The Hamiltonian for test particle motion in Kerr spacetime is separable in Boyer–Lindquist coordinates. Using Hamilton–Jacobi theory one can derive a fourth constant of the motion known as Carter's constant.
Note that in natural units , , and all have units of length. This line element describes the Kerr–Newman metric. Here, is to be interpreted as the mass of the black hole, as seen by an observer at infinity, is interpreted as the angular momentum, and the electric charge. These are all meant to be constant parameters, held fixed. The name of the discriminant arises because it appears as the discriminant of the quadratic equation bounding time-like motion of particles orbiting the black hole, i.e. defining the ergosphere.
The coordinate transformation from Boyer–Lindquist coordinates , , to Cartesian coordinates , , is given by
The contorsion tensor gives the difference between a connection with torsion, and a corresponding connection without torsion. By convention, Riemann manifolds are always specified with torsion-free geometries; torsion is often used to specify equivalent, flat geometries.
The spin connection is useful, because it provides an intermediate way-point for computing the curvature two-form:
It is also the most suitable form for describing the coupling to spinor fields, and opens the door to the twistor formalism.
All six components of the spin connection are non-vanishing. These are:
The Riemann tensor written out in full is quite verbose; it can be found in Frè. The Ricci tensor takes the diagonal form:
Notice the location of the minus-one entry: this comes entirely from the electromagnetic contribution. Namely, when the electromagnetic stress tensor has only two non-vanishing components: and , then the corresponding energy–momentum tensor takes the form