Spinning particles in vacuum spacetimes of different curvature types

Abstract

We consider the motion of spinning test particles with nonzero rest mass in the "pole-dipole" approximation, as described by the Mathisson-Papapetrou-Dixon (MPD) equations, and examine its properties in dependence on the spin supplementary condition added to close the system. The MPD equation of motion is decomposed in the orthonormal tetrad whose time vector is given by the four-velocity $V^\mu$ chosen to fix the spin condition (the "reference observer") and the first spatial vector by the corresponding spin; such projections do not contain the Weyl scalars $\Psi_0$ and $\Psi_4$ obtained in the associated Newman-Penrose (NP) null tetrad. One natural choice of the remaining two spatial basis vectors is shown to follow "intrinsically"; it is realizable if the particle's four-velocity and four-momentum are not parallel. To see how the problem depends on the curvature type, one first identifies the first vector of the NP tetrad $k^\mu$ with the highest-multiplicity principal null direction of the Weyl tensor, and then sets $V^\mu$ so that $k^\mu$ belong to the spin-bivector eigenplane. In spacetimes of any algebraic type but III, it is possible to rotate the tetrads so as to become "transverse", namely so that $\Psi_1$ and $\Psi_3$ vanish. If the spin-bivector eigenplane could be made coincide with the real-vector plane of any of such transverse frames, the motion would consequently be fully determined by $\Psi_2$ and the cosmological constant; however, this can be managed in exceptional cases only. Besides focusing on specific Petrov types, we derive several sets of useful relations valid generally and check whether/how the exercise simplifies for some specific types of motion. The particular option of having four-velocity parallel to four-momentum is advocated and a natural resolution of nonuniqueness of the corresponding reference observer $V^\mu$ is suggested.Comment: 20 pages, no figure

Spinning test body orbiting around a Schwarzschild black hole: Circular dynamics and gravitational-wave fluxes

“…where the second term is known as the hidden momentum, i.e. p µ hidden := p µ − mv µ (8) (see, e.g., [30,31]). As discussed below, the OKS SSC is characterized by p µ hidden = 0.…”

“…In the following we briefly introduce the SSCs used in this work. A thorough analysis of these conditions can be found in [16] and [30].…”

“…The OKS SSC was proposed in [16,20] and revised recently in [30]; we work with the latter version. The main idea of the OKS SSC is to exploit the freedom in the choice of the future-pointing time-like vector V µ to impose desirable features in the EOM, like the cancellation of the hidden momentum.…”

“…For the derivation of Eqs (25), (26), see [30]. Note that the OKS SSC does not specify a unique worldline itself unless an initial V µ has been set.…”

Spacetime dynamics of spinning particles: Exact electromagnetic analogies

“…(31). This choice may actually be cast as a spin supplementary condition [39] (for its detailed discussion, see [33,39,125]). It is especially favored for pole-dipole particles in purely gravitational systems, because it leads to particularly simple equations: the momentum-velocity relation is simply P α = mU α , and S αβ is parallel transported, DS αβ /dτ = 0, cf.…”