Killing energy ($E=-K_a{u}^a$)

For a geodesic tangent $u$ and a stationary Killing vector field $K$, the Killing energy is $E=-g(K,u)$. It is conserved along the geodesic. When $K$ is future timelike and $u$ is future causal, $E>0$ for a nonzero tangent; it need not be positive where $K$ becomes spacelike.