3.6 In summary
4.1.1 Basic theorems
Given a positive definite bilinear form (i.e. a scalar product) on V , represented by the matrix bij,
a norm can be defined:
F (v) = (bijvivj)1/2. (4.2)
In this case, the Finsler metric tensor defined in (4.1) is given by bij and the norm is Riemannian.
However, not all the norms are defined through scalar products, as we shall see later.
The homogeneity property of the norm plays often an important role. In particular, Euler’s theorem and its consequences provide some useful properties that can be exploited in the analysis. The discussion of these issues can be found in appendix A. Euler’s theorem can be cast in the following form.
Euler’s Theorem Let f be a real-valued function on RD, differentiable away from the zero
vector. The two following statements are equivalent. A. f is positively homogeneous of degree s,
f (λv) = λsf (v), ∀ λ > 0. B. The function f satisfies the following partial differential equation
vi∂f
∂vi = sf (v).
Here we shall mention a rather important consequence of the homogeneity property of the norm that deserves some comments. The matrix defined in (4.1) is homogeneous of degree zero:
gij(λv) = gij(v), (4.3)
as one can easily check. This fact has an immediate implication. Since the matrix g is homogeneous of degree zero, in general it is not continuous at v = 0: if we take two non-collinear vectors u, w (v 6= αw, for some α), such that gij(v) 6= gij(w), then taking the limit of gij(λv), gij(λw) for λ → 0
we obtain two different results, for the homogeneity property of the matrix g. Hence, the limit of gij(v), v → 0 does not exist and hence the tensor gij is not continuous in v = 0. There is of course
a case in which the continuity holds on the entire space V , and it is the case in which gij(v) = bij,
i.e. a norm induced by a scalar product.
It follows from the homogeneity of the norm that the norm itself can be always written as:
F (v) = (gij(v)vivj)1/2. (4.4)
This follows from homogeneity and Euler’s theorem. Indeed gij(v)vivj = 1 2 ∂2F2 ∂vi∂vjv ivj =1 2 ∂F2 ∂viv i= F2,
where we have used Euler’s theorem twice.
It can be shown that the validity of the triangular inequality is equivalent to the fact that the matrix (4.1) is positive definite. This statement is a part of a more general theorem on Finsler norms (see, for instance [206], section 1.2B)
Theorem Let F be a positively homogeneous, nonnegative real-valued function on RD such that
it is C∞on RD\ 0, and the matrix defined by (4.1) is positive definite. Then the function is positive
on RD\0 (F (v) > 0, ∀v 6= 0), it obeys the triangular inequality as well as the fundamental inequality
wi∂F
∂vi(v) ≤ F (w), (4.5)
the equality holding if and only if w = αv, for some α > 0.
The proof is omitted. It can be found for instance in [206], section 1.2B. This theorem is crucial for two reasons. The first one is that it gives a direct computational criterion to decide whether a function is a norm, reducing the task of checking the triangle inequality to a check of the positive definiteness of a matrix. The second important consequence comes from a closer inspection of the inequality (4.5). Let us elaborate a bit on this.
From Euler’s theorem we know that vi∂
viF (v) − F (v) = 0, hence (4.5) can be rewritten as
F (v) + ∂viF (v)(wi− vi) ≤ F (w)
This has a nice graphical interpretation. It is telling that at any point v, the graph of the function F is convex. In particular the homogeneity property implies that the graph is a convex cone emanating from the origin of RD+1.
Moreover, given that the norm is positive for every vector different from zero, (4.5) can be rewritten as
gij(v)wivj≤ F (w)F (y), (4.6)
which is nothing else than the generalization of the Cauchy–Schwarz inequality usually discussed for scalar products.
The fundamental inequality has some deep implications in issues on global Finsler geometry, which are irrelevant for the present discussion. It has however an interesting application in the definition of angle. In Euclidean geometry the metric tensor defines the angle θ(w, u) between the two vectors w, u through:
cos(θ(w, u)) = bijw
iuj
(bmnumun)1/2(bhkwhwk)1/2
. (4.7)
This is a direct consequence of the Cauchy–Schwarz inequality: the right hand side is always bounded in modulus. Notice that the angle is symmetric in the two vector arguments, θ(w, u) = θ(u, w). For a Finsler norm, we can define the notion of angle between vectors in the same way:
cos (θ(w, u)) = gij(w)w
iuj
F (w)F (u). (4.8)
The crucial difference with the angle defined by bilinear forms (to which this definition reduces when the norm comes from a Euclidean scalar product) is that it is not symmetric in the vector arguments any more. In general we have that θ(w, u) 6= θ(u, w). The implications of this will be discussed later in this chapter, when we will speak about orthogonality.
Disformal transformations Looking at equation (4.4) one could imagine to be able to define a Finsler norm giving the matrix gij(v) from the beginning. However
this is not as simple as it seems. Indeed, a general symmetric matrix gij(v), even
homogeneous of degree zero, is not necessarily a second derivative of a scalar function of the vector argument. Comparing (4.4) with (4.1) one can get a necessary condition for gij(v): ∂gjm(v) ∂vi v m+∂gim(v) ∂vj v m= −1 2 ∂2g mn(v) ∂vi∂vj v mvn.
To see this, one could imagine to give a generalization to Finsler norms of conformal transformations of the metric, namely gij(v) = Ω2(v)bij. It is easily found that, in
order for gij to be the matrix of the second derivatives of the Finsler norm obtained
from it, Ω(v)2 is constrained to satisfy
∂Ω2 ∂vi∂vj = −2 bimvm bhkvhvk ∂Ω2 ∂vj + bjmvm bhkvhvk ∂Ω2 ∂vi .
Therefore, the generalization of conformal transformations to Finsler norms is prop- erly defined only in terms of the norm itself, ˜F (v) = Ω(v)F (v), with Ω > 0 a posi- tively homogeneous function of degree zero. The corresponding metrics will not be conformally related. Instead, the two metrics will be related by a kind of disformal transformation, ˜ gij(v) = Ω2(v)gij(v) + ∂Ω2 ∂vigjk(v)v k+∂Ω2 ∂vjgik(v)v k+1 2 ∂Ω2 ∂vi∂vjF 2(v),
which is similar to the disformal transformations often encountered in generalized (bi-metric) theories of gravity, where the physical and the gravitational metrics are related by
˜
gµν = Ω2(x)gµν+ Vµ(x)Vν(x), (4.9)
where Ω(x) and Vxare parametrizing the deformation of the metric. See for instance
[209] for a discussion of the possible role of these transformations in gravity.