The Kerr–Newman metric describes the spacetime geometry around a mass that is electrically charged and rotating. It is a vacuum solution that generalizes the Kerr metric (which describes an uncharged, rotating mass) by additionally taking into account the energy of an electromagnetic field, making it the most general asymptotically flat and stationary solution of the Einstein–Maxwell equations in general relativity. As an electrovacuum solution, it only includes those charges associated with the magnetic field; it does not include any free electric charges. The Kerr–Newman metric is primarily of theoretical interest. Astronomical objects have axes for rotation and for magnetic fields, but the metric is only valid for co-aligned axes. The model lacks description of infalling baryonic matter, light (null dusts) or dark matter, and thus provides an incomplete description of stellar mass black holes and active galactic nuclei. The solution however is of mathematical interest and provides a fairly simple cornerstone for further exploration.
History In December of 1963, Roy Kerr and Alfred Schild found the Kerr–Schild metrics that gave all Einstein spaces that are exact linear perturbations of Minkowski space. In early 1964, Kerr looked for all Einstein–Maxwell spaces with this same property. By February of 1964, the special case where the Kerr–Schild spaces were charged (including the Kerr–Newman solution) was known but the general case where the special directions were not geodesics of the underlying Minkowski space proved very difficult. The problem was given to George Debney to try to solve but was given up by March 1964. About this time Ezra T. Newman found the solution for charged Kerr by guesswork. In 1965, Ezra "Ted" Newman found the axisymmetric solution of Einstein's field equation for a black hole which is both rotating and electrically charged. This formula for the metric tensor is called the Kerr–Newman metric. It is a generalisation of the Kerr metric for an uncharged spinning point-mass, which had been discovered by Roy Kerr two years earlier.
Overview of the solution
Newman's result represents the simplest stationary, axisymmetric, asymptotically flat solution of Einstein's equations in the presence of an electromagnetic field in four dimensions. It is sometimes referred to as an "electrovacuum" solution of Einstein's equations. The solution contains a singularity in the shape of a ring. The multipole structure of the solution suggests that the solution represents the field of a ring of charge rotating about its axis of symmetry. Similarly the Kerr solution represents the field of a ring of mass. However, for this simple view to be mathematically correct, charge (or mass in the Kerr case) needs to be distributed around the singular ring of the solution to break the multivalued behavior. Any Kerr–Newman source has its rotation axis aligned with its magnetic axis. Thus, a Kerr–Newman source is different from commonly observed astronomical bodies, for which there is a substantial angle between the rotation axis and the magnetic moment. Specifically, neither the Sun, nor any of the planets in the Solar System has its magnetic field dipole aligned with its spin axis. Thus, while the Kerr solution describes the gravitational field of the Sun and planets, the magnetic fields necessarily arise by a different process.
Limiting cases The Kerr–Newman metric can be seen to reduce to other exact solutions in general relativity in limiting cases. It reduces to
the Kerr metric as the charge Q goes to zero; the Reissner–Nordström metric as the angular momentum J (or a = J/M) goes to zero; the Schwarzschild metric as both the charge Q and the angular momentum J (or a) are taken to zero; and Minkowski space if the mass M, the charge Q, and the rotational parameter a are all zero. The four related solutions may be summarized by the following table:
where Q is the body's electric charge and J is its spin angular momentum. Taking the gravitational constant G to be zero in the Kerr–Newman solution gives an electromagnetic field from a rotating charged disk with a boundary in a Minkowski space. The Kerr–Newman solution itself is a special case of more general exact solutions of the Einstein–Maxwell equations. The more general solutions include a cosmological constant, a Newman, Unti, Tamburino (NUT) parameter, and a magnetic charge.
Metric field The Kerr–Newman metric describes the 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:
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