Abstract
The relative geodesic motion in static (and spherically symmetric) local charts on the (1+3)-dimensional de Sitter spacetimes is studied in terms of conserved quantities. The Lorentzian isometries are derived, relating the coordinates of the local chart of a fixed observer with the coordinates of a mobile chart considered as the rest frame of a massive particle freely moving on a timelike geodesic. The time dilation and Lorentz contraction are discussed pointing out some notable features of the de Sitter relativity in static charts.
Highlights
The simplest (1 + 3)-dimensional spacetimes of special or general relativity are vacuum solutions of the Einstein equations whose geometry is determined only by the value of the cosmological constant Λ
In the comoving local charts we used so far, the de Sitter (dS) relativity seems to be closer to the Einstein special relativity since here we have translations and conserved momenta such that at least in the conformal Euclidean chart all the geodesic trajectories are rectilinear along the momentum direction [6,10]
Concluding we can say that the dS relativity in the conformal charts is closer to the Einstein special relativity having only rectilinear geodesics along the momentum directions, while in static charts the dS relativity is symmetric with the Anti-de Sitter (AdS) one
Summary
The simplest (1 + 3)-dimensional spacetimes of special or general relativity are vacuum solutions of the Einstein equations whose geometry is determined only by the value of the cosmological constant Λ. All these spacetimes have highest possible isometries [2] representing a good framework for studying the role of the conserved quantities with physical meaning in quantum theory [3,4,5,6] or for describing the classical relative geodesic motion [7,8,9] With their help we constructed recently the dS relativity [10] in comoving charts [11] and the AdS relativity [12] in static and spherically symmetric local charts that complete our image of the special relativity in spacetimes with maximal symmetry. In this manner we obtain a version of the dS relativity in static charts which is perfectly symmetric with the AdS one with respect to the change of the hyperbolic functions into trigonometric ones
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