  We study field equations for a weak anisotropic model on the tangent Lorentz
bundle $TM$ of a spacetime manifold. A geometrical extension of General
Relativity (GR) is considered by introducing the concept of local anisotropy,
i.e. a direct dependence of geometrical quantities on observer $4-$velocity. In
this approach, we consider a metric on $TM$ as the sum of an h-Riemannian
metric structure and a weak anisotropic perturbation, field equations with
extra terms are obtained for this model. As well, extended Raychaudhuri
equations are studied in the framework of Finsler-like extensions. Canonical
momentum and mass-shell equation are also generalized in relation to their GR
counterparts. Quantization of the mass-shell equation leads to a generalization
of the Klein-Gordon equation and dispersion relation for a scalar field. In
this model the accelerated expansion of the universe can be attributed to the
geometry itself. A cosmological bounce is modeled with the introduction of an
anisotropic scalar field. Also, the electromagnetic field equations are
directly incorporated in this framework.
