Yeheng Tong

Interdisciplinary Center for Theoretical Study,University of Science and Technology of China,Hefei,Anhui 230026,China

Peng Huanwu Center for Fundamental Theory,Hefei,Anhui 230026,China

Abstract Riemannian geometry,as a basis for general relativity,can be obtained from the more general Finsler geometry in terms of the Cartan connection and Chern connection,as discussed frequently in the literature.However,there are other gravity theories that can be made to be equivalent to general relativity but are based on non-Riemannian geometry.Famous examples are the Teleparallel and Symmetric Teleparallel gravity theories.In this paper,we show how to obtain the geometry for Teleparallel gravity from Finsler geometry in terms of a ‘Teleparallel type’ connection.

Keywords: Finsler geometry,teleparallel gravity,Finsler connection

1.Introduction

However,there are other frameworks that can be used to hatch gravity theories but are not based on the Riemannian geometry.One example is the ‘Teleparallel’ framework,in which the connection is constrained to be curvature-freeRλρμν=0and metric-compatible ?αgμν=0,so the gravity is attributed to the torsionTλμν[5–8].Another similar example is the so-called‘Symmetric Teleparallel’framework,in which the connection is constrained to be curvature-freeRλρμν=0 and torsion-freeTλμν=0,and the gravity is attributed entirely to the non-metricity tensor Qαμν=?αgμν[9,10].Within both frameworks,gravity models can be constructed to be equivalent to GR,i.e.the Teleparallel equivalent General Relativity (TEGR) model [5–8] in the Teleparallel framework and Symmetric Teleparallel equivalent General Relativity (STGR) model [9,10] in Symmetric Teleparallel framework,respectively.Interestingly,these two frameworks can be used to hatch some modified gravity models which are very difficult (if not impossible) to be constructed within the Riemannian geometry,for instance,the Nieh–Yan modified Teleparallel Gravity [11,12] in the Teleparallel framework,and the parity-violating gravity models discussed in [13,14]within the Symmetric Teleparallel framework.All these frameworks including the Riemannian geometry can be considered as special cases of the metric-affine geometry under different constraints.

There are frameworks more general than metric-affine geometry.The conception of ‘metric’ may also be generalized.In Riemannian geometry,the length of the world line of a particle is.In a general case,the length of the world line of a particle may be ds=F(x,dx) as long as the function F is smooth enough and homogeneous for the displacement dx[15,16].Such generalization leads us to the so-called Finsler geometry.Some interesting phenomena may come out from such a generalization,e.g.,the geodesic equations are direction-dependent now,or a curve is a geodesic curve from A to B,but not from B to A.Also,more abundant connection structures are contained in Finsler geometry and it may induce the normal metric-affine theory with a specially selected section of the tangent bundle of the base manifold.Hence,it would exist at least three different kinds of connection structure: the ‘GR type’ that may lead to Riemannian geometry,‘Teleparallel type’ and‘Symmetric Teleparallel type’ lead to corresponding frameworks,respectively.The ‘GR type’ at least includes two connections: the so-called ‘Cartan connection’ [17] and‘Chern connection’ [18],as discussed a lot in the literature.On the other hand,the other two types of connection structures are rarely discussed.In this paper,we come up with an example of ‘Teleparallel type’ connection and show how to obtain a generalized teleparallel gravity model from this connection structure.

In this paper,we will use the metric signature{+,-,-,-}.As usual,the Greek letters μ,ν,ρ,…=0,1,2,3 are tensor indicators and Capital Latins A,B,C,…=0,1,2,3 are internal space indicators.In sections 2 and 3 we review the basic structure of Finsler Geometry.In section 4,we present our connection as an example of ‘Teleparallel type’ connection.In section 5,we use a toy model based on our connection to compare with TEGR or GR,and show the similarities and differences between them.

2.Basics of Finsler geometry

2.1.Metric structure

A Finsler space is an n-dimensional manifold Mnwith a Finsler metric functionF:TM? R+.The function F satisfies the following properties [15,16]:

1.Regularity: F is C∞on the entire tangent bundle except the origin TM{0}.

2.Positive homogeneity: F(x,λy)=λF(x,y) for all λ ＞0.

3.Signature: The Hessian Matrix

has a Lorentzian signature.3The original requirement of the Hessian matrix in Finsler Geometry is to be positive definite.However,considering the connection between the Hessian matrix and metric tensor,we change the requirement to having a Lorentzian signature.

Since for any point x ∈M with local coordinate (xμ),any tangent vector can be expanded as(xμ,yλ) can be chosen as a local coordinate on TM.Then a function F defined on TM can be locally expressed as F=F(xμ,yλ).

Besides the fundamental tensor (1),one can define the so-called Cartan tensor [15,16]:

It is clear that the Cartan tensor is totally symmetric.Moreover,the homogeneity leads to the following identities:

The Finsler Function F defines the length of a curve C(t)in the base manifold M:

It is easy to see that when taking,the fundamental tensor(1)is just aμν(x),and the length of a curve is the same as that in (pseudo-)Riemann geometry.So the fundamental tensor (1) plays a similar role in Finsler geometry to the metric tensor in (pseudo-)Riemann geometry,while the Cartan tensor (2) implies the difference between Finsler geometry and (pseudo-)Riemann Geometry.

2.2.(α,β)-metric

A family of special metric functions is the so-called (α,β)-metric.These metric functions satisfy F=F(α,β),whereα=and β=Aμ(x)yμ.The homogeneity then requires that F(λα,λβ)=λF(α,β).Denoting that

where yμ?aμνyν,we have the fundamental tensor

and Cartan tensor

where the lower indices α,β of L indicate the partial differentiation with respect to α and β.

where the arguments of functions f are all β/α.When f ≡1,(pseudo-)Riemann geometry is recovered.

As examples,we have two simpler cases:Randers metric[19] F=α+β or f(x)=1+x,in which

2.3.Connection structure: basis and non-linear connection

Since the local coordinate on TM has been chosen to be (xμ,yλ),the natural basis on TM is also determined [16]:

so the transformations of basis (12) are [16]

Hence they are not covariant under a coordinate transformation on the base manifold.In order to receive a group of covariant bases,a non-linear connectionshould be introduced,and a group of covariant bases are

with transformations

These bases have the following commutative relations:

In addition,we have the dual basis of (13):

2.4.Connection structure: linear connection on the bundle

The Vertical BundleV is the linear space spanned by the basisin (13),and its complement space is called the Horizontal BundleH,i.e.

Then a linear connection can be defined onV [15,16]:

Such a connection induces two covariant derivatives of a vector X=Xμ(x,y)?μ: the horizontal covariant derivative?HX

and the vertical covariant derivative ?VX

One can continue this connection to the entire tangent bundle if and only if the Deflection condition

is satisfied [16].At this time,we have the connection

and two covariant derivatives (17,18) defined on the entire TM.

2.5.Curvature and torsion

On TM,we have an outer differential of a function f(x,y):

Now from connection(16)one can immediately figure out the curvature and torsion with Cartan Structure equations.

Curvature 2-form is defined as

with the tensor structure

Again,such a neat structure requires the Deflection condition(19).Then from the Cartan Curvature Structure equations we have [15,16]

There is something different about the torsion.The tensor structure of the torsion is

with 2 torsion 2-forms

Then from the Cartan Curvature Structure equations we have

3.Finsler-Bathrel geometry

One can always induce a (pseudo-)Riemannian geometry from a Finsler geometry with a determined vector field.From this section on,we will use ‘^’ to distinguish the parameters defined on the base manifold M from the ones on TM with similar names.For example,the curvature 2-form on M is denoted as

The (pseudo-)Riemannian geometry induced from a given Finsler geometry is constructed as follows.We always have a natural map [16]

While a vector field Y(x) on M,or a section of TM,restricts the map (24) to

And the connection is the pull back of linear connection(16)[15,16]:

The tensor is the pull back of τλ:

And the non-metricity tensor is

One can examine that(28-30)are indeed constructed with the connection (27).

4.The Finsler-Teleparallel connection

We have a group of connections that would lead to Riemannian geometry which hosts GR when the metric is restricted to be the Riemannian metric,and they have been discussed under different cases.On the other hand,as we know,there are other frameworks to host Teleparallel and Symmetric Teleparallel gravities.Then a question comes out that whether these frameworks can be popularized in Finsler Geometry.We will focus on the popularization of the framework hosting the Teleparallel gravity in this paper.

whereηAB=diag {1,-1,-1,-1}is the flat metric,EAμis a matrix and its inverse is expressed asEAλ.As the natural generalization of ones in Teleparallel Geometry,equations(31),(32)imply the existence of local flat spacetime at each point and keep Einstein’s Equivalent Principle.And from (31),we have the connection coefficients

withEAμto be determined.Here we have taken the Weitzenb?ck condition.

We should point out that condition (32) is not easy to satisfy,because gμνis not a fundamental parameter,but is derived from the Finsler metric.Thus (32) would impose restrictions on the matrixEAμ.In the following sections,we will try to determineEAμ.

4.1.The non-linear connection

We will discuss the (α,β)-metric only,and denote that

where eAμis the tetrad associated with aμν.Identities(3)lead to equation

Now we can make up the non-linear connection from the Deflection condition (19).From (33) it is easy to derive

Then to contract it with yμwe have

Finally,we have the non-linear connection

Up to now we did not specify the form of the matrixmρμ,and the result(42)is independent of it.So a conclusion comes that as long as the matrixEAμtakes the form (37) and the Deflection condition is satisfied,the non-linear connection always takes the form (42).

From now on,we will rewrite the 1-form

Then we have

So the non-linear connection (42) has a simpler form:

4.2.Matrix mαμ and linear connection

We now need to determine the matrixmαμby the metriccompatible condition (32).The fundamental tensor (1) of a Finsler metric F=αf is

It is not easy to find a matrix to satisfy(32).However,we may have a connection structure,which slightly breaks the metric-compatibility on TM,but keep this condition after being pulled back to the base manifold M.In this way,we have a satisfying matrix:

and it leads to

Compared with equation (46),there is a slight difference ff″pμpν,which would be seen that have no effect after being pulled back to the base manifold M.And the inverse of this matrix is

With the matrices (47,49),the linear connection is determined from (33):

wherehλμ?hμ ν aλν.

Then the two covariant derivatives of the fundamental tensor (46) can be calculated.The calculation details can be found in the appendix,and here we just give the conclusions directly:

where {*p} refers to the terms that are proportional to pμ,pμpν,or higher order of pμ.Thus we see that the two covariant derivatives are both proportional to pμor its higher order.We will show that they vanish when pulled back to the base manifold in the following sections,which means that the metric-compatible condition is satisfied on the base manifold.

4.3.The induced Finsler–Bathrel geometry

Then the non-linear connection is

And the induced tetrads are

The induced metric is

these two results are indeed equal to each other.Finally,the result is

and induced torsion is

One can directly verify that this connection (57) is flat and compatible with the metric (56).On the other hand,the covariant derivative of metric (56) can be also written as

So we have a connection of the Teleparallel type that is induced from Finsler geometry,and we may call it Finsler–Teleparallel–Bathrel geometry.Within this geometry,one may come up with a group of Teleparallel gravity models.These models should take the tetrad on the base manifoldeAμand the vector field Aλas fundamental variables,for they are defined on the base manifold,i.e.the physical spacetime;while the Finsler metric F(α,β),or the function f,should rely on the model selection,and considered not to be a fundamental variable.

5.A toy model based on Finsler-Teleparallel-Bathrel connection

As a simple example,we lay out a toy model in which the induced metric(56)is dynamical,while the primordial metric aμνis‘geometrical’,i.e.it is aμνinstead ofthat couples to the matter fields.Such convention is to remain the dynamics of matter unchanged.Hence,this simple model has the action

is the torsion scalar determined by the induced tetradAμ.The action of matter Smkeeps the same as that in GR;as an example,considering a scalar field to mimic the matter field,with the following action

where Gαβis the Einstein Tensor of gμν,and Tμνis the normal energy-momentum tensor.Obviously,the form of the function f,or the Finsler metric,greatly influences the evolution of spacetime,and the special case f ≡1 just gives the Einstein field equations in GR.

6.Conclusion

with connection coefficients

We also used a toy model with action (60) to show this,withbeing dynamical and aμνcoupled with matter.We showed that when f ≡1 or F=α,both the action and the EoMs are induced to the same as ones in TEGR or GR,so the model based on the connection(16)can indeed be considered as a generalization of Teleparallel gravity.

A group of gravity models can be built in terms of this scheme.These models can have new predictions beyond GR and how they are constrained by current observations are questions deserving of further studies.We will visit these problems in the future.

Acknowledgments

We thank the anonymous referees for useful suggestions,Prof.Mingzhe Li for instructions and Dr.Haomin Rao and Dr.Dehao Zhao for helpful discussions.

This work is supported in part by NSFC under Grant No.12075231 and 12047502.

Appendix.Proof of equations (51) and (52)

For vertical covariant derivative,using (9),we have

so it is proportional to pμor its higher order.

The horizontal covariant derivative is more complicated.The following results are needed:

Noticing that the arguments of functions f,g,…are β/α,we have

Use results (A2–A5) we have

So the horizontal covariant derivative is