Gravitational mass of composite systems

The equivalence principle in combination with the special relativistic equivalence between mass and energy, $E=m{c}^2$, 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.

Figure 1

A bound N-particle system on a world line $Q^{\mu }(t)$. The primed coordinates describe the frame in which the CM is at rest, under the conditions (3). This frame defines the physical internal properties of the system and the correct gravitational mass.