# The origin of the energy-momentum conservation law

## TL;DR

This paper explores the deep connection between the action-reaction principle and energy-momentum conservation across various theories, highlighting their equivalence and the impact of topology on energy definitions, especially in general relativity.

## Contribution

It demonstrates the equivalence of action-reaction and energy-momentum conservation principles in multiple theories and explains the ambiguity of energy in topologically nontrivial spacetimes.

## Key findings

- Energy-momentum conservation aligns with action-reaction in Maxwell-Lorentz and Yang-Mills-Wong theories.
- In general relativity, energy and momentum become ambiguous due to nontrivial topologies.
- The total energy in systems like black holes depends on coordinate choices and topological features.

## Abstract

The interplay between the action-reaction principle and the energy-momentum conservation law is revealed by the examples of the Maxwell-Lorentz and Yang-Mills-Wong theories, and general relativity. These two statements are shown to be equivalent in the sense that both hold or fail together. Their mutual agreement is demonstrated most clearly in the self-interaction problem by taking account of the rearrangement of degrees of freedom appearing in the action of the Maxwell-Lorentz and Yang-Mills-Wong theories. The failure of energy-momentum conservation in general relativity is attributed to the fact that this theory allows solutions having nontrivial topologies. The total energy and momentum of a system with nontrivial topological content is found to be ambiguous, coordintization-dependent quantities. For example, the energy of a Schwarzschild black hole may take any positive value greater than, or equal to, the mass of the body whose collapse is responsible for arising this black hole. We draw the analogy to the paradoxial Banach-Tarski theorem; the measure becomes a poorly defined concept if initial three-dimensional bounded sets are rearranged in topologically nontrivial ways through the action of free non-Abelian isometry groups.