Rest frames and relativistic effects on de Sitter spacetimes

Abstract

It is shown that the Nachtmann boosting method of introducing coordinates on de Siter manifolds can be completed with suitable gauge transformations able to keep under control the transformation under isometries of the conserved quantities. With this method, the rest local charts (or natural frames) are defined pointing out the role of the conserved quantities in investigating the relative geodesic motion. The advantages of this approach can be seen from the applications presented here. For the first time, the simple kinematic effects, the electromagnetic field of a free falling charge and the binary fission are solved in terms of conserved quantities on the expanding portion of the de Sitter spacetime.

Simple kinematic effects on de Sitter expanding universe

“…In Ref. [10] we have shown that we can define the rest frame {t , x } of the mobile m as the local chart in which this stays at rest in the origin x = 0. This means that for an observer O , comoving with the rest frame, the mobile m has the momentum P = 0.…”

“…A particular case studied in Ref. [10] is when the measurement is performed in the origin of the mobile frame, M ≡ O (with ξ = 0 and κ = κ(t * )). that hold only for small time intervals such that ωδt 1.…”

“…In Ref. [10] we solved completely the problem of the relative motion on de Sitter spacetimes considering the (conformal) Euclidean and de Sitter-Pailevé local charts as inertial natural frames where each geodesic is determined by the conserved momentum in a certain position at a given moment. Obviously, the coordinates of these local charts as well as the conserved quantities on geodesics are related among themselves through de Sitter isometries that take over the role of the Poincaré isometries in special relativity.…”

“…Among the examples briefly outlined in Ref. [10] the time dilation and Lorentz contraction on the de Sitter expanding portion were studied only in the particular case when the time and length are measured near the origin of the mobile frame. Here we would like to continue this study, analyzing how these effects depend on the relative position of the measurement point with respect to the mobile frame.…”

De sitter relativity in static charts

“…All these spacetimes have highest possible isometries [1] representing thus a good framework for studying the role of the conserved quantities with physical meaning [2,3,4] in studying the relative geodesic motion. With their help we constructed recently the dS relativity [5] in comoving charts [6] and the AdS relativity [7] in central charts (i. e. static and spherically symmetric) that complete our image about the special relativity in spacetimes with maximal symmetry.…”