Has been the spin magnetic dipole moment value of the free electron to be Lorentz invariant ever experimentally verified?

Question

As far as I know all experiments measure the rest value of the spin magnetic dipole moment of a free electron either indirectly at relativistic speeds near the speed of light measuring for example the g-factor in a synchrotron at constant translational relativistic speed at about one Bohr magneton or at non-relativistic speeds directly as a split distance of the quantized electron magnetic moment hitting a detection screen at the Stern-Gerlach experiment.

However, the synchrotron relativistic measurement is measuring the g-factor and not directly the magnetic moment force as a split distance as we have in the case of the SG experiment. My argument is that the synchrotron experiment is unsuitable to verify any invariance of the spin magnetic moment of the electron at relativistic speeds compared to non-relativisitc since it relies indirectly at the measurement of the g-factor that has proven many times that is a Lorentz invariant parameter in a vacuum.

Is there an alternative experiment method than can measure directly the spin magnetic dipole moment of the electron as a spatial displacement like the SG experiment without the use of the g-factor but at relativistic speeds that will prove experimentally the invariance of the spin magnetic dipole moment of the electron with translational relativistic speeds (i.e. Lorentz invariant)?

Or is actually the spin magnetic dipole moment of the electron not Lorentz invariant and a Lorentz correction must be applied in such a hypothetical experiment that would calculate its rest value?

Answer 1

The magentic dipole moment is part of a skew symmetric Lorentz tensor $M_{\mu\nu}$ which is defined so that the interaction with an electromagentic field is $\propto M_{\mu\nu}F^{\mu\nu}$. What is called "the magnetic dipole moment vector" has components $\mu_i= \epsilon_{ijk} M_{jk}/2$ where $i,j,k$ are the spatial indices measured in the rest frame of the particle. A moving dipole aquires an electric dipole character through the components $M_{i0}$ becoming non-zero after the boost. All this is taken into account in the theory behind the $g-2$ measurements.