Lie symmetries of nonrelativistic and relativistic motions

Abstract

We study the Lie symmetries of non-relativistic 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 non-relativistic 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 space-time relativistic acceleration, jerk, snap, . . . , is equivalent, in each case, to the vanishing of a d-dimensional spatial vector. These vectors are the d-dimensional non-relativistic 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 non-relativistic 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