  We present a new approach to the Jacobi metric formalism for Lorentzian
spacetimes. We define a new Jacobi metric that inherits the spacetime, null and
timelike geodesic equations for asymptotically flat spacetimes including the
stationary ones. The method provides a way for finding a reduced 2-dimensional
Jacobi metric, which in the static spherically symmetric case coincides with
the already known Jacobi metric. We discuss about the conserved quantities for
both proposals. Applying this formalism to black hole metrics, we show that the
Gaussian curvature of the reduced 2-dimensional Jacobi metric attains an
extremum at the points where the horizons are located giving one of the first
characterizations of horizons in a Riemannian metric coming from a spacetime.
