Conserved deviation energy in constant curvature
= Conserved deviation energy in constant curvature
{title2=$K=|DS|^2-\kappa|S|^2$}
In <constant sectional curvature> $\kappa$, a <deviation vector> orthogonal to a unit <timelike geodesic> obeys $D^2S=\kappa S$, where $D=\nabla_T$. Therefore $K=g(DS,DS)-\kappa g(S,S)$ is constant, since its derivative is $2g(DS,D^2S)-2\kappa g(S,DS)=0$. Its squared length $f=g(S,S)$ satisfies $\ddot f=2K+4\kappa f$. In four spacetime dimensions, $\kappa=R/12$.