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.

With metric signature $(+---)$ and a future timelike stationary Killing vector $K$, positive energy is $E=g(P,K)$ for future physical momentum $P$. A common $(-+++)$ convention instead uses $-g(P,K)$. The normalization of $K$ fixes the observer or asymptotic time scale.