Pseudo-Finsleroid metrics with two axes

Abstract

A pseudo-Finsleroid metric function F of the two-axes structure that involves the vertical axis and the horizontal axis is proposed assuming constancy of the curvature of indicatrix. The curvature is negative and the signature of the Finslerian metric tensor is exactly $(+-\cdots )$. The function F endows the tangent space with the geometry which possesses many interesting Finslerian properties. The use of the angle representation is the underlying method which has been conveniently and successfully applied. The appearance of the positive-definite Finsleroid metric function in the horizontal sections of the tangent space is established.

Appendices

A Appendix: Unit vectors and metric tensor

To study the geometrical aspects of the pseudo-Finsleroid space, we need to clarify the structure of the entailed metric tensor. Choosing a fixed tangent space, it is convenient to represent the involved vectors and tensors with the help of their components with respect to the orthonormal base frame which enters the pseudo-Riemannian metric tensor $a_{ij}$ according to the expansion [see (4)]. Given a chain dependence

where , we can introduce the representation of the form

where ${{\widehat{V}}}=V^*(x,r)$. We subject the function to the condition of the first order homogeneity with the respect to the arguments $w_1,w_2,w_3$. Under these general conditions, the concise and convenient representation for the angular metric tensor $h_{ij}$ can be derived.

Indeed, let us use the notation

The identity

$$\begin{aligned} r=r_1w_1+ r_2w_2+ r_3w_3 \end{aligned}$$

(32)

holds fine due to the homogeneity implied. Evaluating the covariant unit vector components yields with

and , or

(33)

[because of (32)]. Recall that the variables , , and $w_3=i_{\{3\}}/b$ involve $b=y^0$ as a common normalizing factor, so that

in agreement with (33).

With this preparation, all the derivatives can readily be found. Since $V={{\widehat{V}}}$ or $V=V^*$ up to change of the variables, we obtained, first, that , , , and . After that, we can use and conclude that , etc.

By following this method, we can find all the components of the angular metric tensor . The result reads as

(34)

which in turn specifies all the components of the Finsler metric tensor .

B Appendix: Verification of the key representation for $h_{ij}$

To proceed, let us write the examined representation (24) as follows:

(35)

We should clarify the explicit structure of the objects $\eta _i=\eta _rr_i$, , $\phi _i=\phi _tt_i$, where the subscripts r, f, t mean differentiations. It is convenient to use the set , in which the subscripts 1, 2 and 3 mean differentiations with respect to $w_1,w_2$, and $w_3$. First of all, we apply the expansion . Because of the identity $r_1w_1+r_2w_2+r_3w_3=r$ (see (32)), we can write simply

(36)

The identity holds fine.

From (29) it follows that

(37)

and from (6) we obtain

which entails $f_1w_1+f_2w_2+f_3w_3=0$. Thus the expansion

(38)

obeying the property is completely known. For definiteness, we shall take $C_{11}=p$ and $C_{2}=C_{17}=1$. Finally, for the polar angle $\phi$, we have $\phi =\arctan t$, $\phi _t=1/(1+t^2)$, $t=w_1/w_2\equiv i/j$, from which it follows that , and

(39)

. We also have at our disposal the equalities $f=w_{\perp } p/w_3$ and [see (18) and (23)].

Inserting (36), (38), and (39) in (35) yields the following explicit representation:

(40)

Let us compare the contraction

with in (34). By taking $V_{rr}$ and $\eta _r$ from (22), we immediately conclude that , which means that the key representation (35) is true when contracting by the vector pair $b^ib^j\!$.

Let us examine another case

by comparing the right-hand side with the representation in (34). Taking into account (22), the expected equality reduces to

Insert , which yields

Since

we have simply

This value of the quantity precisely coincides with what is obtained after differentiation of $r_1$ indicated in (37). So the equality is true.

By such a method the examined equality (35) can be verified by considering the contractions by all the possible pairs of the orthonormal frame vectors . This proves the validity of the key representation (24).

C Appendix: Axial metric limit

Let us put $p=1$ in the functions introduced in the beginning of Sect. 3. We have

It follows that

and

In this way, we obtain the angle functions

which entail the representations

so that

The function V introduced in Sect. 3 reduces to

(we have taken the integration constant $C_1$ to be 1). Let us square this function:

Convenient cancelation is possible here. We come to

Since and , we have $r=\sqrt{(w_1)^2+(w_2)^2+(w_3)^2}\equiv |w|$. Using the notation

(41)

and , we can write

(42)

This $V^2$ does coincide with the respective function used in [2,3,4,5,6] and indeed $bV=F_{\{H;b\}}$ is valid.