  The equivalence principle in combination with the special relativistic
equivalence between mass and energy, $E=mc^2$, is one of the cornerstones of
general relativity. However, for composite systems a long-standing result in
general relativity asserts that the passive gravitational mass is not simply
equal to the total energy. This seeming anomaly is supported by all explicit
derivations of the dynamics of bound systems, and is only avoided after
time-averaging. Here we rectify this misconception and derive from first
principles the correct gravitational mass of a generic bound system in an
external gravitational field. Our results clarify a lasting conundrum in
general relativity and show how the weak and strong equivalence principles
naturally manifest themselves for composite systems. The results are crucial
for describing new effects due to the quantization of the interaction between
gravity and composite systems.
