Mathisson–Papapetrou–Tulczyjew–Dixon equations in ultra-relativistic regime and gravimagnetic moment

Abstract

Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations in the Lagrangian formulation correspond to the minimal interaction of spin with gravity. Due to the interaction, in the Lagrangian equations instead of the original metric g emerges spin-dependent effective metric G = g + h(S). So we need to decide, which of them the MPTD particle sees as the space-time metric. We show that MPTD equations, if considered with respect to original metric, have unsatisfactory behavior: the acceleration in the direction of velocity grows up to infinity in the ultra-relativistic limit. If considered with respect to G, the theory has no this problem. But the metric now depends on spin, so there is no unique space-time manifold for the Universe of spinning particles: each particle probes his own three-dimensional geometry. This can be improved by adding a non-minimal interaction of spin with gravity through gravimagnetic moment. The modified MPTD equations with unit gravimagnetic moment have reasonable behavior within the original metric.Equations of motion of a rotating body in a curved background are formulated usually in the multipole approach to describe the body [1][2][3][4][5][6][7][8]. We consider the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations, which describe the body in the pole-dipole approximation, in the form studied by Dixon (for the relation of the Dixon equations with those of Papapetrou and Tulczyjew see p. 335 in [4] as well as the recent works [7,8]):(our S µν is twice that of Dixon). They are widely used now in computations of spin effects in compact binaries and rotating black holes [9][10][11][12][13][14][15][16], so our results may be relevant in this framework. In the multipole approach, x µ (τ ) is called the representative point (centroid) of the body, the antisymmetric spin-tensor S µν (τ ) is associated with the inner angular momentum and the vector P µ (τ ) is called momentum.A few words about our notation. As we will discuss the ultra-relativistic limit, we do not assume the proper-time parametrization, that means that we do not add the equation g µνẋ µẋν = −c 2 to the system (1). Our variables are taken in an arbitrary parametrization τ , andẋ µ = , and so on. The notation for the scalar functions constructed from second-rank tensors are θS = θ µν S µν , S 2 = S µν S µν . In the present work we continue the analysis of the MPTD equations on the base of Lagrangian formulation developed in the recent work [17], and discuss the necessity of generalizing the formalism by including an interaction of the spin with gravity through non vanishing gravimagnetic moment. We discuss the behavior of the MPTD-particle in ultra-relativistic limit, when the speed of the particle approximates the speed of light. Since we are interested in the influence of the spin on the particle's trajectory, we eliminate the momenta from the MPTD equations, thus obtaining a second-order equation for the representative point x µ (τ ). To achieve this, we compute the derivative of the spin supplementary condition (SSC), ∇(S µν ...

Charged spinning black holes as accelerators of spinning particles

“…Note that the velocity vector u µ and the canonical momentum vector P µ of the spinning test particle are not parallel [22,25,[28][29][30], and the canonical momentum P µ satisfies P µ P µ = −m 2 which indicates that the canonical momentum vector keeps timelike along the trajectory. However, the velocity vector u µ of the spinning test particle might transform to be spacelike from timelike [22,25,28].…”

“…See some more details in Refs. [29,30]. For simplicity, we do not consider the reaction in this paper.…”

“…Because the velocity vector u µ of the spinning test particle might transform to be spacelike from timelike [22,25,[28][29][30], we can use the velocity vector u µ to determine whether the movement of particles is spacelike or timelike. The corresponding velocity u µ can be solved by using the equations of motion (5) and (6) based on [28] as follows…”

“…In Refs. [22,25,[28][29][30] show that the velocity vector u µ and the canonical momentum vector P µ of the spinning test particle are not parallel, and the velocity vector u µ might transform to be spacelike from timelike along the trajectory. So it is important to clarify the relation between the divergent region and the superluminal region in spin-angular space.…”

Probing noncommutativities of phase space by using persistent charged current and its asymmetry

“…Nevertheless, noncommutative property can also appear in a real quantum system [6,7]. Because the noncommutative extension is fundamental, all the ordinary quantum physics are deformed with corrections parametrized by the constant θ μν , for instance, the breaking of rotational symmetry [8,9], distortion of energy levels of the atoms [10][11][12], corrections on the spin-orbital interactions [13][14][15][16][17][18][19][20], and the quantum speed of relativistic charged particles [21][22][23][24][25] and fluid dynamics [26,27]. Most importantly, the spatial component of the noncommutative parameter (θ μν ), θ ij , behaves like a permanent magnetic dipole moment and gives a contribution ϵ ijk θ ij B k to the energy of system.…”