Abstract
A set of world-line deviation equations is derived in the framework of Mathisson–Papapetrou–Dixon description of pseudo-classical spinning particles. They generalize the geodesic deviation equations. We examine the resulting equations for particles moving in the space–time of a plane gravitational wave.
Highlights
The geodesic deviation equation, D2nμ Dτ 2= −Rμανβ vαnν vβ (1)is one of the well-known equations of general relativity
We show the role of the space-time curvature on the motion of test particles and has important applications, namely, it is used for calculating relative accelerations of nearby particles in an observer-independent manner, and may be integrated to give the Lyapunov exponent in the study of chaotic behaviour of particle’s orbits
The equations we have found describe the world-line deviations in the framework of MPD equations determining the effect of the spin-curvature coupling on relative accelerations of nearby particles
Summary
Is one of the well-known equations of general relativity. This equation shows the role of the space-time curvature on the motion of test particles and has important applications , namely, it is used for calculating relative accelerations of nearby particles in an observer-independent manner, and may be integrated to give the Lyapunov exponent in the study of chaotic behaviour of particle’s orbits. This idea have been used in [1] to obtain generalized geodesic deviation equations by considering expansions containing higher orders of λ These generalized equations have been applied, for example, in [1] to the problem of closed orbital motion of test particles in the Kerr space-time and in [2] to the orbital motion in Schwarzchild metric. When forces other than gravity are present, or the particle has some internal structure, it would no longer move along geodesics in general In these situations a ”world-line deviation” equation may be obtained by modifying the geodesic deviation equation by taking the effect of the matter field or internal structure into. D Dτ and ∇ represent covariant derivative, the space-time signature would be (− + ++), and [μ, ν] stands for μν − νμ
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