Geodesics in the (anti-)de Sitter spacetime
Abstract
A class of exact solutions of the geodesic equations in (anti-)de Sitter AdS4 and dS4 spacetimes is presented. The geodesics for test particles in AdS4 and dS4 spacetimes are respectively sinusoidal and hyperbolic sine world lines. The world line for light rays is straight lines as known. In particular, the world lines of test particles are not dependent on their energy. Spontaneous symmetry breaking of AdS4 spacetime provides a physical explanation for arising of the virtual particle and antiparticle pairs in vacuum. Interestingly, the energy of a pair and the time its particles moving along their geodesics can be related by a relation similar to Heisenberg uncertainty one pertaining quantum vacuum fluctuations. The sinusoidal geodesics of AdS4 spacetime can describe the world lines of the virtual particles and antiparticles. The hyperbolic sine geodesics of dS4 spacetime can explain why galaxies move apart with positive accelerations.
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