The Euler-Arnold equation describes the geodesics on a Lie group on which an inner product is induced, through left group translation, by a given inner product defined on the corresponding Lie algebra (that is, the tangent space of the Lie group at the identity). In this paper, this approach is applied to compute the geodesic curves on the Lorentz Lie group of relativistic transformations.

This is analogous to the geodesic description of different important physical systems, and indeed can be seen as a toy model for ideal fluids as geodesics on the group of incompressible vector fields. In order to obtain a positive metric on the Lorentz group that gives an integrable system, we define a positive non-invariant inner product on the Lorentz Lie bi-algebra. Then, we derive and solve the geodesic equations of motion in a convenient system of coordinates that takes advantage of the Lie bi-algebra structure.