Dilatonic probe, force balance and gyromagnetic ratio

Following the Papapetrou–Dixon–Wald procedure we derive the equation of motion for a dilatonic test body (probe) with the dilaton coupling α=p/(2+p) in four dimension. Since the dilatonic freedom sometimes comes from extra dimensions, it is best to derive the EOM by a dimensional reduction from (p+4)-dimensions. We discuss about the force balance up to the gravitational spin–spin interactions via the probe technique. The force balance condition yields the saturation of a Bogomolʹnyi bound and the gyromagnetic ratio of the test body.