In Lee's Intro to Riemannian manifolds I learned that for a Riemannian manifold (M,g)
(1)The first variation of the arc-lenght functional is given by equation 6.1 (see below)
(2)The second variation of the arc-lenght functional is 10.19 (see below)
(3)The index form of $\gamma$ is related to the second variation by Corollary 10.23 (see below)
(4)The sign of I gives me infomation about $\gamma$.It tells me that : if I(V,V)<0 for every proper normal vector field along $\gamma \implies \gamma $ is not minimizing.(Also from corollary 10.23) (I am not sure if the index form tells anything else about $\gamma$)
My question is what changes if $\gamma$ is a timelike/spacelike geodesic in a Lorentzian manifold? Are formulas (1) and (2) the same and are the conclusion (3) and (4) still valid?
