  Following the lines of a celebrated result by R. Bott (Comm. Pure Appl. Math.
9, 1956) we study the Morse index of the iterated of a closed geodesic in
stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian
geodesic that admits a timelike periodic Jacobi field. Given one such closed
geodesic $\gamma$, we prove the existence of a locally constant integer valued
map $\Lambda_\gamma$ on the unit circle with the property that the Morse index
of the iterated $\gamma^N$ is equal, up to a correction term
$\epsilon_\gamma\in\{0,1\}$, to the sum of the values of $\Lambda_\gamma$ at
the $N$-th roots of unity. The discontinuities of $\Lambda_\gamma$ occur at a
finite number of points of the unit circle, that are special eigenvalues of the
linearized Poincar\'e map of $\gamma$. We discuss some applications of the
theory.
