Schwarzschild-Randers solution on a Lorentz tangent bundle

In this work, we extend for the first time the spherically symmetric Schwarzschild and Schwarzschild-De Sitter solutions with a Randers-type perturbation which is generated by a covector $A_\gamma$. This gives a locally anisotropic character to the metric and induces a deviation from the Riemannian models of gravity. A natural framework for this study is the Lorentz tangent bundle of a spacetime manifold. We apply the generalized field equations of this framework to the perturbed metric and derive the dynamics for the covector $A_\gamma$. Finally, we find the timelike, spacelike and null paths on the Schwarzschild-Randers spacetime and we compare them with the geodesics of general relativity. The obtained solutions are new and they enrich the corresponding literature.