Lie symmetries of nonrelativistic and relativistic motions

Abstract

We study the Lie symmetries of nonrelativistic and relativistic higher order constant motions in d spatial dimensions, i.e., constant acceleration, constant rate-of-change-of-acceleration (constant jerk), and so on. In the nonrelativistic case, these symmetries contain the z = 2/N Galilean conformal transformations, where N is the order of the differential equation that defines the constant motion. The dimension of this group grows with N. In the relativistic case the vanishing of the (d + 1)-dimensional spacetime relativistic acceleration, jerk, snap, ..., is equivalent, in each case, to the vanishing of a d-dimensional spatial vector. These vectors are the d-dimensional nonrelativistic ones plus additional terms that guarantee the relativistic transformation properties of the corresponding (d + 1)-dimensional vectors. In the case of acceleration there are no corrections, which implies that the Lie symmetries of zero acceleration motions are the same in the nonrelativistic and relativistic cases. The number of Lie symmetries that are obtained in the relativistic case does not increase from the four-derivative order (zero relativistic snap) onwards. We also deduce a recurrence relation for the spatial vectors that in the relativistic case characterize the constant motions.