The mass of an object is given by the square root of the norm of the four momentum, which is equivalent to the energy the object has in the centre of momentum frame.
However in the centre of momentum frame, an object can still have angular momentum and hence rotational energy. Am I correct then in thinking then that an object's mass increases with more angular momentum?
Angular momentum itself does not contribute to the invariant mass, however an object with angular momentum will have KE in its center of momentum frame, and this KE is part of the invariant mass.
It is the KE that provides this “additional” mass, not the angular momentum. So, for example, two objects with the same angular momentum but different amounts of rotational KE will have different amounts of “additional” mass. Of course, for a given object the rotational KE and the angular momentum are closely related, so it is natural but imprecise to say that angular momentum affects the mass. To be precise it is the rotational KE that affects the mass.
