Publikace ÚTF

Spinning particles in general relativity: Momentum-velocity relation for the Mathisson-Pirani spin condition

Costa, L., F., O., Lukes-Gerakopoulos, G. and Semerák, O.

The Mathisson-Papapetrou-Dixon (MPD) equations, providing the “pole-dipole” description of
spinning test particles in general relativity, have to be supplemented by a condition specifying the
worldline that will represent the history of the studied body. It has long been thought that the MathissonPirani (MP) spin condition—unlike other major choices made in the literature—does not yield an explicit
momentum-velocity relation. We derive here the desired (and very simple) relation and show that it is in
fact equivalent to the MP condition. We clarify the apparent paradox between the existence of such a
definite relation and the known fact that the MP condition is degenerate (does not specify a unique
worldline), thus shedding light on some conflicting statements made in the literature. We then show how,
for a given body, this spin condition yields infinitely many possible representative worldlines, and derive a
detailed method how to switch between them in a curved spacetime. The MP condition is a convenient
choice in situations when it is easy to recognize its “nonhelical” solution, as exemplified here by bodies in
circular orbits and in radial fall in the Schwarzschild spacetime.
journal:Phys. Rev. D

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