Kerr–Newman metric

The Kerr–Newman metric describes the spacetime geometry around a mass which is electrically charged and rotating.

Because observed astronomical objects do not possess an appreciable net electric charge[citation needed] (the magnetic fields of stars arise through other processes), the Kerr–Newman metric is primarily of theoretical interest.

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.

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.

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.

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.

[5] 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.

[6] Specifically, neither the Sun, nor any of the planets in the Solar System have magnetic fields aligned with the spin axis.

[8] Like the Kerr metric for an uncharged rotating mass, the Kerr–Newman interior solution exists mathematically but is probably not representative of the actual metric of a physically realistic rotating black hole due to issues with the stability of the Cauchy horizon, due to mass inflation driven by infalling matter.

Although it represents a generalization of the Kerr metric, it is not considered as very important for astrophysical purposes, since one does not expect that realistic black holes have a significant electric charge (they are expected to have a minuscule positive charge, but only because the proton has a much larger momentum than the electron, and is thus more likely to overcome electrostatic repulsion and be carried by momentum across the horizon).

The Kerr–Newman metric defines a black hole with an event horizon only when the combined charge and angular momentum are sufficiently small:[9] An electron's angular momentum J and charge Q (suitably specified in geometrized units) both exceed its mass M, in which case the metric has no event horizon.

Thus, there can be no such thing as a black hole electron — only a naked spinning ring singularity.

[11] A 2009 paper by Russian theorist Alexander Burinskii considered an electron as a generalization of the previous models by Israel (1970)[12] and Lopez (1984),[13] which truncated the "negative" sheet of the Kerr-Newman metric, obtaining the source of the Kerr-Newman solution in the form of a relativistically rotating disk.

, replacing the singularity by a flat regular space-time, the so called "bubble".

Assuming that the Lopez bubble corresponds to a phase transition similar to the Higgs symmetry breaking mechanism, Burinskii showed that a gravity-created ring singularity forms by regularization the superconducting core of the electron model [14] and should be described by the supersymmetric Landau-Ginzburg field model of phase transition: By omitting Burinsky's intermediate work, we come to the recent new proposal: to consider the truncated by Israel and Lopez negative sheet of the KN solution as the sheet of the positron.

[15] This modification unites the KN solution with the model of QED, and shows the important role of the Wilson lines formed by frame-dragging of the vector potential.

As a result, the modified KN solution acquires a strong interaction with Kerr's gravity caused by the additional energy contribution of the electron-positron vacuum and creates the Kerr–Newman relativistic circular string of Compton size.

The Kerr–Newman metric can be seen to reduce to other exact solutions in general relativity in limiting cases.

It reduces to Alternately, if gravity is intended to be removed, Minkowski space arises if the gravitational constant G is zero, without taking the mass and charge to zero.

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.

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 ε0 is the vacuum permittivity.

The electromagnetic potential in Boyer–Lindquist coordinates is[17][18] while the Maxwell tensor is defined by In combination with the Christoffel symbols the second order equations of motion can be derived with where

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 static electric and magnetic fields are derived from the vector potential and the scalar potential like this: Using the Kerr–Newman formula for the four-potential in the Kerr–Schild form, in the limit of the mass going to zero, yields the following concise complex formula for the fields:[23] The quantity omega (

) 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 Émile Appell.

[24] The total mass-equivalent M, which contains the electric field-energy and the rotational energy, and the irreducible mass Mirr are related by[25][26] which can be inverted to obtain In order to electrically charge and/or spin a neutral and static body, energy has to be applied to the system.

If for example the rotational energy of a black hole is extracted via the Penrose processes,[27][28] the remaining mass–energy will always stay greater than or equal to Mirr.