The Kerr–Newman metric is the most general
asymptotically flat and
stationary solution of the
Einstein–Maxwell equations in
general relativity that describes the spacetime geometry in the region surrounding an electrically charged and rotating mass. It generalizes the
Kerr metric by taking into account the field energy of an
electromagnetic field, in addition to describing rotation. It is one of a large number of various different
electrovacuum solutions; that is, it is a solution to the Einstein–Maxwell equations that account for the field energy of an
electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed
vacuum solutions.
This solution has not been especially useful for describing astrophysical phenomena because observed astronomical objects do not possess an appreciable net
electric charge,[citation needed] and the magnetic fields of stars arise through other processes. As a model of realistic black holes, it omits any description of infalling
baryonic matter, light (
null dusts) or
dark matter, and thus provides at best an incomplete description of
stellar mass black holes and
active galactic nuclei. The solution is of theoretical and mathematical interest as it does provide a fairly simple cornerstone for further exploration.[citation needed]
The Kerr–Newman solution is a special case of more general exact solutions of the Einstein–Maxwell equations with non-zero
cosmological constant.[1]
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.[2][3] 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.[4]
Four related solutions may be summarized by the following table:
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.
Any Kerr–Newman source has its rotation axis aligned with its magnetic axis.[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. Thus, while the Kerr solution describes the gravitational field of the Sun and planets, the magnetic fields arise by a different process.
If the Kerr–Newman potential is considered as a model for a classical electron, it predicts an electron having not just a magnetic dipole moment, but also other multipole moments, such as an electric quadrupole moment.[7] An electron quadrupole moment has not yet been experimentally detected; it appears to be zero.[7]
In the G = 0 limit, the electromagnetic fields are those of a charged rotating disk inside a ring where the fields are infinite. The total field energy for this disk is infinite, and so this G = 0 limit does not solve the problem of infinite
self-energy.[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]
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. Lopez's truncation regularized the Kerr-Newman metric by a cutoff at :, 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
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.
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. In this case, the electric and magnetic fields are more complicated than simply the
fields of a charged magnetic dipole; the zero-gravity limit is not trivial.[citation needed]
The metric
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.
Notice that k is a
unit vector. Here M is the constant mass of the spinning object, Q is the constant charge of the spinning object, η 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
Notice that 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 (c = 1). In order to provide a complete solution of the
Einstein–Maxwell equations, the Kerr–Newman solution not only includes a formula for the metric tensor, but also a formula for the electromagnetic potential:[19][22]
At large distances from the source (R ≫ a), these equations reduce to the
Reissner–Nordström metric with:
In the Kerr–Schild form of the Kerr–Newman metric, the determinant of the metric tensor is everywhere equal to negative one, even near the source.[1]
Electromagnetic fields in Kerr–Schild form
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 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]
In order to electrically charge and/or spin a neutral and static body, energy has to be applied to the system. Due to the
mass–energy equivalence, this energy also has a mass-equivalent; therefore M is always higher than Mirr. 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.
Important surfaces
Setting to 0 and solving for gives the inner and outer
event horizon, which is located at the Boyer–Lindquist coordinate
Repeating this step with gives the inner and outer
ergosphere
Equations of motion
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[29][30]
with for the total energy and for the axial angular momentum. is the
Carter constant:
where is the poloidial component of the testparticle'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 2π), and
the gravitational time dilation component. The local radial escape velocity for a neutral particle is therefore
References
^
abStephani, Hans et al. Exact Solutions of Einstein's Field Equations (Cambridge University Press 2003). See
page 485 regarding determinant of metric tensor. See
page 325 regarding generalizations.
^
abDebney, G. C.; Kerr, R. P.; Schild, A. (1969). "Solutions of the Einstein and Einstein‐Maxwell Equations". Journal of Mathematical Physics. 10 (10): 1842–1854.
Bibcode:
1969JMP....10.1842D.
doi:
10.1063/1.1664769.. Especially see equations (7.10), (7.11) and (7.14).
^Berman, Marcelo. “Energy of Black Holes and Hawking’s Universe” in Trends in Black Hole Research, page 148 (Kreitler ed., Nova Publishers 2006).
^Burinskii, A.
“Kerr Geometry Beyond the Quantum Theory” in Beyond the Quantum, page 321 (Theo Nieuwenhuizen ed., World Scientific 2007). The formula for the vector potential of Burinskii differs from that of Debney et al. merely by a gradient which does not affect the fields.
^Bhat, Manjiri; Dhurandhar, Sanjeev; Dadhich, Naresh (1985). "Energetics of the Kerr–Newman black hole by the penrose process". Journal of Astrophysics and Astronomy. 6 (2): 85–100.
Bibcode:
1985JApA....6...85B.
doi:
10.1007/BF02715080.
S2CID53513572.