Pseudo focal points along Lorentzian geodesics and Morse index

Given a Lorentzian manifold (M, g), a geodesic υ in M and a timelike Jacobi field Υ along υ, we introduce a special class of instants along υ that we call ΥY- pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Υ-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Υ is obtained as the restriction of a globally defined timelike Killing vector field.