# Elementary example of energy and momentum of an extended physical system in special relativity

## TL;DR

This paper explains a paradox in special relativity where the energy and momentum of a rotating system of two masses connected by a string oscillate for moving observers, highlighting the string's contribution to total energy and momentum.

## Contribution

It demonstrates how the binding mechanism in extended systems affects total energy and momentum in special relativity, using a simple rotor example.

## Key findings

- Oscillations in energy and momentum depend on the observer’s frame of reference.
- The string's contribution to energy and momentum is essential, regardless of its lightness.
- The approach generalizes to other extended systems in relativistic physics.

## Abstract

An instructive paradox concerning classical description of energy and momentum of extended physical systems in special relativity theory is explained using an elementary example of two point-like massive bodies rotating on a circle in their center-of-mass frame of reference, connected by an arbitrarily light and infinitesimally thin string. Namely, from the point of view of the inertial observers who move with respect to the rotating system, the sums of energies and momenta of the two bodies oscillate, instead of being constant in time. This result is understood in terms of the mechanism that binds the bodies: the string contributes to the system total energy and momentum no matter how light it is. Its contribution eliminates the unphysical oscillations from the system total four-momentum. Generality of the relativistic approach, applied here to the rotor example, suggests that in every extended physical system its binding mechanism contributes to its total energy and momentum.