Gravitational mass of composite systems

TL;DR

This paper clarifies the gravitational mass of composite systems in general relativity, resolving a long-standing misconception and demonstrating how equivalence principles apply to bound systems, with implications for quantum gravity effects.

Contribution

The paper derives the correct gravitational mass of composite systems from first principles, clarifying a longstanding issue in general relativity and elucidating the role of equivalence principles.

Findings

Correct gravitational mass derived for bound systems

Clarification of the equivalence principle in composite systems

Implications for quantum gravity interactions

Abstract

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.