form of the Kerr–Newman metric, the determinant of the metric tensor is everywhere equal to negative one, even near the source. Electromagnetic fields in Kerr–Schild form The electric and magnetic fields can be obtained in the usual way by differentiating the four-potential to obtain the electromagnetic field strength tensor. It will be convenient to switch over to three-dimensional vector notation. The static electric and magnetic fields are derived from the vector potential and the scalar potential like this: Using the Kerr–Newman formula for the four-potential in the Kerr–Schild form, in the limit of the mass going to zero, yields the following concise complex formula for the fields: The quantity omega () in this last equation is similar to the Coulomb potential, except that the radius vector is shifted by an imaginary amount. This complex potential was discussed as early as the nineteenth century, by the French mathematician Paul Émile Appell. Irreducible mass The total mass-equivalent M, which contains the electric field-energy and the rotational energy, and the irreducible mass Mirr are related by which can be inverted to obtain In order to electrically charge and/or spin a neutral and static body, energy has to be applied to the system. Due to the mass–energy equivalence, this energy also has a mass-equivalent; therefore M is always higher than Mirr. If for example the rotational energy of a black hole is extracted via the Penrose processes, the remaining mass–energy will always stay greater than or equal to Mirr. Important surfaces Setting to 0 and solving for gives the inner and outer event horizon, which is located at the Boyer–Lindquist coordinate Repeating this step with gives the inner and outer ergosphere Equations of motion For brevity, we further use dimensionless natural units of , with the Coulomb constant , where reduces to and to , and the equations of motion for a testparticle of charge become with for the total energy and for the axial angular momentum. is the Carter constant: where is the poloidial component of the testparticle's angular momentum, and the orbital inclination angle. and are also conserved quantities. is the frame dragging induced angular velocity. The shorthand term is defined by The relation between the coordinate derivatives and the local 3-velocity is for the radial, for the poloidial, for the axial and for the total local velocity, where is the axial radius of gyration (local circumference divided by 2π), and the