Pole-dipole model of massless particles

Zero-rest-mass particles in a gravitational field are considered to have energy-momentum tensors ${T}^{\mathrm{ij}}$ satisfying ${{T}^{\mathrm{ij}}}_{; j}=0$ and also ${{T}^{i}}_{i}=0$. The latter is satisfied by a massless electromagnetic pulse, and also by a neutrino. In analogy with the theory of material spinning particles in general relativity, moments of these equations are taken and cut off at the dipole level. The moments of the first equation yield the usual poledipole equations. The moments of the second equation yield the auxiliary conditions, which turn out to be ${p}_{i}{v}^{i}=0$ and ${v}_{i}{S}^{\mathrm{ki}}=0$, where ${v}^{i}$ is the velocity, ${p}^{i}$ is the momentum, and ${S}^{\mathrm{ki}}$ is the spin. The special cases of arbitrary spin in flat space, zero spin in curved spaces, and the eikonal approximation are treated and shown to give null geodesic trajectories. However, such trajectories do not seem to be a necessary consequence in the general case.

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