Pole-dipole model of massless particles. II

Abstract

The pole-dipole equations derived in a previous paper for massless particles (whose defining relation is that the energy-momentum tensor has zero trace) are examined. If the reference point X' describing the motion is not somewhere on the disk perpendicular to the three-velocity through the energy center, then X' describes a null geodesic with no assumptions. The property that X' is the particle's energy center in some local reference frame (called a C-frame) is shown to be a constant of the motion. If the energy of the particle in the C-frames is not zero, then the trajectory is again a null geodesic. The conditions on the curvature needed to have the momentum parallel to the four-velocity are determ~ned. Helicity properties are also discussed.

Spacetime dynamics of spinning particles: Exact electromagnetic analogies

“…Eqs. (50), H αβ = 0 ⇒ DP α /dτ = 0. Hence we have (C6) as the equation of motion, with trivial solution a α = 0 ⇒ P α = mU α , i.e., the gyroscope moves along a radial geodesic.…”

“…The former seems the most natural choice, as it amounts to computing the center of mass in its proper frame, i.e., in the frame where it has zero 3-velocity. It also arises in a natural fashion in some derivations [46,47] (see also [48]), and has been argued [49][50][51] to be the only one that can be applied in the case of massless particles. It turns out, however, that it does not determine the worldline uniquely.…”

Mathisson’s helical motions for a spinning particle: Are they unphysical?

“…However in the presence of gravitational and electromagnetic fields, the Tulczyjew-Dixon solution no longer coincides with any of Mathisson's solutions, 7 and it turns out, as exemplified in several applications in [31], that in some more complex setups it is the Mathisson-Pirani condition that provides the simplest and clearest description. This condition arises also in a natural fashion in a number of treatments [14,19] (see also [25]); for massless particles, it has been argued in [39,40] that it is actually the only one that can be applied. And for the case of the equation for the spin evolution (3b), it is always the Mathisson-Pirani condition that yields the simplest and physically more sound description: in the absence of electromagnetic field (or other external torques), S is Fermi-Walker transported; i.e., the gyroscope's axis is fixed relative to a nonrotating frame, which is the natural, expected result.…”

How does the photon’s spin affect gravitational wave measurements?

“…For the Mathisson-Pirani SSC S µ νẊ ν = 0, it ism that stays conserved. Setting this to zero for a massless particle, implies that the particle follows a null geodesic, while the momentum vector is spacelike [19][20][21][22]. For the Tulczyjew SSC S µ ν P ν = 0, it is m that is conserved.…”

“…In the massless case, the choice of the SSC is even more subtle than in the massive one. Two main arguments have been given in favor of the Mathisson-Pirani SSC: i) Maxwell equations minimally coupled to gravity yield null geodesics in the geometric-optic limit [24], like do the MPD equations together with this SSC [19][20][21] (with just one type of counterexample given in [20]). (ii) Imposing conformal invariance of the theory, in particular the tracelessness of the energy-momentum tensor, implies (a slight generalization of) the Mathisson-Pirani constraint [22,25,26].…”