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- Title
Observers with constant proper acceleration, constant proper Jerk, and beyond.
- Authors
Pons, Josep M.; de Palol, Ferran
- Abstract
We discuss in Minkowski spacetime the differences between the concepts of constant proper n-acceleration and of vanishing (n + 1) -acceleration. By n-acceleration we essentially mean the higher order time derivatives of the position vector of the trajectory of a point particle, adapted to Minkowski spacetime or eventually to curved spacetime. The 2-acceleration is known as the Jerk, the 3-acceleration as the Snap, etc. As for the concept of propern-acceleration we give a specific definition involving the instantaneous comoving frame of the observer and we discuss, in such framework, the difficulties in finding a characterization of this notion as a Lorentz invariant statement. We show how the Frenet–Serret formalism helps to address the problem. In particular we find that our definition of an observer with constant proper acceleration corresponds to the vanishing of the third curvature invariant κ 3 —thus the motion is three dimensional (3d) in Minkowski spacetime—together with the constancy of the first and second curvature invariants and the restriction κ 2 < κ 1 , the particular case κ 2 = 0 being the one commonly referred to in the literature. We generalize these concepts to curved spacetime, in which the notion of a 2d trajectory is replaced by the vanishing of the second curvature invariant κ 2 . Under this condition, the concept of constant proper n-acceleration coincides with that of the vanising of the (n + 1) -acceleration and is characterized by the fact that the first curvature invariant κ 1 is a (n - 1) -degree polynomial of proper time. We discuss several possible definitions of the uniformly accelerated observer and we illustrate some of our results with examples in Minkowski, de Sitter and Schwarzschild spacetimes.
- Subjects
CURVED spacetime; POLYNOMIAL time algorithms; DEFINITIONS; PARTICLE tracks (Nuclear physics); SPACETIME
- Publication
General Relativity & Gravitation, 2019, Vol 51, Issue 6, pN.PAG
- ISSN
0001-7701
- Publication type
Article
- DOI
10.1007/s10714-019-2562-x