The geodesic motion in a Lorentzian spacetime can be described by trajectories in a 3-dimensional Riemannian metric. In this article we present a generalized Jacobi metric obtained from projecting a Lorentzian metric over the directions of its Killing vectors. The resulting Riemannian metric inherits the geodesics for asymptotically flat spacetimes including the stationary and axisymmetric ones. The method allows us to find Riemannian metrics in three and two dimensions plus the radial geodesic equation when we project over three different directions. The 3-dimensional Riemannian metric reduces to the Jacobi metric when static, spherically symmetric and asymptotically flat spacetimes are considered. However, it can be calculated for a larger variety of metrics in any number of dimensions. We show that the geodesics of the original spacetime metrics are inherited by the projected Riemannian metric. We calculate the Gaussian and geodesic curvatures of the resulting 2-dimensional metric, we study its near horizon and asymptotic limit. We also show that this technique can be used for studying the violation of the strong cosmic censorship conjecture in the context of general relativity.

洛伦兹时空中的测地线运动可以用 3 维黎曼度量中的轨迹来描述。在本文中，我们提出了一个广义的雅可比度量，该度量是通过将洛伦兹度量投影到其 Killing 向量的方向上获得的。生成的黎曼度量继承了渐近平坦时空的测地线，包括稳态和轴对称时空。当我们在三个不同的方向上投影时，该方法允许我们在三维和二维以及径向测地线方程中找到黎曼度量。当考虑静态、球对称和渐近平坦的时空时，三维黎曼度量简化为雅可比度量。但是，可以针对任意数量的维度中的更多种类的量度计算它。我们表明，原始时空度量的测地线由投影的黎曼度量继承。我们计算所得二维度量的高斯曲率和测地线曲率，研究其近地平线和渐近极限。我们还表明，该技术可用于研究广义相对论背景下对强宇宙审查猜想的违反。