Carroll geodesics

TL;DR

This paper develops a Carrollian analog of geodesic equations using effective field theory, revealing rich dynamics and unique features like a single unstable circular orbit and a perfect mirror behavior of Carroll black holes.

Contribution

It introduces the first derivation of Carrollian geodesics, including electric and magnetic contributions, and analyzes their behavior on a Carroll-Schwarzschild background.

Findings

Carrollian geodesics exhibit rich dynamics with coupled electric and magnetic effects.

There is only one unstable circular orbit at the Carroll extremal surface.

Carroll black holes act as perfect mirrors for geodesics.

Abstract

Using effective field theory methods, we derive the Carrollian analog of the geodesic action. We find that it contains both `electric' and `magnetic' contributions that are in general coupled to each other. The equations of motion descending from this action are the Carrollian pendant of geodesics, allowing surprisingly rich dynamics. As an example, we derive Carrollian geodesics on a Carroll-Schwarzschild background and discover an effective potential similar to the one appearing in geodesics on Schwarzschild backgrounds. However, the Newton term in the potential turns out to depend on the Carroll particle's energy. As a consequence, there is only one circular orbit localized at the Carroll extremal surface, and this orbit is unstable. For large impact parameters, the deflection angle is half the value of the general relativistic light-bending result. For impact parameters slightly bigger than the Schwarzschild radius, orbits wind around the Carroll extremal surface. For small impact parameters, geodesics get reflected by the Carroll black hole, which acts as a perfect mirror.