Geodesics in the (anti-)de Sitter spacetime

TL;DR

This paper derives exact geodesic solutions in (anti-)de Sitter spacetimes, revealing how test particles and light follow specific world lines, and connects these to quantum vacuum phenomena and cosmic acceleration.

Contribution

It provides explicit geodesic solutions in (anti-)de Sitter spaces and links them to quantum vacuum fluctuations and cosmic acceleration explanations.

Findings

Test particles in AdS$_4$ follow sinusoidal geodesics.

Test particles in dS$_4$ follow hyperbolic sine geodesics.

Light rays follow straight-line geodesics.

Abstract

A class of exact solutions of the geodesic equations in (anti-)de Sitter AdS$_4$ and dS$_4$ spacetimes is presented. The geodesics for test particles in AdS$_4$ and dS$_4$ 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 AdS$_4$ 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 AdS$_4$ spacetime can describe the world lines of the virtual particles and antiparticles. The hyperbolic sine geodesics of dS$_4$ spacetime can explain why galaxies move apart with positive accelerations.