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The Kerr–Newman metric describes the spacetime geometry around a mass which is electrically charged and rotating. It is a
vacuum solution In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or n ...
which generalizes the
Kerr metric The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...
(which describes an uncharged, rotating mass) by additionally taking into account the energy of an
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
, making it the most general asymptotically flat and stationary solution of the Einstein–Maxwell equations in
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. As an
electrovacuum solution In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the ...
, it only includes those charges associated with the magnetic field; it does not include any free electric charges. Because observed astronomical objects do not possess an appreciable net
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
(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 In particle physics, a baryon is a type of composite subatomic particle that contains an odd number of valence quarks, conventionally three. Protons and neutrons are examples of baryons; because baryons are composed of quarks, they belong to ...
, light ( null dusts) or
dark matter In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
, and thus provides an incomplete description of stellar mass black holes and
active galactic nuclei An active galactic nucleus (AGN) is a compact region at the center of a galaxy that emits a significant amount of energy across the electromagnetic spectrum, with characteristics indicating that this luminosity is not produced by the stars. Such e ...
. The solution however is of mathematical interest and provides a fairly simple cornerstone for further exploration. The Kerr–Newman solution is a special case of more general exact solutions of the Einstein–Maxwell equations with non-zero
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is a coefficient that Albert Einstein initially added to his field equations of general rel ...
.


History

In December of 1963, Roy Kerr and
Alfred Schild Alfred Schild (September 7, 1921 – May 24, 1977) was a leading Austrian American physicist, well known for his contributions to the Golden age of general relativity (1960–1975). Biography Schild was born in Istanbul on September 7, 1921. His ...
found the Kerr–Schild metrics that gave all Einstein spaces that are exact linear perturbations of
Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...
. 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 In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
g_ \! is called the Kerr–Newman metric. It is a generalisation of the
Kerr metric The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...
for an uncharged spinning point-mass, which had been discovered by Roy Kerr two years earlier. Four related solutions may be summarized by the following table: : where ''Q'' represents the body's
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
and ''J'' represents its spin
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
.


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 An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
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. 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 In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
. Specifically, neither the
Sun The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...
, nor any of the
planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s in the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
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. An electron quadrupole moment has not yet been experimentally detected; it appears to be zero. 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. Like the
Kerr metric The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...
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 A rotating black hole is a black hole that possesses angular momentum. In particular, it rotates about one of its axes of symmetry. All currently known celestial objects, including planets, stars (Sun), galaxies, and black holes, spin about one ...
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 hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s have a significant
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
(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: :J^2/M^2 + Q^2 \leq M^2. 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. Such a metric has several seemingly unphysical properties, such as the ring's violation of the
cosmic censorship hypothesis The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity. Singularities that arise in the solutions of Einstein's equations are typical ...
, and also appearance of causality-violating closed timelike curves in the immediate vicinity of the ring. A 2009 paper by Russian theorist Alexander Burinskii considered an electron as a generalization of the previous models by Israel (1970) and Lopez (1984), 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 : r= r_e=e^2/2M , 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 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. 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.


Limiting cases

The Kerr–Newman metric can be seen to reduce to other
exact solutions in general relativity In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying approximations of the equations, though the starting point for that derivation may be a ...
in limiting cases. It reduces to * the
Kerr metric The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...
as the charge ''Q'' goes to zero; * the
Reissner–Nordström metric In physics and astronomy, the Reissner–Nordström metric is a Static spacetime, static solution to the Einstein–Maxwell equations, Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, s ...
as the angular momentum ''J'' (or ''a'' =  ) goes to zero; * the
Schwarzschild metric In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
as both the charge ''Q'' and the angular momentum ''J'' (or ''a'') are taken to zero; and *
Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...
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.


The metric

The Kerr–Newman metric describes the geometry of
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
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 geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
in a particular set of
spherical coordinates In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point ...
, also called
Boyer–Lindquist coordinates In the mathematical description of general relativity, the Boyer–Lindquist coordinates are a generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole. Th ...
: :c^ d\tau^ = -\left(\frac + d\theta^2 \right) \rho^2 + \left(c \, dt - a \sin^2 \theta \, d\phi \right)^2 \frac - \left(\left(r^2 + a^2 \right) d\phi - a c\, dt \right)^2 \frac, where the coordinates are standard
spherical coordinate system In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point ...
, and the length scales: :a = \frac\,, :\rho^=r^2+a^2\cos^2\theta\,, :\Delta=r^2-r_\textr+a^2+r_Q^2\,, have been introduced for brevity. Here ''r''s is the Schwarzschild radius of the massive body, which is related to its total mass-equivalent ''M'' by :r_ = \frac, where ''G'' is the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
, and ''r''''Q'' is a length scale corresponding to the
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
''Q'' of the mass :r_^ = \frac, where ''ε''0 is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
.


Electromagnetic field tensor in Boyer–Lindquist form

The electromagnetic potential in Boyer–Lindquist coordinates is :A_=\left( \frac,0,0,-\frac \right) while the Maxwell tensor is defined by :F_ = \frac - \frac \ \to \ F^=g^ \ g^ \ F_ In combination with the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metri ...
the second order
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
can be derived with :, where q is the charge per mass of the test particle.


Kerr–Schild coordinates

The Kerr–Newman metric can be expressed in the Kerr–Schild form, using a particular set of
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
, proposed by
Kerr Kerr may refer to: People *Kerr (surname) *Kerr (given name) Places ;United States *Kerr Township, Champaign County, Illinois *Kerr, Montana, A US census-designated place *Kerr, Ohio, an unincorporated community *Kerr County, Texas Kerr Co ...
and Schild in 1965. The metric is as follows. See equations (7.10), (7.11) and (7.14). :g_ = \eta_ + fk_k_ \! :f = \frac\left Mr - Q^2 \right/math> :\mathbf = ( k_ ,k_ ,k_ ) = \left( \frac , \frac, \frac \right) :k_ = 1. \! Notice that k is a
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
. Here ''M'' is the constant mass of the spinning object, ''Q'' is the constant charge of the spinning object, ''η'' is the
Minkowski metric In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model ...
, and ''a'' = ''J''/''M'' is a constant rotational parameter of the spinning object. It is understood that the vector \vec is directed along the positive z-axis, i.e. \vec = a \hat. The quantity ''r'' is not the radius, but rather is implicitly defined by the relation :1 = \frac + \frac. Notice that the quantity ''r'' becomes the usual radius ''R'' :r \to R = \sqrt 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: :A_ = \frack_ At large distances from the source (''R'' ≫ ''a''), these equations reduce to the
Reissner–Nordström metric In physics and astronomy, the Reissner–Nordström metric is a Static spacetime, static solution to the Einstein–Maxwell equations, Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, s ...
with: :A_ = \frack_ 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. See page 485 regarding determinant of metric tensor. See page 325 regarding generalizations.


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. :A_ = \left(-\phi, A_x, A_y, A_z \right) \, The static electric and magnetic fields are derived from the vector potential and the scalar potential like this: :\vec = - \vec \phi \, :\vec = \vec \times \vec \, 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: :\vec + i\vec = -\vec\Omega\, :\Omega = \frac \, The quantity omega (\Omega) in this last equation is similar to the
Coulomb potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
, 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 :''M. P. Appell is the same person: it stands for Monsieur Paul Appell''. Paul Émile Appell (27 September 1855 in Strasbourg – 24 October 1930 in Paris) was a French mathematician and Rector of the University of Paris. Appell polynomials and ...
.


Irreducible mass

The total mass-equivalent ''M'', which contains the electric field-energy and the
rotational energy Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the ob ...
, and the irreducible mass ''M''irr are related byEq. 57 in : M_ = \frac\sqrt which can be inverted to obtain : M = \frac 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 In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame. The two differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist Albert Einstei ...
, this energy also has a mass-equivalent; therefore ''M'' is always higher than ''M''irr. If for example the rotational energy of a black hole is extracted via the Penrose processes, the remaining mass–energy will always stay greater than or equal to ''M''irr.


Important surfaces

Setting 1 / g_ to 0 and solving for r gives the inner and outer
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive c ...
, which is located at the Boyer–Lindquist coordinate :r_^ = \frac \pm \sqrt. Repeating this step with g_ gives the inner and outer ergosphere :r_^ = \frac \pm \sqrt.


Equations of motion

For brevity, we further use nondimensionalized quantities normalized against G, M, c and 4\pi\epsilon_0, where a reduces to Jc/G/M^2 and Q to Q/(M\sqrt), and the equations of motion for a test particle of charge q become :\dot t \Delta \rho^2 = \csc ^2 \theta \ ( (a \ \Delta \sin ^2 \theta -a \ (a^2+r^2) \sin ^2 \theta )-q \ Q \ r \ (a^2+r^2) \sin ^2 \theta +E ((a^2+r^2)^2 \sin ^2 \theta -a^2 \Delta \sin ^4 \theta )) :\dot r \rho^2= \pm \left(((r^2+a^2) \ E - a \ L_z - q \ Q \ r)^2-\Delta \ (C+r^2)\right)^ :\dot \theta \rho^2 = \pm \left(C-(a \cos \theta)^2-(a \ \sin^2 \theta \ E-L_z)^2/\sin^2 \theta\right)^ :\dot \phi \rho^2 \ \Delta \ \sin^2\theta= E \ (a \ \sin^2 \theta \ (r^2+a^2)-a \ \sin^2 \theta \ \Delta)+L_z \ (\Delta-a^2 \ \sin^2 \theta)-q \ Q \ r \ a \ \sin^2 \theta with E for the total energy and L_z for the axial angular momentum. C is the Carter constant: :C = p_^ + \cos^\theta \left( a^(\mu^2 - E^) + \frac\right) = a^2 \ (\mu^2-E^2) \ \sin^2 \delta + L_z^2 \ \tan^2 \delta = , where p_ = \dot \theta \ \rho^2 is the poloidial component of the test particle's angular momentum, and \delta the orbital inclination angle. :L_z = p_=-g_ -g_ - q \ A_ = \frac+\frac = and :E = -p_t =g_ +g_ + q \ A_ = \sqrt + \Omega \ L_z +\frac = with \mu^2=0 and \mu^2=1 for particles are also conserved quantities. :\Omega = -\frac = \frac is the frame dragging induced angular velocity. The shorthand term \chi is defined by :\chi = \left(a ^2+r^2\right)^2-a ^2 \ \sin ^2 \theta \ \Delta. The relation between the coordinate derivatives \dot r, \ \dot \theta, \ \dot \phi and the local 3-velocity v is :v^ = \dot r \ \sqrt for the radial, :v^ = \dot \theta \ \sqrt for the poloidial, :v^ = \sqrt \left(L_z \ \Sigma - a \ q \ Q \ r \left( 1-\mu^2 v^2 \right) \sin^2 \theta \right)\cdot(\bar \ \Sigma )^ for the axial and :v = \frac = \sqrt for the total local velocity, where :\bar R = \sqrt = \sqrt \ \sin \theta is the axial radius of gyration (local circumference divided by 2π), and :\varsigma = \sqrt = \frac the gravitational time dilation component. The local radial escape velocity for a neutral particle is therefore :v_=\frac .


References


Bibliography

*


External links

* co-authored by Ezra T. Newman himself
SR Made Easy, chapter 11: Charged and Rotating Black Holes and Their Thermodynamics
{{DEFAULTSORT:Kerr-Newman metric Exact solutions in general relativity Equations Metric tensors Gravitational singularities