR for DSR1 Amelino (2001, 2002) in a “1 + 1” spacetime. The MDR for DSR1 can
be expressed as

$p_{0}^{2}-p_{1}^{2}-\ell p_{0}p_{1}^{2}=m^{2},$ (52)

where $\ell$ is the LV scale. The action takes the form

$I=\int\left(\dot{x}^{\mu}p_{\mu}-\lambda\left(p_{0}^{2}-p_{1}^{2}-\ell
p_{0}p_{1}^{2}-m^{2}\right)\right)d\tau,$ (53)

and the corresponding Hamilton equations are

$\displaystyle\dot{x}^{0}$ $\displaystyle=\lambda\left(2p_{0}-\ell
p_{1}^{2}\right),$ (54) $\displaystyle\dot{x}^{1}$
$\displaystyle=\lambda\left(-2p_{1}-2\ell p_{0}p_{1}\right).$

Working at the leading order of $\ell$, inverting the above equation, one
obtains

$\displaystyle p_{0}$
$\displaystyle=\frac{\dot{x}^{0}}{2\lambda}+\frac{\ell}{2}\frac{\left(\dot{x}^{1}\right)^{2}}{4\lambda^{2}}$
(55) $\displaystyle p_{1}$
$\displaystyle=\frac{\dot{x}^{1}}{-2\lambda}+\ell\frac{\dot{x}^{0}\dot{x}^{1}}{4\lambda^{2}};$

thus, the Lagrangian can be expressed at the leading order of $\ell$ as

$L=\frac{\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}}{4\lambda}+\lambda
m^{2}+\ell\frac{\dot{x}^{0}\left(\dot{x}^{1}\right)^{2}}{8\lambda^{2}}.$ (56)

Minimizing the above equation with respect to $\lambda$, one obtains

$\lambda(\dot{x})=\frac{1}{2}\frac{\sqrt{\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}}}{m}+\frac{\ell}{2}\frac{\dot{x}^{0}\left(\dot{x}^{1}\right)^{2}}{\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}}.$
(57)

Thus, the momentum can be expressed by velocities as

$\displaystyle
p_{0}=\frac{m\dot{x}^{0}}{\sqrt{\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}}}-m^{2}\frac{\ell}{2}\frac{\left(\dot{x}^{1}\right)^{2}\left(\left(\dot{x}^{0}\right)^{2}+\left(\dot{x}^{1}\right)^{2}\right)}{\left(\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}\right)^{2}},$
(58) $\displaystyle
p_{1}=-\frac{m\dot{x}^{1}}{\sqrt{\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}}}+m^{2}\ell\frac{\dot{x}^{1}\left(\dot{x}^{0}\right)^{3}}{\left(\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}\right)^{2}},$

and the Lagrangian is

$L=m\left(\sqrt{\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}}+\frac{\ell}{2}m\frac{\dot{x}^{0}\left(\dot{x}^{1}\right)^{2}}{\left(\dot{x}^{0}\right)^{2}-\left(\dot{x}^{1}\right)^{2}}\right).$
(59)

Finally, we obtain the Finsler norm associated with the MDR of DSR1 at the
leading order of $\ell$ as

$F(x,y)=\sqrt{\left(y^{0}\right)^{2}-\left(y^{1}\right)^{2}}+\frac{\ell}{2}m\frac{y^{0}\left(y^{1}\right)^{2}}{\left(y^{0}\right)^{2}-\left(y^{1}\right)^{2}},$
(60)

where $y^{\mu}=\dot{x}^{\mu}$ are the velocities. For more examples, see Refs.
Girelli (2007); Lobo (2017, 2017, 2021); JieZ (2022).

In Ref. Lobo (2021), Lobo and Pfeifer found a general formula connecting
Finsler norms and general forms of MDRs in curled spacetime. For an MDR
described as

$m^{2}=g^{\mu\nu}(x)p_{\mu}p_{\nu}+\epsilon h(x,p),$ (61)

where $\epsilon$ is a perturbation parameter counting the first nontrivial
contribution to the MDR, the corresponding Finsler norm can be expressed at
the leading order of $\epsilon$ as

$F(x,\dot{x})=\sqrt{g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}}\left[1-\epsilon\frac{h(x,\bar{p}(x,\dot{x}))}{2m^{2}}\right],$
(62)

where

$\bar{p}_{\mu}(x,\dot{x})=m\frac{\dot{x}_{\mu}}{\sqrt{g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}}}=m\frac{g_{\mu\nu}(x)\dot{x}^{\nu}}{\sqrt{g_{ab}(x)\dot{x}^{a}\dot{x}^{a}}}.$
(63)

For example, for an $n$th order polynomial modification

$\begin{gathered}{h(x,p)=h^{a_{1}a_{2}\ldots a_{n}}(x)p_{a_{1}}p_{a_{2}}\ldots
p_{a_{n}},}\\\ {\Rightarrow
h(x,\bar{p}(x,\dot{x}))=m^{n}\frac{h_{a_{1}a_{2}\ldots
a_{n}}(x)\dot{x}^{a_{1}}\dot{x}^{a_{2}}\ldots\dot{x}^{a_{n}}}{\left(g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}\right)^{\frac{n}{2}}},}\end{gathered}$
(64)

and the Finsler norm is

$F(x,\dot{x})=\sqrt{g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}}-\epsilon
m^{n-2}\frac{h_{a_{1}a_{2}\ldots
a_{n}}(x)\dot{x}^{a_{1}}\dot{x}^{a_{2}}\ldots\dot{x}^{a_{n}}}{2\left(g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}\right)^{\frac{n-1}{2}}}.$
(65)

From Equation (51), one can obtain the relation

$p_{\mu}=m\frac{\partial
F}{\partial\dot{x}^{\mu}}=m\frac{g_{\mu\nu}\dot{x}^{\nu}}{F},$ (66)

where $g_{\mu\nu}$ is the Finsler metric of $F(x,\dot{x})$ defined in Equation
(3). Inverting the above equation, and using the inverse metric $g^{\mu\nu}$,
one can recover the MDR $\mathcal{M}(p)=m^{2}$ as

$m^{2}=g^{\mu\nu}(\dot{x}(p))p_{\mu}p_{\nu}.$ (67)

From Equation (46) one can obtain

$\dot{x}^{\mu}=\lambda\frac{\partial\mathcal{M}(p)}{\partial p_{\mu}},$ (68)

thus the traditional relation

$v_{i}=\frac{\dot{x}^{i}}{\dot{x}^{0}}=\frac{\partial p_{0}}{\partial p_{i}}$
(69)

holds.

From Equations (60) and (65), we can see that the mass of the particle $m$
plays an important role in the Finsler norm. Keeping the MDR unchanged and
letting $m\rightarrow 0$, we can see that the above Finsler norm becomes a
trivial Riemann norm. Thus, the procedure fails when meeting MDRs with zero
mass, such as photons. In addition, since the Finsler norm is an order-1
homogeneous function of $\dot{x}$, any other parameters with mass dimensions
in the Finsler norm should be present as combinations to cancel their
dimensions. To insert the LV scale into the Finsler norm, we need other
parameters to cancel the dimension introduced by the LV scale. Since we
construct the Finsler norm from an MDR and without any assumption introducing
dimensional physical constants, the only parameter to cancel the dimension
from the LV scale is the mass of the particle. For example, in the MDR as
Equation (52), the dimension of the LV scale $\ell$ is finally canceled in the
Finsler norm by $m$ as a combination of $m\ell$ in Equation (60). Therefore,
it is not feasible to construct Finsler norms from MDRs for massless particles
without other assumptions. Another feature we should notice is that even with
the same MDR, different particles with different masses meet different Finsler
structures.

###  3.2 Physical Influences from Finsler Geometry

####  3.2.1 Time Dilation in Finsler Geometry

Time dilation is an important phenomenon in special relativity. It enlarges
the lifetime of high-energy particles in high-energy particle physics
experiments. In Finsler geometry, the length element is modified and the time
dilation formula is modified. Lobo and Pfeifer realized this phenomenon Lobo
(2021, 2022), and here we review their results.

In Finsler spacetime, the proper time of a massive particle experiencing
between events $A$ and $B$ along a time-like curve can be defined as

$\Delta\tau_{AB}=\int_{A}^{B}F(x^{\mu},\dot{x}^{\mu})d\tau.$ (70)

Noticing that $x^{0}=t$ is the time coordinate of the local observer, the time
dilation can be expressed by

$\Delta\tau=F(x^{\mu},\frac{dx^{\mu}}{dx^{0}})\cdot\Delta t,$ (71)

where $\Delta\tau$ is the proper time duration of the particle and $\Delta t$
is the observed time duration. Taking the MDR of DSR1 as an example, the
Finsler norm is expressed as Equation (60); thus, the time dilation can be
expressed as

$\Delta\tau=\frac{\Delta
t}{\gamma}\left[1+\frac{\ell}{2}m\gamma\left(\gamma^{2}-1\right)\right],$ (72)

where $\gamma$ is the usual velocity Lorentz factor as

$\gamma:=\frac{1}{\sqrt{1-v^{2}}},$ (73)

with $v^{i}:=dx^{i}/dx^{0}$ and $v^{2}=\delta_{ij}v^{i}v^{j}$. At the leading
order of $\ell$, the above time dilation can be expressed as

$\Delta
t=\gamma\left[1-\frac{\ell}{2}m\gamma\left(\gamma^{2}-1\right)\right]\cdot\Delta\tau.$
(74)

Considering that

$\displaystyle
p_{0}=m\frac{\partial}{\partial\dot{x}^{0}}F(x,\dot{x})=m\gamma-\frac{\ell}{2}m^{2}\left(\gamma^{2}-1\right)\left(2\gamma^{2}-1\right),$
(75) $\displaystyle
p_{i}=m\frac{\partial}{\partial\dot{x}^{i}}F(x,\dot{x})=-v_{i}\gamma m+\ell
m^{2}v_{i}\gamma^{4},$

we can express the Lorentz factor with the energy of the particle at the
leading order of $\ell$ as

$\gamma=\frac{p_{0}}{m}+\frac{\ell}{2}m\left(1-3\frac{p_{0}^{2}}{m^{2}}+2\frac{p_{0}^{4}}{m^{4}}\right).$
(76)

In the limit of $p_{0}\gg m$, Equation (74) can be expressed as

$\displaystyle\Delta t$
$\displaystyle=\frac{p_{0}}{m}\Delta\tau\left[1+\frac{\ell}{2}\left(\frac{m^{2}}{p_{0}}-2p_{0}+\frac{p_{0}^{3}}{m^{2}}\right)\right]$
(77)
$\displaystyle\approx\frac{p_{0}}{m}\Delta\tau\left[1+\frac{\ell}{2}\frac{p_{0}^{3}}{m^{2}}\right]$
$\displaystyle=\Delta t_{\rm
SR}\left[1+\frac{\ell}{2}\frac{p_{0}^{3}}{m^{2}}\right],$

where $\Delta t_{\rm SR}=\frac{p_{0}}{m}\Delta\tau$ is the observed time in
special relativity. Thus, the limit on $\ell$ of unstable particles can be
made from experiments, i.e., muons Lobo (2021).

####  3.2.2 Arrival Time Delay of Astroparticles in Finsler Geometry

One important influence on particles from LV is the possible speed variation.
By the relation $v=\partial E/\partial p$, the velocities of particles are
modified from the modification of dispersion relations, especially when the
particles have high energies, such as astroparticles. After propagating
cosmological distances, the astroparticles from the same source emitted at the
same time will arrive at the Earth’s equipment at different times because of
the speed variation. This effect opens a window for constraining the LV scale.
Tests on LV based on the arrival time delay of astroparticles can be found in
Refs. Ellis (2003); Amelino (2003); Ellis (2006); Jacob (2007); Biesiada
(2009); Shao (2010, 2010); fermi (2013); Zhang (2015); Pan (2015); Xu (2016,
2016); Zou (2018); Liu (2018); Xu (2018); Huang (2018, 2019); Li (2020); Pan
(2020); Acciari (2020); Chen (2021); JieZ (2021, 2021).

Since the Universe is not flat, there arises the question as to how to
calculate the arrival time differences between different astroparticles with
different energies under the influence of LV. Jacob and Piran Jacob (2008)
derived the time delay formula in the standard model of cosmology. For an MDR
expressed as

$E^{2}=m^{2}+p^{2}\left[1-s_{n}\left(\frac{p}{E_{\mathrm{LV},n}}\right)^{n}\right]=m^{2}+p^{2}+\alpha
p^{n+2},$ (78)

the arrival time delay between massless particles with high energies and low
energies (which is assumed negligible) is

$\Delta
t=\frac{1+n}{2H_{0}}\left(\frac{E_{\mathrm{obs}}}{E_{\mathrm{LV},n}}\right)^{n}\int_{0}^{z}\frac{\left(1+z^{\prime}\right)^{n}\mathrm{~{}d}z^{\prime}}{\sqrt{\Omega_{\mathrm{m}}\left(1+z^{\prime}\right)^{3}+\Omega_{\Lambda}}},$
(79)

where $z$ is the redshift of the source of the two particles, $E_{\text{obs
}}$ is the observed energy of the high-energy particle from Earth equipment,
$\Omega_{\mathrm{m}}$ and $\Omega_{\Lambda}$ are universe constants, and
$H_{0}$ is the current Hubble parameter. Generalized research on the time
delay from homogeneous and isotropic modified dispersion relations in a
Friedmann–Lemaître–Robertson–Walker (FRW) universe is presented in Ref.
Pfeifer (2018). In this work, the MDR in the FRW universe is assumed to be

$H\left(t,p_{t},w\right)=-p_{t}^{2}+a(t)^{-2}w^{2}+\epsilon
h\left(t,p_{t},w\right),$ (80)

where

$w^{2}=p_{r}^{2}\chi^{2}+\frac{p_{\theta}^{2}}{r^{2}}+\frac{p_{\phi}^{2}}{r^{2}\sin^{2}\theta},\quad\chi=\sqrt{1-kr^{2}},$
(81)

$h\left(t,p_{t},w\right)$ can be an arbitrary function of $t,p_{t}$, and $w$,
and $\epsilon$ is an arbitrary perturbation parameter. The time delay formula
can be obtained by solving the Hamilton equations of motion of the above
Hamiltonian. Now that the Finsler spacetimes corresponding to MDRs are
constructed, it is time to reconsider the time delay problem via the
geometrical approach by solving geodesic equations.

In Section 3.1 the Finsler spacetimes corresponding to MDRs were constructed.
However, for MDRs in flat spacetime, the constructed Finsler norm is also
flat. It is not clear how to corporate the additional LV term in the Finsler
metric with a background Riemann metric. In Refs. Lobo (2017, 2017), Lobo et
al. bypassed the difficulty by assuming the form of the MDR in the curved
spacetime at the beginning. In Ref. Lobo (2017), they assumed that the MDR
which formalizes the physics of particles embedded in a de Sitter-like curved
spacetime can be expressed as

$m^{2}=p_{0}^{2}-p_{1}^{2}e^{-2Hx^{0}}+\ell\left(\gamma p_{0}^{3}+\beta
p_{0}p_{1}^{2}e^{-2Hx^{0}}\right),$ (82)

where $H$ is the parameter of curvature, $\ell$ is the deformation parameter
due to Planck-scale effects, and $\beta,\gamma$ are two numerical parameters
of order 1. In Ref. Lobo (2017), they focused on the MDR as

$m^{2}=a^{-2}(\eta)\left(\Omega^{2}-\Pi^{2}\right)+\ell
a^{-3}(\eta)\left(\gamma\Omega^{3}+\beta\Omega\Pi^{2}\right),$ (83)

where $(\eta,x)$ are the so-called conformal time coordinates, $(\Omega,\Pi)$
are their conjugate momenta, $a(\eta)$ is the scale factor of the universe,
$\ell$ is the deformation parameter due to Planck-scale effects, and
$\beta,\gamma$ are two numerical parameters of order 1. By constructing the
corresponding Finsler spacetime and solving the geodesic equations, they
obtained the time delay formula.

Luckily, the FRW metric for the real universe is simple and can be expressed
as

$ds^{2}=dt^{2}-a(t)^{2}(dx^{2}+dy^{2}+dz^{2}),$ (84)

or written in Finsler norm as

$F_{\rm
FRW}=\sqrt{(y^{0})^{2}-(a(x^{0})y^{1})^{2}-(a(x^{0})y^{2})^{2}-(a(x^{0})y^{3})^{2}}.$
(85)

It is natural to embed the effect of the expansion of the universe into the
Finsler norm just by replacing $y^{i}$ with $a(x^{0})y^{i}$, where $i$ is the
space index. It makes sense since $a(x^{0})$ describes how the space expands
and it should be multiplied to every space component in the metric. In Ref.
JieZ (2022), the above strategy was applied. The Finsler norm in FRW universe
corresponding to the MDR in Equation (78) in “1 + 1” spacetime is

$F=\sqrt{\left(y^{0}\right)^{2}-\left(a\left(x^{0}\right)y^{1}\right)^{2}}+\alpha
m^{n}\frac{\left(-a\left(x^{0}\right)y^{1}\right)^{n+2}}{2\left(\sqrt{\left(y^{0}\right)^{2}-\left(a\left(x^{0}\right)y^{1}\right)^{2}}\right)^{n+1}},$
(86)

and the geodesic equations are

$\begin{gathered}{\dot{y}^{0}+a\left(x^{0}\right)a^{\prime}\left(x^{0}\right)\left(y^{1}\right)^{2}+\alpha
m^{n}\frac{(n+2)a^{\prime}\left(x^{0}\right)a\left(x^{0}\right)^{n+1}\left(-y^{1}\right)^{n+2}\left[(n-1)\left(y^{0}\right)^{2}+a\left(x^{0}\right)^{2}\left(y^{1}\right)^{2}\right]}{2\left[\left(y^{0}\right)^{2}-a\left(x^{0}\right)^{2}\left(y^{1}\right)^{2}\right]^{\frac{n+2}{2}}}=0,}\\\
{\dot{y}^{1}+2\frac{a^{\prime}\left(x^{0}\right)}{a\left(x^{0}\right)}y^{0}y^{1}-\alpha
m^{n}\frac{n(n+2)a^{\prime}\left(x^{0}\right)a\left(x^{0}\right)^{n-1}\left(y^{0}\right)^{3}\left(-y^{1}\right)^{n+1}}{2\left[\left(y^{0}\right)^{2}-a\left(x^{0}\right)^{2}\left(y^{1}\right)^{2}\right]^{\frac{n+2}{2}}}=0.}\end{gathered}$
(87)

The solution to the above geodesic equations suggests the same time delay
formula as Equation (79), obtained by Jacob and Piran differently from the
standard model of cosmology.

One thing that should be noticed is that the Finsler norms corresponding to
MDRs strongly rely on the forms of dispersion relations. Even if two forms of
dispersion relations break Lorentz invariance in the same order, from Equation
(65) we can see that the corresponding Finsler norms can be different. Thus,
the geodesic equations are different. However, interestingly, the differences
do not influence the arrival time delay in Finsler spacetimes when the
modified term of the MDR is not zero in the limit $p/E\rightarrow 1$ JieZ
(2023). For the MDR of the form

$E^{2}=m^{2}+p^{2}+\alpha E^{n+2}G\left(\frac{p}{E}\right),$ (88)

the arrival time delay is calculated as

$\Delta t=\alpha
G(1)E_{\mathrm{obs}}^{n}\frac{n+1}{2H_{0}}\int_{0}^{z}\frac{(1+z^{\prime})^{n}}{\sqrt{\Omega_{m}\left(1+z^{\prime}\right)^{3}+\Omega_{\Lambda}}}dz^{\prime}.$
(89)

Thus, if $G(1)\neq 0$, we can define new $\alpha^{\prime}=\alpha G(1)$ and
$G^{\prime}(x)=G(x)/G(1)$, and the above time delay formula is still the same
as Equation (79). If $G(1)=0$, which means the modified term in the MDR
becomes zero in the limit $p/E\rightarrow 1$, there is no time delay. An
example for the $G(1)=0$ is the dispersion relation of DSR2 Smolin (2002,
2003), expressed as

$\frac{E^{2}-p^{2}}{(1-\lambda E)^{2}}=\frac{m^{2}}{(1-\lambda
m)^{2}}=\mu^{2}$ (90)

or $E^{2}=\mu^{2}+p^{2}-2\lambda E(E^{2}-p^{2})$ at the leading order of
$\lambda$.

The derivation of the time delay formula brings success to describing LV with
Finsler geometry. However, determining how to embed a background Riemann
metric into a flat Finsler norm describing LV is still an open question. This
question arises when studying a particle subject to LV moving in curled
spacetimes, e.g., black holes or gravity lensing.

####  3.2.3 Transformation between Inertial Frames and Modified Composition
Laws

The principle of relativity plays an important role in Einstein’s general
relativity. It requires that the equations describing the laws of physics have
the same form in all admissible frames of reference. In Section 3.1, we
construct the Finsler spacetime describing an MDR, and the formulas of many
physical quantities are modified, such as the relations between the momentum
$p_{\mu}$ and the speed $v$. This causes a serious problem: if the composition
law of the momentum is still the traditional adding law such as
$p_{1\mu}+p_{2\mu}=p_{3\mu}+p_{3\mu}$ in one inertial frame, after frame
transformation, the equality can not be expressed as
$p_{1\mu}^{\prime}+p_{2\mu}^{\prime}=p_{3\mu}^{\prime}+p_{3\mu}^{\prime}$,
where $p_{i\mu}^{\prime}$ is the new 4-momentum in the new frame. Thus, a new
composition law of momentum is needed in the Finsler spacetime. A natural idea
is an assumption that the principle of reality still holds in Finsler
spacetime. Here, we review the results of Ref. Lobo (2022).

To start the discussion, here, we take the MDR of DSR1 in (1+1) spacetime as
an example. The MDR can be expressed as

$m^{2}=p_{0}^{2}-p_{1}^{2}-\ell p_{0}p_{1}^{2},$ (91)

the Finsler norm is Equation (60), and the relations between the momentum
$p_{\mu}$ and the speed $v$ are Equation (75). Next, let us consider the
transformation law in two inertial frames. Let us assume that the two inertial
frames are labeled by $S$ and $\tilde{S}$ and move with relative velocity $v$,
and the observers in $S$ and $\tilde{S}$ assign momentum $p_{\mu}$ and
$\tilde{p}_{\mu}$ for a particle. From Equation (75), at the leading order of
$\ell$ the relation between $p_{\mu}$ and $\tilde{p}_{\mu}$ can be assumed to
be

$\displaystyle\tilde{p}_{0}=\gamma\left(p_{0}-vp_{1}\right)+\ell\left[Ap_{0}p_{1}+Bp_{1}^{2}-\frac{1}{2}p_{0}^{2}\left(\gamma^{2}-1\right)\left(2\gamma^{2}-1\right)\right],$
(92)
$\displaystyle\tilde{p}_{1}=\gamma\left(p_{1}-vp_{0}\right)+\ell\left(p_{0}^{2}v\gamma^{4}+Fp_{0}p_{1}+Gp_{1}^{2}\right),$

where $\gamma=\frac{1}{\sqrt{1-v^{2}}}$, and $A,B,F$, and $G$ are functions of
$v$. The principle of relativity commands that
$m^{2}=\tilde{p}_{0}^{2}-\tilde{p}_{1}^{2}-\ell\tilde{p}_{0}\tilde{p}_{1}^{2}$.
Thus, we have the relation

$\displaystyle
B=-\frac{A}{v}-\frac{\left(1-v^{2}\right)^{3/2}-v^{2}-1}{2\left(1-v^{2}\right)},$
(93) $\displaystyle F=-\frac{A}{v},$ $\displaystyle
G=A-\frac{v\left[2v^{2}-\left(1-v^{2}\right)^{3/2}\right]}{2\left(1-v^{2}\right)^{2}}.$

In Ref. Lobo (2022) these parameters are fixed by setting $A=0$ for
simplicity. The reader should keep in mind that more assumptions are needed to
fix all of the parameters. From the above relations and setting $A=0$, the
transformation relations are

$[\Lambda(v,p)]_{\mu}=\tilde{p}_{\mu}=\left\\{\begin{array}[]{l}\tilde{p}_{0}=\gamma\left(p_{0}-vp_{1}\right)+\frac{\ell}{2}\left[p_{1}^{2}\gamma\left(2\gamma^{3}-\gamma-1\right)-p_{0}^{2}\left(\gamma^{2}-1\right)\left(2\gamma^{2}-1\right)\right],\\\
\tilde{p}_{1}=\gamma\left(p_{1}-vp_{0}\right)+\ell
v\left[p_{0}^{2}\gamma^{4}-\frac{p_{1}^{2}}{2}\gamma\left(2\gamma^{3}-2\gamma-1\right)\right],\end{array}\right.$
(94)

where $[\Lambda(v,p)]_{\mu}$ is referred to be the transformed $\mu$-component
of momenta $p$ using boost parameter $v$.

Now, we turn to the modified composition law. The most general form of the
composition law in first-order perturbation is

$\displaystyle(p\oplus q)_{0}=p_{0}+q_{0}+\ell\left(\alpha p_{0}q_{0}+\beta
p_{1}q_{1}+\omega p_{0}q_{1}+\eta p_{1}q_{0}\right),$ (95)
$\displaystyle(p\oplus q)_{1}=p_{1}+q_{1}+\ell\left(\delta p_{1}q_{0}+\epsilon
p_{0}q_{1}+\lambda p_{1}q_{1}+\mu p_{0}q_{0}\right),$

where $(\alpha,\beta,\omega,\eta,\delta,\epsilon,\lambda,\mu)$ are
dimensionless parameters yet to be determined. The momentum composition law
and the translation relation must have a relation as

$\Lambda(v,p\oplus
q)=\Lambda\left(v_{q},p\right)\oplus\Lambda\left(v_{p},q\right),$ (96)

where $v_{p}$ and $v_{q}$ are back-reacting parameters and may depend on the
moment $p$ and $q$, respectively. The most general relations on $v_{p}$ and
$v_{q}$ are

$\displaystyle v_{q}=v+\ell\left(Hq_{0}+Jq_{1}\right),$ (97) $\displaystyle
v_{p}=v+\ell\left(Mp_{0}+Rp_{1}\right).$

The parity invariance demands that the composition law is invariant under
transformations $p_{0}\rightarrow p_{0}$ and $p_{i}\rightarrow-p_{i}$, and
this demands $\omega=\eta=\mu=\lambda=0$. Another thing that should be
considered is the commutative property and the associative property of the
composition law, which means that $p\oplus q=q\oplus p$ and $(p\oplus q)\oplus
r=p\oplus(q\oplus r)$, and these conditions demand $\eta=\omega$ and
$\delta=\epsilon$. These constraints suggest that

$\begin{gathered}{\alpha=\omega=\eta=\mu=\lambda=0,\quad\epsilon=\delta,\quad\beta=1+2\delta,}\\\
{R=J=\frac{1-\gamma}{\gamma}\left(1+\gamma-\frac{\delta}{\gamma}\right),\quad
M=H=-v\left(\gamma+\frac{\delta}{\gamma}\right),}\end{gathered}$ (98)

and the modified composition law

$\left\\{\begin{array}[]{l}(p\oplus q)_{0}=p_{0}+q_{0}+(1+2\delta)\ell
p_{1}q_{1},\\\ (p\oplus
q)_{1}=p_{1}+q_{1}+\delta\ell(p_{0}q_{1}+p_{1}q_{0}).\end{array}\right.$ (99)

Equation (99) suggests one more constraint to obtain the value of $\delta$.
The undeformed spatial momentum conservation requires $\delta=0$, and the
composition law becomes

$\left\\{\begin{array}[]{l}(p\oplus q)_{0}=p_{0}+q_{0}+\ell p_{1}q_{1},\\\
(p\oplus q)_{1}=p_{1}+q_{1}.\end{array}\right.$ (100)

The undeformed energy conservation requires $\delta=-1/2$, and the composition
law becomes

$\left\\{\begin{array}[]{l}(p\oplus q)_{0}=p_{0}+q_{0},\\\ (p\oplus
q)_{1}=p_{1}+q_{1}-\frac{1}{2}\ell(p_{0}q_{1}+p_{1}q_{0}).\end{array}\right.$
(101)

Another case is $\delta=-1$. In this case, the composition law becomes

$\left\\{\begin{array}[]{l}(p\oplus q)_{0}=p_{0}+q_{0}-\ell p_{1}q_{1},\\\
(p\oplus q)_{1}=p_{1}+q_{1}-\ell(p_{0}q_{1}+p_{1}q_{0}).\end{array}\right.$
(102)

This composition law is the one suggested in DSR1 from Refs. Amelino (2002);
Lukierski (2003); Judes (2003). In Ref. Lobo (2022), one kind of conservation
law without the commutative proper was also considered. This kind of
conservation law reads

$\left\\{\begin{array}[]{l}(p\oplus q)_{0}=p_{0}+q_{0},\\\ (p\oplus
q)_{1}=p_{1}+q_{1}-\ell p_{0}q_{1}.\end{array}\right.$ (103)

This composition law is the one from the bicrossproduct $\kappa$-Poincaré
coproduct structure Gubitosi (2013).

##  4 Connections Between Lorentz Violation Theories and Finsler Geometry

###  4.1 Doubly Special Relativity and Finsler Geometry

Doubly special relativity is a novel idea about the fate of Lorentz
invariance. From a phenomenological perspective, DSR is a set of assumptions
that the Lorentz group acts on so that the usual speed of light, c, and a new
momentum scale, $E_{\rm DSR}$, are invariant. Usually, $E_{\rm DSR}$ is taken
as the Planck energy. Soon after the idea was pointed out by Amelino-Camelia
Amelino (2001, 2002), $\kappa$-Poincaré algebra Lukierski (1991, 1991); Majid
(1994); Lukierski (1995) was applied on this kind of DSR Kowalski (2001);
Bruno (2001); Kowalski (2002); Lukierski (2003). This kind of DSR was referred
to as DSR1 later. In DSR1, the brackets of rotations $M_{i}$ boosts $N_{i}$,
and the components of momenta $P_{\mu}$ read

$\begin{gathered}{\left[M_{i},M_{j}\right]=i\epsilon_{ijk}M_{k},\quad\left[M_{i},N_{j}\right]=i\epsilon_{ijk}N_{k},\quad{\left[N_{i},N_{j}\right]=-i\epsilon_{ijk}M_{k},}}\\\
{\left[M_{i},P_{j}\right]=i\epsilon_{ijk}P_{k},\quad\left[M_{i},P_{0}\right]=0,}\\\
{\left[N_{i},P_{j}\right]=i\delta_{ij}\left(\frac{\kappa}{2}\left(1-e^{-2P_{0}/\kappa}\right)+\frac{1}{2\kappa}\vec{P}^{2}\right)-i\frac{1}{\kappa}P_{i}P_{j},}\\\
{\left[N_{i},P_{0}\right]=iP_{i},}\end{gathered}$ (104)

where $\kappa$ acts as the role of $E_{\rm DSR}$. The Casimir of the
$\kappa$-Poincaré algebra reads

$\kappa^{2}\cosh\frac{P_{0}}{\kappa}-\frac{\vec{P}^{2}}{2}e^{P_{0}/\kappa}=M^{2}.$
(105)

Expanding it at the leading order of $\frac{1}{\kappa}$, one obtains

$P_{0}^{2}-\vec{P}^{2}-\ell P_{0}\vec{P}^{2}=m^{2},$ (106)

where $\ell=1/\kappa$ and $m^{2}=2(M^{2}-\kappa^{2})$. From Equation (105),
one can obtain the result that the value of three-momentum $|\vec{P}|=\kappa$
corresponds to infinite energy $P_{0}=\infty$. Thus, in DSR1 there is a
maximal three-momentum $|\vec{P}|=\kappa$, and if a particle has momentum
$|\vec{P}|=\kappa$ for some observers, it has the same momentum for all
observers.

Another realization of DSR, called DSR2, was proposed by Magueijo and Smolin
Smolin (2002, 2003). In DSR2, the Lorentz algebra is not deformed, and the
boosts–momenta generators now have the form

$\left[N_{i},P_{j}\right]=i\left(\delta_{ij}P_{0}-\lambda
P_{i}P_{j}\right),\quad\left[N_{i},P_{0}\right]=i\left(1-\lambda{P_{0}}\right)P_{i},$
(107)

where $\lambda=\frac{1}{E_{\rm DSR}}$. The Casimir of the above algebra reads

$M^{2}=\frac{P_{0}^{2}-\vec{P}^{2}}{(1-\lambda P_{0})^{2}}.$ (108)

From Equation (108), one obtains that $P_{0}=1/\lambda$ corresponds to
infinite momentum, and this energy $1/\lambda$ is invariant in transformations
of frames.

Although the DSR theories are referred to as Lorentz-violation theories, the
symmetry group is still the Lorentz group, but acting nonlinearly on the
energy-momentum sector Smolin (2002); Kowalski (2002); Lukierski (2003). The
nonlinear representations lead to modified conservation laws. In Ref. Judes
(2003), Judes proposed general conservation laws for DSRs. Since a DSR
symmetry group is simply a nonlinear realization of the Lorentz group, a
function between the physical energy–momentum $P_{4}=(E,P)$ and a pseudo-
energy–momentum $\mathcal{P}_{4}=(\epsilon,\pi)$, which transforms similar to
a Lorentz 4-vector, can be established as

$P_{4}=F(\mathcal{P}_{4}),\quad\mathcal{P}_{4}=F^{-1}(P_{4}).$ (109)

The transformations act on the pseudo-energy–momentum in the normal linear
manner $\mathcal{P}_{4}^{\prime}=\mathcal{L}(\mathcal{P}_{4})$, where
$\mathcal{L}$ is the usual Lorentz transformation. Function $F$ satisfies the
dispersion relation

$\left[\epsilon(E,p)\right]^{2}-\left[\pi(E,p)\right]^{2}=\mu_{0}^{2},$ (110)

where $\mu_{0}$ is the Casimir invariant and $\mu_{0}=\epsilon(m,0)$. In the
linear representation, the total energy can be defined as

$\mathcal{P}_{4}^{\rm tot}=\sum_{i}\mathcal{P}_{4}^{i};$ (111)

thus, in the nonlinear representation, the total energy can be expressed as

$P_{4}^{\rm tot}=F\left(\sum_{i}F^{-1}\left(P_{4}^{i}\right)\right).$ (112)

As an example, in the case of DSR2, the relation between $(E,p)$ and
$(\epsilon,\pi)$ can be expressed as

$(\epsilon,\pi)=\frac{(E,p)}{1-\lambda
E},\quad(E,p)=\frac{(\epsilon,\pi)}{1+\lambda\epsilon};$ (113)

thus, the conservation law reads

$\epsilon_{\rm tot}=\sum_{i}\frac{E_{i}}{1-\lambda E_{i}},\quad\pi_{\rm
tot}=\sum_{i}\frac{p_{i}}{1-\lambda E_{i}},$ (114)

and

$E_{\rm tot}=\frac{\sum_{i}E_{i}/\left(1-\lambda
E_{i}\right)}{1+\lambda\sum_{i}E_{i}/\left(1-\lambda E_{i}\right)},\quad
p_{\rm tot}=\frac{\sum_{i}p_{i}/\left(1-\lambda
E_{i}\right)}{1+\lambda\sum_{i}E_{i}/\left(1-\lambda E_{i}\right)}.$ (115)

The formulation of DSR in the energy–momentum space is clearly incomplete, as
it lacks any description of the structure of spacetime. DSR was formulated in
a somehow unusual way: one started with the energy–momentum space and only
then the problem of construction of spacetime was considered. An early
discussion on the connection between DSR and Finsler geometry can be found in
Ref. Mignemi (2007), via the relation between the MDRs of DSRs and Finsler
geometries. In Ref. Amelino (2014), Amelino-Camelia et al. connected DSR1
symmetries with Finsler spacetime. The Finsler norm for DSR1 is Equation (60);
for convenience, here we write it as

$F(x,y)=\sqrt{\left(y^{0}\right)^{2}-\left(y^{1}\right)^{2}}+\frac{A}{2}\frac{y^{0}\left(y^{1}\right)^{2}}{\left(y^{0}\right)^{2}-\left(y^{1}\right)^{2}},$
(116)

where $A=m\ell$ is a dimensionless parameter. Well-defined relationships
between modified relativistic symmetries and Finsler geometries are
established in their work.

However, one huge difference between DSR theories and Finsler geometries can
be seen in Equation (116). In DSR theories, the new momentum scale $E_{\rm
DSR}$ ($\frac{1}{\ell}$ in DSR1) is invariant under transformations in
different frames. This parameter is supposed to be a constant of the universe
at the beginning of the construction of the theories. However, from Equation
(116), the structure of the Finsler spacetime is described by the parameter

$A=m\ell=\frac{m}{E_{\rm DSR}}.$ (117)

This means that if the new momentum scale $E_{\rm DSR}$ is indeed a constant
of the universe, different particles with different masses meet different
Finsler structures. On the other hand, if one assumes that the universe has a
unique Finsler structure, the energy scale $E_{\rm DSR}$ depends on the
particle mass as $E_{\rm DSR}=\frac{m}{A}$ and is not a constant of the
universe.

Another thing that needs to be mentioned is the problem when DSR is applied to
macroscopic objects. In fact, DSR is a nonperturbative theory. In the macro
world, the resting energies of objects can easily achieve $E_{\rm DSR}$, and
this means obvious effects on the modified relations, e.g., the conservation
laws shown as Equation (115). To solve this problem, Magueijo and Smolin
Smolin (2003) proposed that $E_{\rm DSR}$ in a system of $N$ elementary
particles should be replaced by $NE_{\rm DSR}$. This proposal is just an
assumption in their work, but now it has roots in Finsler geometry. We
consider a macro system with $N$ same particles moving with the same speed $v$
and assume that these particles have mass $m$, and they are of DSR1 particles
with an MDR as Equation (106). Thus, they have the same Finsler structure as
Equation (116), with the same constant $A$. Now let us regard these particles
as a whole. Since they have the same Finsler structure and have the same
speed, the whole system should also be under the effect of the same Finsler
structure. This system has mass $Nm$ and Finsler constant $A$; thus, the
momentum scale of the system becomes

$E=\frac{Nm}{A}=NE_{\rm DSR},$ (118)

which is the same as the proposal of Magueijo and Smolin.

###  4.2 Standard-Model Extension and Finsler Geometry

The Standard-Model Extension Colladay (1997, 1998); Kostelecký (2004) is a
comprehensive realistic effective field with Lorentz violation that
incorporates both the Standard Model and general relativity. The connection
between SME and Finsler geometry was established in Refs. Kostelecký (2011,
2012); Colladay (2015); Russell (2015); Schreck (2016); Edwards (2018);
Schreck (2019, 2021).

For a single massive spin-$\frac{1}{2}$ Dirac fermion, the general form of the
quadratic sector of a renormalizable Lorentz- and $CPT$-violating Lagrangian
Colladay (1997, 1998) is

$\mathcal{L}=\frac{1}{2}i\bar{\psi}\Gamma^{\nu}\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\nu}\psi-\bar{\psi}M\psi,$
(119)

where

$\Gamma^{\nu}:=\gamma^{\nu}+c^{\mu\nu}\gamma_{\mu}+d^{\mu\nu}\gamma_{5}\gamma_{\mu}+e^{\nu}+if^{\nu}\gamma_{5}+\frac{1}{2}g^{\lambda\mu\nu}\sigma_{\lambda\mu}$
(120)

and

$M:=m+a_{\mu}\gamma^{\mu}+b_{\mu}\gamma_{5}\gamma^{\mu}+\frac{1}{2}H^{\mu\nu}\sigma_{\mu\nu}.$
(121)

To study the corresponding classical Lorentz-violating kinematics, a topic
central to subjects such as the behavior of quantum wave packets, the analysis
of relativistic scattering, and the motion of macroscopic bodies, one useful
approach is to introduce an analog point-particle system with relativistic
Lagrangian $L$, which leads directly to various results such as the equations
of motion for the classical trajectory, the momentum–velocity connection, and
the dispersion relation. Thus, Finsler geometry is connected with SME via the
relativistic Lagrangian $L$ by

$L=-mF(x,y).$ (122)

The MDR of the Lagrangian Equation (119) was derived in Ref. Kostelecký (2001)
as

$\mathrm{det}\left(\Gamma^{\mu}p_{\mu}-M\right)=0.$ (123)

Expansion of the determinant of this matrix yields

$\displaystyle 0=$
$\displaystyle\frac{1}{4}\left(V^{2}-S^{2}-A^{2}-P^{2}\right)^{2}+4[P(VTA)-S(V\widetilde{T}A)-VTTV+ATTA]$
(124) $\displaystyle+V^{2}A^{2}-(V\cdot
A)^{2}-X\left(V^{2}+S^{2}-A^{2}-P^{2}\right)-2YSP+X^{2}+Y^{2},$

where the scalar quantity is $S=-m+e\cdot p$, the pseudoscalar is $P=f\cdot
p$, the vector is $V_{\mu}=p_{\mu}+(cp)_{\mu}-a_{\mu}$, the axial-vector is
$A_{\mu}=(dp)_{\mu}-b_{\mu}$, and the tensor is
$T_{\mu\nu}=\frac{1}{2}(gp-H)\mu\nu$. The two invariants of $T_{\mu\nu}$ are
denoted as $X:=T_{\mu\nu}T^{\mu v}$ and $Y:=T_{\mu\nu}\widetilde{T}^{\mu v}$,
with the dual defined by $\widetilde{T}_{\mu\nu}:=\frac{1}{2}\epsilon_{\mu
v\alpha\beta}T^{\alpha\beta}$. Note that for vanishing coefficients for
Lorentz violation, the dispersion relation Equation (124) reduces to the usual
form $\left(p^{2}-m^{2}\right)^{2}=0$, which is effectively quadratic. The
quadratic nature is retained when the only nonzero coefficients are
$a_{\mu},c_{\mu\nu},e_{\mu}$, and $f_{\mu}$. However, the dispersion relation
is generically quartic if $b_{\mu},d_{\mu\nu},g_{\lambda\mu\nu}$, or
$H_{\mu\nu}$ are nonzero.

To determine the relativistic Lagrangian $L$ from an MDR $\mathcal{R}(p)=0$,
in addition to the workflow presented in Section 3.1, a method was developed
in Ref. Kostelecký (2010), presented as five key equations:

$\begin{split}&\mathcal{R}(p)=0,\\\ &y^{j}=-y^{0}\frac{\partial
p_{0}}{\partial p_{j}},j=1,2,3,\\\ &L=-p_{\mu}y^{\mu}.\end{split}$ (125)

The first equation of Equation (125) is the MDR we already have. Recall that
$y^{\mu}=\frac{dx^{\mu}}{d\tau}$ is the 4-velocity of the particle and
$p_{\mu}:=-\frac{\partial L}{\partial y^{\mu}}$, the second equation of
Equation (125) means that the traditional relation

$v_{j}=\frac{\partial E}{\partial p_{j}}$ (126)

holds in this situation, where
$v_{j}=\frac{y^{j}}{y^{0}}=\frac{dx^{j}}{dx^{0}}=\frac{dx^{j}}{dt}$ is the
speed of the particle. The third equation of Equation (125) is just from the
homogeneity of $L$, as Equation (4). The five equations can be manipulated to
eliminate the 4-momentum components, leaving a single equation that can be
viewed as a polynomial for $L$.

For the quadratic case of Equation (124), any form of MDR can be written as

$(p+\kappa)\Omega(p+\kappa)=\mu^{2},$ (127)

where $\mu>0$ is a mass-like scalar, $\kappa_{\mu}$ is a constant 4-vector
shift of the momentum, and $\Omega^{\mu\nu}$ is a constant metric-like
symmetric tensor. In the limit of vanishing background fields, $\Omega^{\mu
v}\rightarrow\eta^{\mu\nu}$, $\kappa_{\mu}\rightarrow 0$, and $\mu\rightarrow
m$. For perturbative background fields, $\Omega$ is invertible. The solution
of Equation (125) yields

$L=-\mu\sqrt{y\Omega^{-1}y}+\kappa\cdot y.$ (128)

For the special case of nonzero coefficients $a_{\mu}$, $c_{\mu\nu}$,
$e_{\mu}$ and $,f_{\mu}$, $\Omega=\left(\delta+2c+c^{T}c-ee-ff\right)$. With
additional condition $c_{\mu\nu}=0$, $(\Omega^{-1})_{\mu\nu}$ can be expressed
in finite form and the corresponding Lagrangian is given by

$\displaystyle L=$
$\displaystyle-\mu\left\\{y^{2}+\frac{1}{\Delta}\left[\left(1-f^{2}\right)(e\cdot
y)^{2}+\left(1-e^{2}\right)(f\cdot y)^{2}+2(e\cdot f)(e\cdot y)(f\cdot
y)\right]\right\\}^{\frac{1}{2}}-a\cdot y$ (129)
$\displaystyle+\frac{1}{\Delta}\left[\left(1-f^{2}\right)(m-e\cdot a)-(e\cdot
f)(f\cdot a)\right]e\cdot y$ $\displaystyle+\frac{1}{\Delta}\left[(e\cdot
f)(m-e\cdot a)-\left(1-e^{2}\right)(f\cdot a)\right]f\cdot y,$

where

$\mu=\frac{1}{\sqrt{\Delta}}\left[\left(1-f^{2}\right)(m-e\cdot
a)^{2}-2(e\cdot f)(f\cdot a)(m-e\cdot a)+\left(1-e^{2}\right)(f\cdot
a)^{2}\right]^{\frac{1}{2}}$ (130)

and

$\Delta=(1-e^{2})(1-f^{2})-(e\cdot f)^{2}$ (131)

is the determination of $\Omega$. For the quartic cases when the coefficients
$b_{\mu}$, $d_{\mu\nu}$, $H_{\mu\nu}$, or $g_{\lambda\mu\nu}$ are nonzero,
things are more complicated. When the first nonzero coefficients are $a_{\mu}$
and $b_{\mu}$, the MDR is

$\mathcal{R}(p)=\left[-(p-a)^{2}+b^{2}+m^{2}\right]^{2}-4\left[b\cdot(p-a)\right]^{2}+4b^{2}(p-a)^{2}=0,$
(132)

and the corresponding Lagrangian is

$L=-m\sqrt{y^{2}}-a\cdot y\mp\sqrt{(b\cdot y)^{2}-b^{2}y^{2}}.$ (133)

When $b_{\mu}=0$, the Finsler geometry corresponding to the above Lagrangian
is of Randers. Another case with only $H_{\mu\nu}$ nonzero yields

$L=-m\sqrt{y^{2}}\pm\sqrt{yHHy+2Xy^{2}}.$ (134)

The common feature of Equation (133) and Equation (134) is that if
$a_{\mu}=0$, the corresponding Finsler norms have the form

$F=\sqrt{y^{2}}\pm\sqrt{y^{\mu}s_{\mu\nu}y^{\nu}}.$ (135)

In this case, the Finsler geometry is called bipartite Finsler geometry.
Detailed discussions of bipartite Finsler geometry can be found in Refs.
Kostelecký (2012); Silva (2014); Colladay (2017); Silva (2019). For modified
Dirac field theory at higher orders, it is impossible to write down the
corresponding Lagrangians and Finsler norms explicitly. Instead, perturbative
expansions for Lagrangians are adapted in Refs. Schreck (2019, 2021).

For the case of Lorentz-violating scalar fields, the effective quadratic
Lagrange density of a complex scalar field $\phi(x^{\mu})$ can be written in
the form

$\mathcal{L}\left(\phi,\phi^{\dagger}\right)=\partial^{\mu}\phi^{\dagger}\partial_{\mu}\phi-m^{2}\phi^{\dagger}\phi-\frac{1}{2}\left(i\phi^{\dagger}\left(\hat{k}_{a}\right)^{\mu}\partial_{\mu}\phi+\text{
h.c.
}\right)+\partial_{\mu}\phi^{\dagger}\left(\hat{k}_{c}\right)^{\mu\nu}\partial_{\nu}\phi,$
(136)

where $(\hat{k}_{a})^{\mu}$ and $(\hat{k}_{c})^{\mu\nu}$ are operators
constructed as series of even powers of the partial spacetime derivatives
$\partial_{\alpha}$. The MDR for the above Lagrange density is

$p^{2}-m^{2}-\left(\hat{k}_{a}\right)^{\mu}p_{\mu}+\left(\hat{k}_{c}\right)^{\mu\nu}p_{\mu}p_{v}=0,$
(137)

where the operators $\left(\hat{k}_{a}\right)^{\mu}$ and
$\left(\hat{k}_{c}\right)^{\mu\nu}$ can conveniently be expressed as
expansions in even powers of the $n$-momentum $p_{\mu}$ of the form

$\displaystyle\left(\hat{k}_{a}\right)^{\mu}=\sum_{d\geq
n-1}\left(k_{a}^{(d)}\right)^{\mu\alpha_{1}\alpha_{2}\ldots\alpha_{d-n+1}}p_{\alpha_{1}}p_{\alpha_{2}}\ldots
p_{\alpha_{d-n+1}},$ (138)
$\displaystyle\left(\hat{k}_{c}\right)^{\mu\nu}=\sum_{d\geq
n}\left(k_{c}^{(d)}\right)^{\mu\nu\alpha_{1}\alpha_{2}\ldots\alpha_{d-n}}p_{\alpha_{1}}p_{\alpha_{2}}\ldots
p_{\alpha_{d-n}}.$

The explicit solution for the above MDR from Equation (125) is also
impossible. The implicit solution Edwards (2018) can be written in the form

$\displaystyle L=$ $\displaystyle-\big{[}m^{2}\bar{y}^{2}$ (139)
$\displaystyle+\frac{1}{16}\left(\sum_{d\geq
n+1}(d-n+2)y_{\alpha_{1}}p_{\alpha_{2}}\ldots
p_{\alpha_{d-n+2}}\left(k^{(d)}\right)^{\alpha_{1}\ldots\alpha_{d-n+2}}\right)^{2}$
$\displaystyle+\frac{1}{2}\sum_{d\geq n+1}(d-n)\bar{y}^{2}p_{\alpha_{1}}\ldots
p_{\alpha_{d-n+2}}\left(k^{(d)}\right)^{\alpha_{1}\ldots\alpha_{d-n+2}}\big{]}^{\frac{1}{2}}$
$\displaystyle+\frac{1}{4}\sum_{d\geq
n+1}(d-n+2)y_{\alpha_{1}}p_{\alpha_{2}}\ldots
p_{\alpha_{d-n+2}}\left(k^{(d)}\right)^{\alpha_{1}\ldots\alpha_{d-n+2}},$

with implicit relations

$\displaystyle y^{\mu}=$ $\displaystyle-
Lp_{\nu}\left[\eta^{\mu\nu}+\frac{1}{2}\sum_{d\geq
n+1}(d-n+2)p_{\alpha_{1}}\ldots
p_{\alpha_{d-n}}\left(k^{(d)}\right)^{\alpha_{1}\ldots\alpha_{d-n}\mu\nu}\right]$
(140) $\displaystyle\times\left[m^{2}+\frac{1}{2}\sum_{d\geq
n+1}(d-n)p_{\alpha_{1}}\ldots
p_{\alpha_{d-n+2}}\left(k^{(d)}\right)^{\alpha_{1}\ldots\alpha_{d-n+2}}\right]^{-1}$

and $\bar{y}^{2}=\sqrt{\eta_{\mu\nu}y^{\mu}y^{\nu}}$. The expansion solution
of Equation (139) can be found in Ref. Edwards (2018).

###  4.3 Very Special Relativity and Finsler Geometry

Very special relativity has roots in the early research of Bogoslovsky. By the
year 1977, to explain the discrepancy between the Greisen–Zatsepin–Kuzmin
cutoff (GZK cutoff) and the experimental data, Bogoslovsky suggested the
violation of usual relativistic relations at Lorentz factors $\gamma\geq
5\times 10^{10}$ Bogoslovsky (1977). In calculating the point of the spectrum
cutoff, the Lorentz transformation from the Earth-centered frame of reference
to that of the proton at rest is essentially used, but the relative velocity
of these coordinate systems is close to the velocity of light and it is far
from being evident that the formula of the special relativity theory remains
valid up to velocities infinitely close to the velocity of light at that time.
He suggested the new transformation law at velocities very close to the
velocity of light is essentially different from the old one. Following Ref.
Bogoslovsky (1977), to generalize the Lorentz transformation written in

$\left\\{\begin{array}[]{l}x_{0}^{\prime}=x_{0}\cosh\alpha-x\sinh\alpha,\\\
x^{\prime}=-x_{0}\sinh\alpha+x\cosh\alpha,\end{array}\right.$ (141)

where $\tanh\alpha=V$, he assumed that the wave equation
$\left(\partial^{2}/\partial x_{0}^{2}-\partial^{2}/\partial
x^{2}\right)\varphi=0$ is invariant under the new transformation law. Thus, he
wrote the generalized transformations

$\left\\{\begin{array}[]{l}x_{0}^{\prime}=\exp[-r\alpha]\left(x_{0}\cosh\alpha-x\sinh\alpha\right),\\\
x^{\prime}=\exp[-r\alpha]\left(-x_{0}\sinh\alpha+x\cosh\alpha\right),\end{array}\right.$
(142)

where $r$ is a dimensionless parameter of scale transformations. Equation
(142) can be written in the form

$\left\\{\begin{array}[]{l}x_{0}^{\prime}=\left(\frac{1-V}{1+V}\right)^{r/2}\frac{x_{0}-Vx}{\sqrt{1-V^{2}}},\\\
x^{\prime}=\left(\frac{1-V}{1+V}\right)^{r/2}\frac{x-/x_{0}}{\sqrt{1-V^{2}}}.\end{array}\right.$
(143)

By writing Equation (142) in infinitesimal form, the invariant function

$f(x_{0},x)=\left(\frac{x_{0}-x}{x_{0}+x}\right)^{r}\left(x_{0}^{2}-x^{2}\right)$
(144)

was found; thus, the new differential invariant is

$\mathrm{d}s=\left(\frac{\mathrm{d}x_{0}-\mathrm{d}x}{\mathrm{~{}d}x_{0}+\mathrm{d}x}\right)^{r/2}\sqrt{\mathrm{d}x_{0}^{2}-\mathrm{d}x^{2}},$
(145)

and in four-dimensional spacetime, the generalized form is

$\mathrm{d}s=\left[\frac{\left(\mathrm{d}x_{0}-\vec{v}\cdot\mathrm{d}\vec{x}\right)^{2}}{\mathrm{~{}d}x_{0}^{2}-\mathrm{d}\vec{x}^{2}}\right]^{r/2}\sqrt{\mathrm{d}x_{0}^{2}-\mathrm{d}\vec{x}^{2}},$
(146)

where $|\vec{v}|=1$. In fact, Equation (146) is just a Finsler norm. The
corresponding relativistic kinematics and Finsler geometries were studied by
Bogoslovsky and Goenner later Bogoslovsky (1998, 1999, 1999). The group
theories of the above spacetime were also studied in Ref. Bogoslovsky (1977).

On the other hand, in 2006, Cohen and Glashow pursued a different approach to
the possible failure of Lorentz symmetry, using group theory. They first
introduced the very special relativity Glashow (2006), which means
descriptions of nature whose spacetime symmetries are certain proper subgroups
of the Poincaré group. These subgroups contain spacetime translations together
with at least a two-parameter subgroup of the Lorentz group isomorphic to that
generated by $T_{1}:=K_{x}+J_{y}$ and $T_{2}:=K_{y}-J_{x}$, where $\mathbf{J}$
and $\mathbf{K}$ are the generators of rotations and boosts, respectively.
These commuting generators form a group, $T(2)$, which is isomorphic to the
group of translations in the plane. $T_{1},T_{2}$ with the addition of $J_{z}$
yields a group isomorphic to the three-parameter group of Euclidean motions,
$E(3)$. $T_{1},T_{2}$ with the addition of $K_{z}$ yields one isomorphic to
the three-parameter group of orientation-preserving similarity
transformations, or homotheties, $HOM(2)$. The addition of both $J_{z}$ and
$K_{z}$ yields one isomorphic to the four-parameter similitude group,
$SIM(2)$. The corresponding Lie algebras and their Lie brackets are listed in
Table 1 Ahluwalia (2010). Usually, the special relativity with symmetry
$SIM(2)$ is referred to as VSR.

Table 1:  The four VSR algebras. Designation |  Generators |  Algebra
---|---|---
$\mathfrak{t}(2)$ |  $T_{1},T_{2}$ |  ${\left[T_{1},T_{2}\right]=0}$
$\mathfrak{e}(2)$ |  $T_{1},T_{2},J_{z}$ |  ${\left[T_{1},T_{2}\right]=0,\left[T_{1},J_{z}\right]=-iT_{2},\left[T_{2},J_{z}\right]=iT_{1}}$
$\mathfrak{hom}(2)$ |  $T_{1},T_{2},K_{z}$ |  ${\left[T_{1},T_{2}\right]=0,\left[T_{1},K_{z}\right]=iT_{1},\left[T_{2},K_{z}\right]=iT_{2}}$
$\mathfrak{sim}(2)$ |  $T_{1},T_{2},J_{z},K_{z}$ |  ${\left[T_{1},T_{2}\right]=0,\left[T_{1},K_{z}\right]=iT_{1},\left[T_{2},K_{z}\right]=iT_{2}},$
|  |  ${\left[T_{1},J_{z}\right]=-iT_{2},\left[T_{2},J_{z}\right]=iT_{1},\left[J_{z},K_{z}\right]=0}$

The semidirect product between the translations and $SIM(2)$ gives an eight-
dimensional subgroup of the Poincaré group called $ISIM(2)$, which was first
introduced but not named in 1970 by Kogut and Soper Kogut (1970). In 2007,
Gibbons et al. Gibbons (2007) tried to deform $ISIM(2)$ similar to deforming
the Poincaré group into de Sitter (anti-de Sitter) group Monique (1967), with
noncommutative relation $\left[P_{\mu},P_{\nu}\right]=\frac{1}{3}\Lambda
M_{\mu,\nu}$, where $\Lambda$ is the cosmological constant. They obtained a
one-parameter family of deformations, denoted by $DISIM_{b}(2)$. Different
from de Sitter (anti-de Sitter) group, the translations of $DISIM_{b}(2)$
remain commutative. The nontrivial Lie brackets for the algebra
$\mathfrak{disim}_{b}(2)$ are given by

$\displaystyle{\left[N,P_{\pm}\right]=-(b\pm
1)P_{\pm},\quad\left[N,P_{i}\right]=-bP_{i}\text{, }}$ (147)
$\displaystyle{\left[N,M_{+i}\right]=-M_{+i},\quad\left[J,P_{i}\right]=\epsilon_{ij}P_{j},}$
$\displaystyle{\left[J,M_{+i}\right]=\epsilon_{ij}M_{+i},\quad\left[M_{+i},P_{-}\right]=P_{j}\text{,
}}$ $\displaystyle{\left[M_{+i},P_{j}\right]=-\delta_{ij}P_{+},}$

where $N:=M_{+-}$ and $J:=M_{12}$, with the Minkowski metric to be
$ds^{2}=\eta_{\mu\nu}x^{\mu}x^{\nu}=2dx^{+}dx^{-}+dx^{i}dx^{i}$, and i and j
ranging over the values 1 and 2. The deformed generator $N$ acts as

$x^{i}\rightarrow\lambda^{-b}x^{i},\quad
x^{-}\rightarrow\lambda^{1-b}x^{-},\quad x^{+}\rightarrow\lambda^{-1-b}x^{+};$
(148)

thus, the invariant line element is a Finsler line element

$\displaystyle ds$
$\displaystyle=\left(2dx^{+}dx^{-}+dx^{i}dx^{i}\right)^{(1-b)/2}\left(dx^{-}\right)^{b}$
(149)
$\displaystyle=\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right)^{(1-b)/2}\left(n_{\rho}dx^{\rho}\right)^{b}.$

Equation (149) is almost the same as Equation (146), found by Bogoslovsky in
1977 in different ways. Thus, the general very special relativity is Finsler
geometry, as per the title of Ref. Gibbons (2007). The Finsler norm of the
$DISIM_{b}(2)$ VSR spacetime from Equation (149) is

$F=(-\eta_{\mu\nu}y^{\mu}y^{\nu})^{(1-b)/2}(-n_{\rho}y^{\rho})^{b},$ (150)

and the Lagrangian is

$L=-m\left(-\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\right)^{(1-b)/2}\left(-n_{\rho}\dot{x}^{\rho}\right)^{b}.$
(151)

The dispersion relation of the above Lagrangian is

$\eta^{\mu\nu}p_{\mu}p_{\nu}=-m^{2}\left(1-b^{2}\right)\left(-\frac{n^{\nu}p_{\nu}}{m(1-b)}\right)^{2b/(1+b)},$
(152)

and the corresponding Klein–Gordon equation is

$-\square\phi+m^{2}\left(1-b^{2}\right)\left(\frac{\mathrm{i}n^{\mu}\partial_{\mu}}{m(1-b)}\right)^{2b/(1+b)}\phi=0.$
(153)

The Finsler structure of the $DISIM_{b}(2)$ VSR is quite different from the
Finsler structures in Section 3, as there is no LV energy scale in VSR.

For the case of $SIM(2)$ VSR, one interesting result is that it may provide an
origin of Lorentz violent neutrino mass Glashow (2006). Depending on Ref.
Glashow (2006), a $SIM(2)$-invariant modification of the electrodynamic theory
is established Cheon (2009). The $SIM(2)$-covariant Dirac equation can be
written in the form

$\left[i\gamma^{\mu}\left(\partial_{\mu}+\frac{\lambda}{2}N_{\mu}\right)-m\right]\Psi(x)=0,$
(154)

where $N_{\mu}:=\frac{n_{\mu}}{n\cdot\partial}$ with a chosen preferred null
direction $n^{\mu}=(1,0,0,1)$ and attributes $N\cdot N=0,N\cdot\partial=1$.
Squaring Equation (154), one obtains

$\left[\partial^{\mu}\partial_{\mu}+M^{2}\right]\Psi(x)=0,$ (155)

where $M^{2}=m^{2}+\lambda$. Thus, the physical mass $M$ does not vanish if
$\lambda\neq 0$, even with $m=0$. The field theory of symmetry $SIM(2)$ and
the Very Special Relativity Standard Model were developed by Alfaro et al.
Alfaro (2013, 2014, 2015, 2017, 2018, 2019, 2019, 2019, 2021, 2022);
Haghgouyan (2022); Bufalo (2022); Cheng-Yang (2015); Ilderton (2016) later.
Interestingly, in the VSR field theory, a gauge-invariant photon mass Alfaro
(2019) and graviton mass Alfaro (2022) are allowed.

For the case of $DISIM_{b}(2)$ VSR, the field theory is more complicated. For
photons, a mass term is suggested, given as Gibbons (2007)

$L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}m^{2}\left(\frac{\left(n^{\mu}A_{\mu}\right)^{2}}{A_{\nu}A_{\nu}}\right)^{b}A_{\rho}A^{\rho},$
(156)

and the mass term of the Dirac Lagrangian is suggested by Bogoslovsky
Bogoslovsky (2004) as

$m\left[\left(\frac{\mathrm{i}n_{\mu}\bar{\psi}\gamma^{\mu}\psi}{\bar{\psi}\psi}\right)^{2}\right]^{b/2}\bar{\psi}\psi,$
(157)

where $n_{\mu}$ is the same as in Equation (149).

The VSR in curved spacetime also arouses people’s interest. For the case of
$SIM(2)$ VSR, the VSR to the physical situation in which a cosmological
constant is present is considered by Alvarez and Vidal Alvarez (2008), and the
generalization of $SIM(2)$ VSR to curved spacetimes is considered by Mück Mück
(2008). For the case of $DISIM_{b}(2)$ VSR, since it is connected with Finsler
spacetime in the form

$F(x,y)=\left(\eta_{\mu\nu}y^{\mu}y^{\nu}\right)^{(1-b)/2}\left(n_{\rho}y^{\rho}\right)^{b},$
(158)

geometric methods can be applied. The Finsler cosmology of the $DISIM_{b}(2)$
VSR was considered by Kouretsis et al. Kouretsis (2009). In their
consideration, they replace the Minkowskian metric $\eta_{\mu\nu}$ in Equation
(149) with the FRW one

$a_{\mu\nu}=\operatorname{diag}\left(1,-\frac{a^{2}(t)}{1-kr^{2}},-a^{2}(t)r^{2},-a^{2}(t)r^{2}\sin^{2}\theta\right),$
(159)

and they consider the case of $n^{\mu}=(n(t),0,0,0)$ with $|n^{\mu}|\ll 1$ and
Taylor expansion $n(t)\sim At+B$. In this case, the Friedman equation is
reconsidered and an extra term $\Omega_{X}=-2\frac{A}{BH}b$ is present and
might give a significant contribution to the acceleration compared to the dark
energy parameter $\Omega_{\Lambda}$. In addition, an additional redshift
effect due to the Lorentz violations inherited by the parameter $b$ and the
spurion vector field $n_{\mu}$ is present, and the energy of the massive
nonrelativistic particle will change with time as

$E=E_{0}\exp\left(-\frac{2Ab}{B}t\right).$ (160)

The very special relativity in curved spacetime is of Berwald–Finsler
spacetime was also studied by Fuster et al. Fuster (2018).

##  5 Summary and Discussion

As a generalization of Riemann geometry, Finsler geometry provides more
possibilities for new physics. Different from Riemann geometry, the square of
the length element $ds^{2}$ in Finsler geometry is not necessarily a quadratic
form of $dx^{\mu}$, and thus Finsler norms can be of infinite choices of
forms. Unfortunately or luckily, there are no criteria to constrain what kind
of Finsler geometries should be applied in physics, so research on Finsler
geometries applied in physics springs up in different areas, as per the
examples shown in Section 1.

The application of Finsler geometry in LV studies is a success. On the one
hand, theories on LV, if accompanied by modified dispersion relations, can be
connected with Finsler geometries via the methodology in Section 3.1. On the
other hand, theories without LV energy scales may also be connected to Finsler
geometries via spacetime symmetries, such as VSR in Section 4.3. Finsler
geometry provides descriptions of spacetime for LV theories and has advantages
in kinematics research. In Section 3.2, we reviewed the effects of time
dilation and arrival time delay formula in the Finsler geometry description of
LV. With the assumption of the principle of relativity, we also presented the
transformations between inertial frames and the modified composition laws.
These modified composition laws together with the MDRs may bring different
views on threshold anomalies. Finsler geometry also shows the possibility of
viewing other LV theories from different perspectives. One proposal, proposed
by Magueijo and Smolin Smolin (2003), to address the difficulty of applying
DSR to macroscopic objects, is replacing $E_{\rm DSR}$ with $NE_{\rm DSR}$
when a macroscopic system has $N$ particles. This proposal is just an
assumption in Ref. Smolin (2003), but in Finsler geometry, it is a result of
the analysis of the Finsler structure, as per the discussion in Section 4.1.

Maybe we could progress further by assuming that the Finsler structure from LV
is an attribute of the universe, and that means that all particles and objects
in the universe have the same Finsler structure. Let us assume that the MDRs
of these objects are

$E^{2}=m^{2}+p^{2}+\frac{s}{E_{\rm
LV}^{n}}E^{n+2}G\left(\frac{\vec{p}}{E}\right),$ (161)

where $s=\pm 1$, $E_{\rm LV}$ is the LV scale, and $G(x)$ is an abstract
function. The corresponding Finsler norms are

$F(y)=\sqrt{y^{2}}+\frac{s}{2}\left(\frac{m}{E_{\rm
LV}}\right)^{n}\frac{(y^{0})^{n+2}}{\left(\sqrt{y^{2}}\right)^{n+1}}G\left(-\frac{\vec{y}}{y^{0}}\right),$
(162)

where $y^{2}=(y^{0})^{2}-\delta_{ij}y^{i}y^{j}$ and $i,j$ are spatial indexes.
The assumption on the Finsler structures suggests that $m/E_{\rm LV}$ is a
constant for all objects, or

$E_{\rm LV,p}=Cm_{\rm p},$ (163)

where different particles are identified by the index $\rm p$ and $C$ is a
dimensionless constant of the universe. This means that for massive particles,
the LV scales for different particles are proportional to their masses, and
the heavier the particle is, the higher the LV scale is and the less obvious
the LV effects are. Of course, this inference is too radical, but it conforms
to the phenomenological cognition of LV. Or, we can assume the weak version of
the inference, that the LV scales of particles and the masses of particles are
positively correlated for massive particles. Of course, more tests are needed
to examine the assertion.

Although Finsler geometry performs successfully in the studies of LV, it is
not advisable to regard Finsler geometry as the final theory of LV. It works
well when dealing with spacetime and gravity, but it lacks the description of
other interactions. Another challenge is that we lack a methodology to
construct the Finsler norms that describes LV directly based on some basic
assumptions. Worse still, we cannot construct Finsler norms from MDRs for
massless particles. These problems mean that, at present, Finsler geometry can
be considered an effective theory when dealing with LV, and more research is
needed in applying Finsler geometry in LV studies.

Both J.Z. and B.-Q.M. contributed extensively to this project. All authors
have read and agreed to the published version of the manuscript.

This work was supported by the National Natural Science Foundation of China,
grant number 12075003.

The authors declare no conflicts of interest.

Not applicable.

Not applicable.

No new data were created or analyzed in this study. Data sharing is not
applicable to this article.

Abbreviations The following abbreviations are used in this manuscript:

DSR | Doubly special relativity
---|---
FRW | Friedmann–Lemaître–Robertson–Walker
GZK | Greisen–Zatsepin–Kuzmin
LV | Lorentz violation
MDR | Modified dispersion relation
QG | Quantum gravity
SME | Standard-Model Extension
VSR | Very special relativity

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