Observers with constant proper acceleration, constant proper Jerk, and beyond

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 proper n-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 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 trajectory in a plane 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 illustrate some of our results with examples in Minkowski, de Sitter and Schwarzschild spacetimes.1 From now on, by constantly accelerated observer we mean the observer with constant proper acceleration.Proper in the sense of being described in the instantaneous frame comoving with the observer. 2 We learn in [4] that "This terminology goes back to a 1932 advertisement of Kellogg's Rice Crispies which 'merrily Snap, crackle, and pop in a bowl of milk' ". Here our use of these concepts is unrelated to the standard use in cosmology as higher order time derivatives of the scale factor in the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric

On cosmological expansion and local physics

“…ordinary derivative for the covariant one. Or the concept of the constantly accelerated observer, that can be brought to GR by keeping the requirement of constancy [19,32,33] for some curvature scalars that generalize the Frenet-Serret formalism [34]. These cases bear in common that only the point and its neighborhood in a world line trajectory are necessary ingredients.…”

“…Next we should compute the Jerk, defined as the Fermi-Walker covariant derivative of the 4-acceleration [18,19] and implement the equation Jerk = 0, which describes the constantly accelerated trajectory. Due to the symmetry of our setting, one can check that the vanishing of the Jerk is equivalent to the constancy of the norm of the acceleration.…”

“…It is obvious therefore that the rubber balloon picture, as long as it is used to support the expansion of 3-space, is deeply flawed, misleading to say the least, and plain wrong to use more accurate words, despite its pedagogical role for instance in the quick explanation of the cosmological redshift. 19 One can trace the problem with this picture to the confusion and mixing of two different scales (Stages 4 and 2 in Fig. 1): One, the cosmological scale in which we apply the approximation of a continuous and homogeneous distribution of matter, and another, much lower scale, in which the distribution becomes discrete, essentially galaxies, DM and -vacuum, 20 and thus homogeneity is only considered at a discrete level.…”

On cosmological expansion and local physics

“…Next we should compute the Jerk, defined as the Fermi-Walker covariant derivative of the 4-velocity [20,21] and implement the equation Jerk = 0, which describes the constantly accelerated trajectory. Due to the symmetry of our setting, one can check that the vanishing of the Jerk is equivalent to the constancy of the norm of the acceleration.…”

“…Take for instance the geodesic motion, which is extended to GR by basically replacing the ordinary derivative for the covariant one. Or the concept of the constantly accelerated observer, that can be brought to GR by keeping the requirement of constancy [21,31,32] for some curvature scalars that generalize the Frenet-Serret formalism [33]. These cases bear in common that only the point and its neighborhood in a world line trajectory are necessary ingredients.…”