Spinning test particles in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>γ</mml:mi></mml:math> spacetime
Abstract
We consider the motion of spinning particles in the field of a well-known vacuum static axially symmetric spacetime, known as $\ensuremath{\gamma}$ metric, that can be interpreted as a generalization of the Schwarzschild manifold to include prolate or oblate deformations. We derive the equations of motion for spinning test particles by using the Mathisson-Papapetrou-Dixon equations together with the Tulczyjew spin-supplementary condition and restricting the motion to the equatorial plane. We determine the limit imposed by superluminal velocity for the spin of the particle located at the innermost stable circular orbits (ISCOs). We show that the particles on ISCO of the prolate $\ensuremath{\gamma}$ spacetime are allowed to have higher spin than the corresponding ones in the oblate case. We determine the value of the ISCO radius depending on the signature of the spin-angular momentum, $\mathrm{s}\ensuremath{-}\mathrm{L}$ relation, and show that the value of the ISCO with respect to the nonspinning case is bigger for $\text{sL}<0$ and smaller for $\text{sL}>0$. The results may be relevant for determining the properties of accretion disks and constraining the allowed values of quadrupole moments of astrophysical black hole candidates.