geometry of spacetime for a rotating charged black hole with mass M, charge Q and angular momentum J. The formula for this metric depends upon what coordinates or coordinate conditions are selected. Two forms are given below: Boyer–Lindquist coordinates, and Kerr–Schild coordinates. The gravitational metric alone is not sufficient to determine a solution to the Einstein field equations; the electromagnetic stress tensor must be given as well. Both are provided in each section. Boyer–Lindquist coordinates One way to express this metric is by writing down its line element in a particular set of spherical coordinates, also called Boyer–Lindquist coordinates: where the coordinates (r, θ, ϕ) are standard spherical coordinate system, and the length scales: have been introduced for brevity. Here rs is the Schwarzschild radius of the massive body, which is related to its total mass-equivalent M by where G is the gravitational constant, and rQ is a length scale corresponding to the electric charge Q of the mass where 1/(4πε0) is the Coulomb constant. Electromagnetic field tensor in Boyer–Lindquist form The electromagnetic potential in Boyer–Lindquist coordinates is while the Maxwell tensor is defined by In combination with the Christoffel symbols the second order equations of motion can be derived with where is the charge per mass of the testparticle. Kerr–Schild coordinates The Kerr–Newman metric can be expressed in the Kerr–Schild form, using a particular set of Cartesian coordinates, proposed by Kerr and Schild in 1965. The metric is as follows. Notice that k is a unit vector. Here M is the constant mass of the spinning object, Q is the constant charge of the spinning object, η is the Minkowski metric, and a = J/M is a constant rotational parameter of the spinning object. It is understood that the vector is directed along the positive z-axis, i.e. . The quantity r is not the radius, but rather is implicitly defined like this: Notice that the quantity r becomes the usual radius R when the rotational parameter a approaches zero. In this form of solution, units are selected so that the speed of light is unity (c = 1). In order to provide a complete solution of the Einstein–Maxwell equations, the Kerr–Newman solution not only includes a formula for the metric tensor, but also a formula for the electromagnetic potential: At large distances from the source (R ≫ a), these equations reduce to the Reissner–Nordström metric with: In the Kerr–Schild