The determination of the parameters of “spherical” motion in a kerr field

The problem of the existence of class M geodesic motions on the surface r=const, all of whose mechanical parameters in the specified Kerr field are determined only from stability conditions, is posed and investigated. A system of equations which determines this class is derived and solved. It is shown that in the general case this motion does not cover the entire surface r=const and is restricted by the condition θ0 ≤θ ≤ π}- θ0. Simple algebraic expressions are found for all the parameters of these configurations-energy, momentum, radius, and the angle θ0-as functions of the specific angular momentuma of the Kerr field. It is shown that these motions can exist only in Kerr fields with a value of the parametera larger or equal to rg/2. In a Kerr field with a fixed value ofa there exist only two configurations with the indicated properties. In conclusion, the properties of the M-solutions associated with the appearance of configurations with negative energies and negativeness of g00 within the limits of some configurations and values ofa larger than rg/2 are discussed. It is noted that the negative values of the energy occur only in those configurations within whose limits g00 rg/2 if one does not made the assumption, unnecessary within the framework of the problem under discussion, that the sources of the Kerr field are in the region r=0.