DrStupid said:
That is quite surprising. To my knowledge trajectories around a BH depend on its angular momentum. In order to make this happen without affecting the trajectory of the BH itself, the differences in energy and momentum would need to be emitted as gravitational waves. Is that the case?
Another point is the relativistic velovity dependence of the gravitational force. The gravitational mass is different from rest mass for relativistic velocities. In "Measuring the active gravitational mass of a moving object" [American Journal of Physics 53, 661 (1985)] DW Olson and RC Guarino derived a factor of ##\gamma \cdot \left( {1 + \beta ^2 } \right)## for hyperbolic trajectories (just to give an example). Keeping the same weight for rotating and non-rotating bodies would require that this effect cancels always out over all parts of a rotating body (at least with center of mass at rest) - independend from geometry and angular velocity. Is that the case?
This raises a number of question I have given some thought to.
To understand these issues, we need to separate notions:
1) A body's mass (within domain of GR where one can talk of a body with adequately defined mass), however it arises (rotating or not, to any degree). We don't care how internal components may add non-linearly to constitute mass, just whether we can speak of a reasonably isolated body whose mass can be measured. Then we can define effective passive gravitational mass by how it responds to the background metric of a much more massive body (of any nature, rotating or not). This same body's effects on much tinier test particles, far away, can be used to define an active gravitational mass. Far away, because we want the net influence of the body, not issues of energy/pressure distribution within it producing complex near field. One may also similarly define inertial mass.
2) Velocity dependent interaction (and even acceleration dependent interactions). The latter are discussed in Steve Carlip's well known paper "Aberration and the Speed of Gravity", where in he shows that (analyzed from a force point of view), a moving body generates both velocity and acceleration dependent gravitational forces, this being how the 'apparent speed' of gravity may appear to be near infinite. That a moving charge produces velocity dependent force on another test charge in no way is said to change the charge of the moving charge. Similarly, that a moving body produces velocity and acceleration dependent gravitational interactions with a test body doesn't change what the effective mass of the moving body is - it is the mass as in (1) measured in a quasilocal rest frame of the body. Guarino is discussing the velocity dependent effect.
3) Then, the question of how one accumulates component information to explain empirical mass (1), is a separate question. In the case of pressure-less, non interacting, uncharged, dust body, with insignificant self gravitation, then even if internal motions are relativistic, the invariant mass (= sum of particle total energies
in the center of momentum frame) will be equal to the mass in (1). Beyond this, you would need to use either a quasilocal mass integrating the stress energy tensor in a nontrivial way (for example, Bartnik mass), or, more commonly, treat the body as embedded in isolation in asymptotically flat spacetime and compute its ADM mass. Another option is komar mass if the body may be treated as producing a stationary metric, which is true for a uniformly spinning disk in equilibrium. Generally, whenever several of these are valid, they come out the same.
A further issue is that the question of orbits near a Kerr BH is probing near field of an extended body. Nothing about the equivalence principle says probing the near field of different same mass bodies with different SET distributions will fail to detect differences. While a BH is all vaccuum, it may be considered the remnant of different collapses for different BH parameters (angular momentum, for our purposes), and its having different near fields is irrelevant to the EP.
The EP would come into play when we ask, even for the near field of an extreme Kerr BH, if the motion of a tiny (very low mass) non-rotating BH would be different from the motion of a tiny Kerr BH. Per numerous claims (see for example, Clifford Will's living review article on testing GR), GR is the unique theory that predicts no violations of even the strong equivelence principle, ever. Experiment so far is consistent with this, but coverage of extreme regimes is very limited. Note that a charged body is not considered valid for the EP because it is never isolated from its own field. However, collections of balanced extreme charges, with extreme motion, such there is no external field, are perfectly ok for the EP (this, of course, is ordinary matter).
Thus, I return to may claim that there is no sense in which the weight (passive gravitational mas) and mass (inertial or active gravitational) will differ, when they are definable at all. As for computing the mass of a spinning body, integrating relativistic energy in COM frame will only be valid when pressure and stress (in natural units) and self gravity are all insignificant. But that is a separate issue from weight and mass being different.
