# Energy in spherically symmetric space times

In deriving the equations of motion for geodesics in spherically symmetric spacetimes through Hamiltonian formalism, we can find some constants of motion, namely, $$E$$ and $$L$$, the energy per unit of rest mass and angular momentum measured by a Schwarzschild observer. Nevertheless, it's not clear to me the real difference between the energy $$E$$ (related to timelike killing field) and the Hamiltonian $$\mathscr{H}$$, once the Hamiltonian is usually interpreted as the total energy of the system?

• Total energy of what system? With or without the energy of spacetime itself? – Qmechanic Jan 8 at 7:33

The Hamiltonian $$\mathscr{H}$$ of a geodesic is equal to the norm of the 4-velocity/4-momentum (depending on what affine parameter is being used). Conservation of the Hamiltonian simply expresses that the norm of the 4-velocity or 4-momentum (i.e. the particle mass squared) is conserved along the geodesic. This is true in any spacetime.