In this work, we extend for the first time the spherically symmetric Schwarzschild and Schwarzschild–De Sitter solutions with a Finsler–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 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, we solve the timelike ones numerically and we compare them with the classic geodesics of general relativity. The obtained solutions are new and they enrich the corresponding literature.

在这项工作中，我们首次扩展了由Finsler-Randers型扰动产生的球对称Schwarzschild和Schwarzschild-De Sitter解，该扰动由向量cos \（A_ \ gamma \）生成。这为度量提供了局部各向异性的特征，并引起了与黎曼引力模型的偏离。这项研究的自然框架是时空流形的Lorentz切线束。我们将广义场方程应用于扰动量度，并推导辅矢量\（A_ \ gamma \）的动力学。最后，我们在Schwarzschild–Randers时空上找到时空，时空和空路径，我们用数值方法求解了时空，并将它们与广义相对论的经典测地线进行了比较。所获得的解决方案是新的，它们丰富了相应的文献。