Resultados Estadísticos

Having discussed the properties of a norm, we can pass to the definition of a Finsler space. Definition A Finsler space is a pair (M , F ), where M is a manifold of dimension D and F−(−)

is a pointwise defined norm, i.e. Fm(−) : TmM → R, ∀m ∈ M .

We can conclude therefore that Riemannian spaces are special cases of Finsler spaces, where the norm is defined by a scalar product. It is important to make this statement more explicit, making a comparison with other generalization of Riemannian geometry.

Through a norm one can define a notion of distance. Indeed, one can define the notion of length of an arc of curve in the obvious way:

ℓ(C) = Z τ1

τ0

Due to the homogeneity property of the norm this definition is reparametrization invariant. Take two points p, q ∈ M , and consider all the curves connecting these two points. The minimum1 _of

the lengths of all these curves defines a distance between the two points.

We can hence say that Finsler geometry is a generalization of Riemannian geometry, but it is still based on the existence of a distance function. It generalizes the way in which distances are defined. Moreover, the linearity properties, often used in Riemannian geometry due to the fact that the metric tensor is bilinear, cannot be used in Finsler geometry, where the metric tensor is not defining a bilinear form, in general. Hence, Finsler geometry is fundamentally different from metric-affine theories [210], where the geometry of a manifold is described in terms of a metric tensor and an independent affine connection. The analysis of the geodesic equation will make this statement rigorous.

Let us make a side remark. As we have shown, the metric tensor and the Cartan tensor are not continuous at v = 0. Therefore, in defining the various structures needed in Finsler geometry it is customary to remove the zero section of the tangent bundle. The resulting bundle is called the slit tangent bundle,

˘

T M ≡ T M \ 0. (4.13)

This bundle is particularly useful since typically Finslerian quantities are regular on it, since the continuity problems typically arise on the zero section. Therefore, most of the definitions are more transparent when given in terms of the slit tangent bundle.

4.1.4

The indicatrix

Before giving some examples, it is useful to introduce the notion of indicatrix. The indicatrix Im

is defined as Im= {v ∈ TmM : Fm(v) = 1}. In Riemannian geometry the indicatrix is an ellipsoid

(solution of a quadratic equation). It is interesting to note that due to the homogeneity property of the norm, the latter can be specified just in terms of the indicatrix (see for instance [211], chapter 15). Given a closed convex hypersurface J in TmM with center in v = 0 (i.e. symmetric under

reflections through the origin of the tangent space), and such that every ray emanating from v = 0 intersects J exactly once, we can define a norm F(J)_{in the following way. For any vector w ∈ T}

mM

there is a vector λw = u ∈ J, for our hypothesis that J intersects each ray exactly once. Hence, one can define the norm of the vector w as

F(J)_{(w) =} 1

λ. (4.14)

In order for the norm to be defined in this way to obey the triangular inequality, the indicatrix must be convex. If it is desired, one can relax the property of symmetry around v = 0. This corresponds to the definition of positively homogeneous Finsler norms.

This construction allows us to have a clear visual picture of the relationships between Rieman- nian and Finslerian geometries. While Riemannian metrics correspond to all the ellipsoids contain- ing the origin, Finsler norms correspond to all the closed and convex hypersurfaces, containing the origin and intersecting all the rays emanating from it just once.

1_{Actually, to be rigorous, one should take the Infimum, i.e. the maximum lower bound to the length of a curve}

Therefore we can say that while Riemannian geometry boils down to Euclidean geometry in a sufficiently small neighborhood, Finsler geometry can be distinguished from Euclidean geometry even in a single point. We will make this qualitative statement more rigorous in the rest of the chapter. What we can say already here is that Finsler geometry is a locally anisotropic extension of Riemannian geometry.

4.2

Examples

We have already seen that Riemannian spaces are particular cases of Finsler spaces where the norm is defined through a positive definite quadratic form. It is useful at this point to make a short list of other simple examples of Finsler structures. Other examples can be found in the book by Asanov [207].

4.2.1

Randers spaces

Let us assume to have a Riemannian space, with metric bij, and a vector field Ai. The function:

F (v) = (bijvivj)1/2+ Aivi, (4.15)

defines a positively homogeneous norm (pseudo-norm). The metric tensor is given as usual by: gij = bij+ AiAj+1 2 bikvkAj+ bjkvkAi (bmnvmvn)1/2 +1 2 bijAkvk (bmnvmvn)1/2 − 1 2 bihvhbjlvlAkvk (bmnvmvn)3/2 . (4.16)

There are some conditions for this norm to represent a Finsler norm. The first one is positivity: F (v) > 0, ∀v 6= 0. The fact that bij satisfies the Cauchy–Schwarz inequality implies that:

|Aivi| ≤ (AmAnbmn)1/2(vhvkbhk)1/2. (4.17)

The positivity condition therefore leads to

(bijvivj)1/2> −Aivi≥ −sgn(Aivi)(AmAnbmn)1/2(vhvkbhk)1/2. (4.18)

Hence, if Aiviis positive the inequality is trivially verified. If the sign of Aivi is −1, the inequality

holds if and only if AiAjbij< 1.

The second condition to be checked is the positive definiteness of the matrix (4.16). We refer to [206] for the detailed discussion of this point.

The Finsler spaces of this class are called Randers spaces. They were discussed by Randers [212] with the purpose of introducing some extra structures besides the pseudo-Riemannian metric attached to spacetime in order to implement in a geometrical way the notion of “arrow of time”. Of course, in order to do this, one should be able to rigorously extend the definitions of Finsler geometry to what would be the generalization of Lorentzian signature for Finsler spaces. The discussion of this issue is postponed to the next chapter, where applications of Finsler geometry to physics will be discussed in detail.

There is another obvious instance in which Randers spaces do appear in physics (with the abovementioned caveat about the Lorentzian signature): indeed the Lagrangian for a charged point

particle moving in a given spacetime and external electromagnetic field takes exactly the same form of (4.15). However, given that particles with different charges will see different Finsler structures, for the analysis of the system it is simpler to speak about the motion of particles in a Lorentzian manifold under the influence of an external electromagnetic field rather than motion of particles in a multi-Finsler structure2_.

4.2.2

Berwald–Moor

Despite the Randers structures are pretty familiar to physicists, albeit under other names, they were not the first Finslerian structures to be introduced, hystorically. Riemann itself introduced the quartic line element, back in his dissertation “ ¨Uber die Hypothesen, welche der Geometrie zu Grunde liegen” in 1854 [213]:

ds4= Qijhkdxidxjdxhdxk. (4.19)

However, he later focused on the simpler case of quadratic line elements, what we know today as Riemannian geometry. It was Finsler that later started developing the theory of more general metric spaces.

Nowadays these spaces are called Berwald–Moor spaces. They are defined through totally symmetric rank four tensors Qijhk. The norm is:

F (v) = (Qijhkvivjvhvk)1/4. (4.20)

The corresponding Finsler metric can be easily derived: gij(v) = 3 Qijhkvhvk (Qmnhkvmvnvhvk)1/2 − 2 QilhkvlvhvkQjlhkvlvhvk (Qmnhkvivjvhvk)3/2 . (4.21)

In this case, positivity is guaranteed from the very definition, provided that the tensor Qijhk

is nondegenerate (i.e. there is no vector ¯v such that Qijhkv¯i¯vj¯vhv¯k = 0). On the other hand, the

positive definiteness must be carefully checked.

As a particular case one could consider some sort of bi-metric Finsler structures, where the rank four tensor is the symmetrized product of two Riemannian metrics

Qijhk= b(ijchk). (4.22)

As in the case of Randers spaces, these structures do appear in physics, as we have seen in the introductory chapter, for instance in the study of the propagation of light rays or sound waves in crystals.

4.2.3

(α, β) spaces

The class of spaces discussed by Randers can be further generalized. These spaces are characterized by two structures, a bilinear form aij and a vector field bi. These two objects define two homogenous

2_{A multi-Finsler structure is the Finslerian counterpart of a multi-metric structure. Instead of having a manifold}

functions on the tangent space:

α(v) =qaijvivj β(v) = bivi. (4.23)

An (α, β) space is a special Finsler space where the norm is specified as F (v) = α(v)Φ β(v)

α(v) 

, (4.24)

where Φ, apart from the constraints due to the definition of a norm, is arbitrary.

A particular example, which has been already encountered in the previous chapter and that will be discussed in the following, is

f (v) = (α(v))1−b(β)b. (4.25)

As we have mentioned in the previous chapter, this is the class of Finsler norms introduced in very special relativity.