Schwarzschild: Energy Conservation at Finite r

[kent davidge]

Let $u^\alpha$ and $p^\alpha$ denote a massive particle's four velocity and four momentum, respectively. Also, let $\xi^\alpha = (1,0,0,0)$ be a time like Killing vector. Since $g_{00} \xi^0 u^0 = g_{00} p^0 / m = -(1 - 2m / r) E / m$ is conserved, if we let $r \longrightarrow \infty$ we have that $E / m$ is conserved. This is the particle's energy per mass. But how to think about that term when $r$ is finite? What's the quantity that's being conserved there?

 

[Orodruin]

A GR analogue of total energy in this case. If you look at the weak field limit with non-relativistic velocities relative to $\xi$, you can expand in small quantities and obtain the Newtonian total energy = kinetic + potential.