When we calculate the relativistic angular momentum of a particle in the direction of the $z$-axis, what relativistic mass should we use? My hypothesis is that the relativistic mass used for such a calculation does not depend on the particle’s speed in the $z$-direction; rather the relativistic mass used for such a calculation should depend on the speed of the particle on the $xy$-plane, i.e., the Lorentz factor should only use the square of the speed on the $xy$-plane. Is my hypothesis correct?
Relativistic angular momentum in a related discussion suggests that the component of the relativistic angular momentum along the direction of motion between two inertial frames is the same for both frames. It can be inferred from this that the relativistic mass used for calculating the relativistic angular momentum in that direction is independent of the momentum in that direction. Your advice will be much appreciated.
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