  Given the class of Finsler spaces with Lorentzian signature $(M,L)$ on a
manifold $M$ endowed with a timelike vector field $\mathcal{X}$ satisfying
$g_{(x,y)}(\mathcal{X},\mathcal{X})<0$ at any point $(x,y)$ of the slit tangent
bundle, a pseudo-Riemannian metric defined on $M$ of signature $n-1$ is
associated to the fundamental tensor $g$. Furthermore, an affine, torsion free
connection is associated to the Chern connection determined by $L$. The
definition of the average connection does not make use of $\mathcal{X}$.
Therefore, there is no direct relation between these two averaged objects.
