In this paper we define a generalized Bondi–Sachs (GBS) coordinate system. We utilize the null geodesics with zero angular momentum in the Kerr spacetime to transform the metric into GBS form. We write down the Kerr metric components explicitly in these new coordinates. The flat (M = 0) subcase of the Kerr spacetime can be shown to be coordinatized by null geodesics emanating from a ring. The curved (M ≠ 0) subcase with a = 0 is the Schwarzschild spacetime in outgoing Eddington–Finkelstein coordinates. We analyse the important physical properties of the Kerr spacetime in the new coordinates. We conclude with the suggestion that a radiating generalization of the Kerr spacetime is most likely to be found in GBS coordinates.

The Curzon metric has an invariantly defined surface, given by the vanishing of the cubic invariant of the Weyl tensor. On a spacelike slice, this surface has the topology of a 2-sphere, and surrounds the singularity of the metric. In Weyl coordinates, this surface is defined by R=m where R= square root ( rho 2+z2). The two points where the z-axis, the axis of symmetry, cuts this surface, z=m and -m, have the interesting property that, at both of them, the Riemann tensor vanishes. The Weyl tensor is of Petrov type D at all non-singular points of the z-axis, rho =0, except at the two points where it is zero. Off the axis, the Weyl tensor is of type I(M+), and the metric asymptotically tends to flat spacetime away from the source. The principal null directions of the Weyl tensor are shown to be everywhere independent of the angular basis vector delta / delta phi , and their projections into a t=constant, phi =constant plane are presented graphically.

The conservation and impermeability conditions are reformulated utilising differential forms and generalised to spacetime. The thermodynamic and electromagnetic potential vorticity scalar fields are defined for relativistic magnetohydrodynamics and their evolution equations are derived.