  For smooth solutions to Maxwell's equations sourced by a smooth
charge-current distribution $j_a$ in stationary, asymptotically flat
spacetimes, one can prove an energy conservation theorem which asserts the
vanishing of the sum of (i) the difference between the final and initial
electromagnetic self-energy of the charge distribution, (ii) the net
electromagnetic energy radiated to infinity (and/or into a black hole/white
hole), and (iii) the total work done by the electromagnetic field on the charge
distribution via the Lorentz force. A similar conservation theorem can be
proven for linearized gravitational fields off of a stationary, asymptotically
flat background, with the second order Einstein tensor playing the role of an
effective stress-energy tensor of the linearized field. In this paper, we prove
the above theorems for smooth sources and then investigate the extent to which
they continue to hold for point particle sources. The ``self-energy'' of point
particles is ill defined, but in the electromagnetic case, we can consider
situations where, initially and finally, the point charges are stationary and
in the same spatial position, so that the self-energy terms should cancel.
Under certain assumptions concerning the decay behavior of source-free
solutions to Maxwell's equations, we prove the vanishing of the sum of the net
energy radiated to infinity and the net work done on the particle by the
DeWitt-Brehme radiation reaction force. We also obtain a similar conservation
theorem for angular momentum in an axisymmetric spacetime. In the gravitational
case, we argue that similar conservation results should hold for freely falling
point masses whose orbits begin and end at infinity.
