The KerrâÂÂNewman 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 EinsteinâÂÂMaxwell 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 KerrâÂÂNewman 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.
In December of 1963, Roy Kerr and Alfred Schild found the KerrâÂÂSchild metrics that gave all Einstein spaces that are exact linear perturbations of Minkowski space. In early 1964, Kerr looked for all EinsteinâÂÂMaxwell spaces with this same property. By February of 1964, the special case where the KerrâÂÂSchild spaces were charged (including the KerrâÂÂNewman 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 KerrâÂÂNewman 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.
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 KerrâÂÂNewman source has its rotation axis aligned with its magnetic axis. Thus, a KerrâÂÂNewman 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.
The KerrâÂÂNewman metric can be seen to reduce to other exact solutions in general relativity in limiting cases. It reduces to
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 to be zero in the KerrâÂÂNewman solution gives an electromagnetic field from a rotating charged disk with a boundary in a Minkowski space. The KerrâÂÂNewman solution itself is a special case of more general exact solutions of the EinsteinâÂÂMaxwell equations. The more general solutions include a cosmological constant, a Newman, Unti, Tamburino (NUT) parameter, and a magnetic charge.
The KerrâÂÂNewman 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: BoyerâÂÂLindquist coordinates, and KerrâÂÂSchild 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.
One way to express this metric is by writing down its line element in a particular set of spherical coordinates, also called BoyerâÂÂLindquist coordinates:
where are standard spherical coordinates, and the length scales:
have been introduced for brevity. Here r<sub>s</sub> 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 r<sub>Q</sub> is a length scale corresponding to the electric charge Q of the mass
where õ<sub>0</sub> is the vacuum permittivity.
The electromagnetic potential in BoyerâÂÂLindquist 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-to-mass ratio of the test particle.
The KerrâÂÂNewman metric can be expressed in the KerrâÂÂSchild form, using a particular set of Cartesian coordinates, proposed by Kerr and Schild in 1965. The metric is as follows.
Here, k is a unit vector. Here M is the constant mass of the spinning object, Q is the constant charge of the spinning object, ÷ 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 by the relation
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 (). In order to provide a complete solution of the EinsteinâÂÂMaxwell equations, the KerrâÂÂNewman solution not only includes a formula for the metric tensor, but also a formula for the electromagnetic potential:
At large distances from the source (), these equations reduce to the ReissnerâÂÂNordström metric with:
In the KerrâÂÂSchild form of the KerrâÂÂNewman metric, the determinant of the metric tensor is everywhere equal to negative one, even near the source.
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 KerrâÂÂNewman formula for the four-potential in the KerrâÂÂSchild form, in the limit of the mass going to zero, yields the following concise complex formula for the fields:
The quantity omega (é) 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 ÃÂmile Appell.
The total mass-equivalent M, which contains the electric field-energy and the rotational energy, and the irreducible mass M<sub>irr</sub> are related by
which can be solved 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 massâÂÂenergy equivalence, this energy also has a mass-equivalent; therefore M is always higher than M<sub>irr</sub>. If for example the rotational energy of a black hole is extracted via the Penrose processes, the remaining massâÂÂenergy will always stay greater than or equal to M<sub>irr</sub>.
Setting to 0 and solving for gives the inner and outer event horizon, which is located at the BoyerâÂÂLindquist coordinate
Repeating this step with gives the inner and outer ergosurface
The region between the event horizon and the ergosurface is called the ergosphere. Within the ergosphere all local light cones are tilted in the direction of rotation.
For brevity, we further use nondimensionalized quantities normalized against , , and , where reduces to and to , and the equations of motion for a test particle 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 test particle's angular momentum, and the orbital inclination angle.
and
with and for particles 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 2ÃÂ), and
the gravitational time dilation component. The local radial escape velocity for a neutral particle is therefore
For the set of parameters the solution is called "extremal"; in the super-extremal regime, , there is no event horizon and the interior singularity is observable as a naked singularity. That is, the KerrâÂÂNewman metric defines a black hole with an event horizon only when the combined charge and angular momentum are sufficiently small compared to the mass.
An electron's rotational parameter a and charge length scale r<sub>Q</sub> both far exceed its Schwarzschild radius r<sub>s</sub>, which implies that there is no event horizon, so a "black hole electron" would necessarily be a naked spinning ring singularity. Such a metric has several seemingly unphysical properties, such as the ring's violation of the cosmic censorship hypothesis and appearance of causality-violating closed timelike curves in the immediate vicinity of the ring.
In an analysis of the KerrâÂÂNewman solution for a rotating charged body, Brandon Carter remarked that the solutions predict a gyromagnetic ratio of 2, equal to that predicted by the relativistic quantum-mechanical Dirac equation, and close to that experimentally observed for the electron. This led to the application of the KerrâÂÂNewman metric to create non-quantum models of the electron based on general relativity. The electron and all similar particles have large charge-to-mass ratios, which have KerrâÂÂNewman parameters in the regime where there is no event horizon but there is a naked singularity. For example, for the electron the charge-to-mass ratio and the spin effects are both orders of magnitude larger than this limit.
If the KerrâÂÂNewman potential is considered as a classical model for an electron (an electron black hole), it predicts an electron having not only a magnetic dipole moment, but also other multipole moments, such as an electric quadrupole moment. However, no experimental measurements of such a quadrupole moment have been reported.