Relativistic particles with torsion in three-dimensional non-vacuum spacetimes

Abstract

In this paper we analyze trajectories of spacelike curves which are critical points of a Lagrangian depending on its total torsion. We focus on two important families of spacetimes, Generalized Robertson-Walker and standard static spacetimes. For the former, we show that such trajectories are those with constant curvature. For the latter we also obtain a characterization in terms of the curvature of the trajectory, but in this case measured with an appropriate conformal metric.

Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems

“…On the surface N we have zβ = 0, so za are the coordinates of N. The Poisson tensor ωij = {z i , zj } of M n in these coordinates has the following special form: {z a , zb } = ω ab (z β , zc ), {z β , zi } = {K β , zi } = 0, for any i. Since ωij obeys (40) and ( 41 we can write the first equality in (79) as follows:…”

“…As we saw in Sect. II, Poisson manifold can be defined choosing a contravariant tensor with the properties (40) and (41). Here we discuss another way, which works for the construction of a nondegenerate Poisson structures on even-dimensional manifolds.…”

On the dynamics of relativistic particles with torsion in warped product spacetimes

“…We thus need to compute W(v) and W(κ 2 ). Most of the computation proceeds like in the 2 + 1 case, as found in [11]. The main difference is as follows: the second curvature is given by…”

“…The other terms in W(κ 2 ) and W(v) are expanded using the same techniques. For completeness, we provide some steps of the calculation, which can be found in [11]. For W(v), we use that…”

“…Thus, our aim in this work is to study the classical dynamics of the torsion functional in fourdimensional backgrounds, extending the three-dimensional study [11]. This is not only justified from a physical viewpoint, but also mathematically, as the resulting equations of motion are much more complicated (see theorem 2.1) and a more sophisticated approach is needed.…”