Modified Einstein and Finsler Like Theories on Tangent Lorentz Bundles

We study modifications of general relativity, GR, with nonlinear dispersion relations which can be geometrized on tangent Lorentz bundles. Such modified gravity theories, MGTs, can be modeled by gravitational Lagrange density functionals $f(\mathbf{R},\mathbf{T},F)$ with generalized/ modified scalar curvature $\mathbf{R}$, trace of matter field tensors $\mathbf{T}$ and modified Finsler like generating function $F$. In particular, there are defined extensions of GR with extra dimensional "velocity/ momentum" coordinates. For four dimensional models, we prove that it is possible to decouple and integrate in very general forms the gravitational fields for $f(\mathbf{R},\mathbf{T},F)$--modified gravity using nonholonomic 2+2 splitting and nonholonomic Finsler like variables $F$. We study the modified motion and Newtonian limits of massive test particles on nonlinear geodesics approximated with effective extra forces orthogonal to the four-velocity. We compute the constraints on the magnitude of extra-accelerations and analyze perihelion effects and possible cosmological implications of such theories. We also derive the extended Raychaudhuri equation in the framework of a tangent Lorentz bundle. Finally, we speculate on effective modelling of modified theories by generic off-diagonal configurations in Einstein and/or MGTs and Finsler gravity. We provide some examples for modified stationary (black) ellipsoid configurations and locally anisotropic solitonic backgrounds.