Addendum to "Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric", Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 27 (2010) 857--876

We give the details of the proof of the equality between the critical groups, with respect the H^1 and C^1 topology, at a non-degenerate critical point of the energy functional of a non-reversible Finsler manifold (M,F), defined on the Hilbert manifold of the H^1 curves connecting two given points on M.