Abstract
We study the Lie symmetries of non-relativistic and relativistic higher order constant motions, in $d$ spatial dimensions, like constant acceleration, constant rate-of-change -of-acceleration (constant jerk), and so on. In the non-relativistic case, these symmetries contain the $z=\frac 2N$ 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.
Highlights
The study of the symmetries of nonrelativistic and relativistic motions in flat spacetime has been the subject of interest through the years; see [1,2]
We start with the simplest case of constant position, which is somehow special, in that it exhibits an infinite number of point symmetries, and has a relation with the Carrollian limit of the relativistic case [14]
The total variation of xi is given by an arbitrary space diffeomorphism ξi 1⁄4 ξiðxÞ, while t can be transformed by an arbitrary, space-dependent diffeomorphism δt 1⁄4 fðt; xÞ. This makes physical sense: space points can be mapped to arbitrary space points and provided the map does not depend on time one still gets a fixed point, while time can arbitrarily be transformed at each point of space
Summary
The study of the symmetries of nonrelativistic and relativistic motions in flat spacetime has been the subject of interest through the years; see [1,2]. A recurrence relation was presented in [5] for these higher order d þ 1 spacetime vectors, it turns out that the study of the symmetries of the corresponding zero motions is better done in terms of a d-spatial vector containing the components of the d þ 1 vector, such that the vanishing of the latter is equivalent to the vanishing of the former These vectors are the nonrelativistic acceleration, nonrelativistic jerk, etc., plus additional terms that guarantee the relativistic transformation properties of the corresponding d þ 1 vectors.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
