# Non--geodesic circular motion of massive spinning test bodies

## TL;DR

This paper demonstrates that spinning test bodies in circular orbit around Earth deviate from geodesic motion due to spin effects, affecting their tangential velocities depending on spin orientation, which has implications for testing the Equivalence Principle.

## Contribution

It provides an exact solution for the circular motion of spinning bodies, highlighting non-geodesic behavior caused by spin, a factor previously neglected in experiments.

## Key findings

- Spinning bodies exhibit different tangential velocities based on spin orientation.
- Circular orbits of spinning bodies show no tangential acceleration.
- Estimated velocity differences for Earth-orbiting bodies are provided.

## Abstract

Recent interest on studying possible violations of the Equivalence Principle has led to the development of space satellite missions testing it for bodies moving on circular orbits around Earth. This experiment establishes that the validity of the Equivalence Principle is independent of the composition of bodies. However, the internal degrees of freedom of the bodies (such as spin) were not taken into account. In this work, it is shown exactly that the circular orbit motion of test bodies does present a departure from geodesic motion when spin effects are not negligible. Using a Lagrangian theory for spinning massive bodies, an exact solution for their circular motion is found showing that the non--geodesic behavior manifests through different tangential velocities of the test bodies, depending on the orientation of its spin with respect to the total angular momentum of the satellite. Besides, for circular orbits, spinning test bodies present no tangential acceleration. We estimate the difference of the two possible tangential velocities for the case of circular motion of spinning test bodies orbiting Earth.