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TheKerr–Newman metric describes thespacetime geometry around a mass that is electrically charged and rotating. It is avacuum solution that generalizes theKerr metric (which describes an uncharged, rotating mass) by additionally taking into account the energy of anelectromagnetic field, making it the most generalasymptotically flat andstationary solution of theEinstein–Maxwell equations ingeneral relativity. As anelectrovacuum 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 infallingbaryonic matter, light (null dusts) ordark matter, and thus provides an incomplete description ofstellar mass black holes andactive galactic nuclei. The solution however is of mathematical interest and provides a fairly simple cornerstone for further exploration.

In December of 1963,Roy Kerr andAlfred Schild found the Kerr–Schild metrics that gave allEinstein spaces that are exact linear perturbations ofMinkowski 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.^[1][2] This formula for themetric tensor is called the Kerr–Newman metric. It is a generalisation of theKerr metric for an uncharged spinning point-mass, which had been discovered by Roy Kerr two years earlier.^[3]

Newman's result represents the simpleststationary,axisymmetric, asymptotically flat solution ofEinstein's equations in the presence of anelectromagnetic 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.^[1]

Any Kerr–Newman source has its rotation axis aligned with its magnetic axis.^[4] Thus, a Kerr–Newman source is different from commonly observed astronomical bodies, for which there is a substantial angle between the rotation axis and themagnetic moment.^[5] Specifically, neither theSun, nor any of theplanets in theSolar 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 Kerr–Newman metric can be seen to reduce to otherexact solutions in general relativity in limiting cases. It reduces to^[6]: 61

• theKerr metric as the chargeQ goes to zero;
• theReissner–Nordström metric as the angular momentumJ (ora =J/M) goes to zero;
• theSchwarzschild metric as both the chargeQ and the angular momentumJ (ora) are taken to zero; and
• Minkowski space if the massM, the chargeQ, and the rotational parametera are all zero.

The four related solutions may be summarized by the following table:

	Non-rotating (J = 0)	Rotating (anyJ)
Uncharged (Q = 0)	Schwarzschild	Kerr
Charged (anyQ)	Reissner–Nordström	Kerr–Newman

whereQ represents the body'selectric charge andJ represents its spinangular momentum.

Taking the gravitational constantG to be zero in the Kerr-Newman solution gives an electromagnetic field from a rotating charged disk with a boundary in aMinkowski space.^[7][8] The Kerr–Newman solution itself is a special case of more general exact solutions of the Einstein–Maxwell equations. The more general solutions include acosmological constant, aNewman, Unti, Tamburino (NUT) parameter, and amagnetic charge.^[9]: 324

The Kerr–Newman metric describes the geometry ofspacetime for a rotating charged black hole with massM, chargeQ and angular momentumJ. The formula for this metric depends upon what coordinates orcoordinate 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.

One way to express this metric is by writing down itsline element in a particular set ofspherical coordinates,^[10] also calledBoyer–Lindquist coordinates:

${\displaystyle c^{2}d{\tau }^2=-\left(\frac{d{r}^2}{\mathrm{\Delta }}+d{\theta }^2\right){\rho }^2+{\left(cdt-a{\sin }^2\theta d\varphi \right)}^2\frac{\mathrm{\Delta }}{{\rho }^2}-{\left(\left(r^2+a^2\right)d\varphi -acdt\right)}^2\frac{{\sin }^2\theta }{{\rho }^2},}$

where(r,θ,ϕ) are standardspherical coordinates, and the length scales:

${\displaystyle a=\frac{J}{Mc},}$

${\displaystyle {\rho }^2=r^2+a^2{\cos }^2\theta ,}$

${\displaystyle \mathrm{\Delta }=r^2-r_{\text{s}}r+a^2+r_Q^2,}$

have been introduced for brevity. Herer_s is theSchwarzschild radius of the massive body, which is related to its total mass-equivalentM by

${\displaystyle r_{\text{s}}=\frac{2GM}{c^2},}$

whereG is thegravitational constant, andr_Q is a length scale corresponding to theelectric chargeQ of the mass

${\displaystyle r_Q^2=\frac{Q^{2}G}{4\pi {ϵ}_0{c}^4},}$

whereε₀ is thevacuum permittivity.

The electromagnetic potential in Boyer–Lindquist coordinates is^[11][12]

${\displaystyle A_{\mu }=\left(\frac{r{r}_Q}{{\rho }^2},0,0,-\frac{ar{r}_Q{\sin }^2\theta }{{\rho }^2}\right)}$

while the Maxwell tensor is defined by

${\displaystyle F_{\mu \nu }=\frac{\mathrm{\partial }{A}_{\nu }}{\mathrm{\partial }{x}^{\mu }}-\frac{\mathrm{\partial }{A}_{\mu }}{\mathrm{\partial }{x}^{\nu }}\to F^{\mu \nu }=g^{\mu \sigma }{g}^{\nu \kappa }{F}_{\sigma \kappa }}$

In combination with theChristoffel symbols the second orderequations of motion can be derived with

${\displaystyle {\ddot{x}}^i=-{\mathrm{\Gamma }}_{jk}^i{\dot{x}}^j{\dot{x}}^k+q{F}^{ik}{\dot{x}}^j{g}_{jk},}$

where${\displaystyle q}$ is the charge-to-mass ratio of the test particle.

The Kerr–Newman metric can be expressed in theKerr–Schild form, using a particular set ofCartesian coordinates, proposed byKerr andSchild in 1965. The metric is as follows.^[13][14][15]

${\displaystyle g_{\mu \nu }={\eta }_{\mu \nu }+f{k}_{\mu }{k}_{\nu }}$

${\displaystyle f=\frac{G{r}^2}{r^4+a^2{z}^2}\left[2Mr-Q^2\right]}$

${\displaystyle \mathbf{k}=(k_x,k_y,k_z)=\left(\frac{rx+ay}{r^2+a^2},\frac{ry-ax}{r^2+a^2},\frac{z}{r}\right)}$

${\displaystyle k_0=1.}$

Here,k is aunit vector. HereM is the constant mass of the spinning object,Q is the constant charge of the spinning object,η is theMinkowski metric, anda = J/M is a constant rotational parameter of the spinning object. It is understood that the vector${\displaystyle \overrightarrow{a}}$ is directed along the positive z-axis, i.e.${\displaystyle \overrightarrow{a}=a\hat{z}}$. The quantityr is not the radius, but rather is implicitly defined by the relation

${\displaystyle 1=\frac{x^2+y^2}{r^2+a^2}+\frac{z^2}{r^2}.}$

The quantityr becomes the usual radiusR

${\displaystyle r\to R=\sqrt{x^2+y^2+z^2}}$

when the rotational parametera 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 theEinstein–Maxwell equations, the Kerr–Newman solution not only includes a formula for the metric tensor, but also a formula for the electromagnetic potential:^[13][16]

${\displaystyle A_{\mu }=\frac{Q{r}^3}{r^4+a^2{z}^2}{k}_{\mu }}$

At large distances from the source (R ≫a), these equations reduce to theReissner–Nordström metric with:

${\displaystyle A_{\mu }=\frac{Q}{R}{k}_{\mu }}$

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.^[9]

The electric and magnetic fields can be obtained in the usual way by differentiating the four-potential to obtain theelectromagnetic field strength tensor. It will be convenient to switch over to three-dimensional vector notation.

${\displaystyle A_{\mu }=\left(-\varphi ,A_x,A_y,A_z\right)}$

The static electric and magnetic fields are derived from the vector potential and the scalar potential like this:

${\displaystyle \overrightarrow{E}=-\overrightarrow{\mathrm{\nabla }}\varphi }$

${\displaystyle \overrightarrow{B}=\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{A}}$

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:^[17]

${\displaystyle \overrightarrow{E}+i\overrightarrow{B}=-\overrightarrow{\mathrm{\nabla }}\mathrm{\Omega }}$

${\displaystyle \mathrm{\Omega }=\frac{Q}{\sqrt{(\overrightarrow{R}-i\overrightarrow{a}{)}^2}}}$

The quantity omega (Ω) in this last equation is similar to theCoulomb 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 mathematicianPaul Émile Appell.^[18]

The total mass-equivalentM, which contains theelectric field-energy and therotational energy, and the irreducible massM_irr are related by^[19][20]

${\displaystyle M_{\mathrm{i}\mathrm{r}\mathrm{r}}=\frac{1}{2}\sqrt{2M^2-r_Q^2{c}^4/G^2+2M\sqrt{M^2-(r_Q^2+a^2)c^4/G^2}}}$

which can be inverted to obtain

${\displaystyle M=\frac{4M_{\mathrm{i}\mathrm{r}\mathrm{r}}^2+r_Q^2{c}^4/G^2}{2\sqrt{4M_{\mathrm{i}\mathrm{r}\mathrm{r}}^2-a^2{c}^4/G^2}}}$

In order to electrically charge and/or spin a neutral and static body, energy has to be applied to the system. Due to themass–energy equivalence, this energy also has a mass-equivalent; thereforeM is always higher thanM_irr. If for example the rotational energy of a black hole is extracted via thePenrose processes,^[21][22] the remaining mass–energy will always stay greater than or equal toM_irr.

Setting${\displaystyle 1/g_{rr}}$ to 0 and solving for${\displaystyle r}$ gives the inner and outerevent horizon, which is located at the Boyer–Lindquist coordinate

${\displaystyle r_{\text{H}}^{\pm }=\frac{r_{\mathrm{s}}}{2}\pm \sqrt{\frac{r_{\mathrm{s}}^2}{4}-a^2-r_Q^2}.}$

Repeating this step with${\displaystyle g_{tt}}$ gives the inner and outerergosurface

${\displaystyle r_{\text{E}}^{\pm }=\frac{r_{\mathrm{s}}}{2}\pm \sqrt{\frac{r_{\mathrm{s}}^2}{4}-a^2{\cos }^2\theta -r_Q^2},}$

or innatural units,

${\displaystyle r_{\text{E}}^{\pm }=M\pm \sqrt{M^2-a^2{\cos }^2\theta -Q^2}.}$

The region between the event horizon and the ergosurface is called theergosphere. Within the ergosphere all local light cones are tilted in the direction of rotation.^[8]: 18

For brevity, we further usenondimensionalized quantities normalized against${\displaystyle G}$,${\displaystyle M}$,${\displaystyle c}$ and${\displaystyle 4\pi {ϵ}_0}$, where${\displaystyle a}$ reduces to${\displaystyle Jc/G/M^2}$ and${\displaystyle Q}$ to$Q/(M\sqrt{4\pi {ϵ}_{0}G})$, and the equations of motion for a test particle of charge${\displaystyle q}$ become^[23][24]

${\displaystyle \dot{t}\mathrm{\Delta }{\rho }^2={\csc }^2\theta (L_z(a\mathrm{\Delta }{\sin }^2\theta -a(a^2+r^2){\sin }^2\theta )-qQr(a^2+r^2){\sin }^2\theta +E((a^2+r^2{)}^2{\sin }^2\theta -a^2\mathrm{\Delta }{\sin }^4\theta ))}$

${\displaystyle \dot{r}{\rho }^2=\pm {\left(((r^2+a^2)E-a{L}_z-qQr{)}^2-\mathrm{\Delta }(C+r^2)\right)}^{1/2}}$

${\displaystyle \dot{\theta }{\rho }^2=\pm {\left(C-(a\cos \theta {)}^2-(a{\sin }^2\theta E-L_z{)}^2/{\sin }^2\theta \right)}^{1/2}}$

${\displaystyle \dot{\varphi }{\rho }^2\mathrm{\Delta }{\sin }^2\theta =E(a{\sin }^2\theta (r^2+a^2)-a{\sin }^2\theta \mathrm{\Delta })+L_z(\mathrm{\Delta }-a^2{\sin }^2\theta )-qQra{\sin }^2\theta }$

with${\displaystyle E}$ for the total energy and${\displaystyle L_z}$ for the axial angular momentum.${\displaystyle C}$ is theCarter constant:

${\displaystyle C=p_{\theta }^2+{\cos }^2\theta \left(a^2({\mu }^2-E^2)+\frac{L_z^2}{{\sin }^2\theta }\right)=a^2({\mu }^2-E^2){\sin }^2\delta +L_z^2{\tan }^2\delta =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t},}$

where${\displaystyle p_{\theta }=\dot{\theta }{\rho }^2}$ is the poloidial component of the test particle's angular momentum, and${\displaystyle \delta }$ the orbital inclination angle.

${\displaystyle L_z=p_{\varphi }=-g_{\varphi \varphi }\dot{\varphi }-g_{t\varphi }\dot{t}-q{A}_{\varphi }=\frac{v^{\varphi }\overline{R}}{\sqrt{1-{\mu }^2{v}^2}}+\frac{(1-{\mu }^2{v}^2)ar\mho q{\sin }^2\theta }{\mathrm{\Sigma }}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.}$

and

${\displaystyle E=-p_t=g_{tt}\dot{t}+g_{t\varphi }\dot{\varphi }+q{A}_t=\sqrt{\frac{\mathrm{\Delta }{\rho }^2}{(1-{\mu }^2{v}^2)\chi }}+\mathrm{\Omega }{L}_z+\frac{\mho qr}{\mathrm{\Sigma }}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.}$

with${\displaystyle {\mu }^2=0}$ and${\displaystyle {\mu }^2=1}$ for particles are also conserved quantities.

${\displaystyle \mathrm{\Omega }=-\frac{g_{t\varphi }}{g_{\varphi \varphi }}=\frac{a\left(2r-Q^2\right)}{\chi }}$

is the frame dragging induced angular velocity. The shorthand term${\displaystyle \chi }$ is defined by

${\displaystyle \chi ={\left(a^2+r^2\right)}^2-a^2{\sin }^2\theta \mathrm{\Delta }.}$

The relation between the coordinate derivatives${\displaystyle \dot{r},\dot{\theta },\dot{\varphi }}$ and the local 3-velocity${\displaystyle v}$ is

${\displaystyle v^r=\dot{r}\sqrt{\frac{{\rho }^2(1-{\mu }^2{v}^2)}{\mathrm{\Delta }}}}$

for the radial,

${\displaystyle v^{\theta }=\dot{\theta }\sqrt{{\rho }^2(1-{\mu }^2{v}^2)}}$

for the poloidial,

${\displaystyle v^{\varphi }=\sqrt{1-{\mu }^2{v}^2}\left(L_z\mathrm{\Sigma }-aqQr\left(1-{\mu }^2{v}^2\right){\sin }^2\theta \right)\cdot (\overline{R}\mathrm{\Sigma }{)}^{-1}}$

for the axial and

${\displaystyle v=\frac{\sqrt{{\dot{t}}^2-{\varsigma }^2}}{\dot{t}}=\sqrt{\frac{\chi (E-L_z\mathrm{\Omega }{)}^2-\mathrm{\Delta }{\rho }^2}{\chi (E-L_z\mathrm{\Omega }{)}^2}}}$

for the total local velocity, where

${\displaystyle \overline{R}=\sqrt{-g_{\varphi \varphi }}=\sqrt{\frac{\chi }{{\rho }^2}}\sin \theta }$

is the axial radius of gyration (local circumference divided by 2π), and

${\displaystyle \varsigma =\sqrt{g^{tt}}=\frac{\chi }{\mathrm{\Delta }{\rho }^2}}$

the gravitational time dilation component. The local radial escape velocity for a neutral particle is therefore

${\displaystyle v_{\mathrm{e}\mathrm{s}\mathrm{c}}=\frac{\sqrt{{\varsigma }^2-1}}{\varsigma }.}$

For the set of parameters${\displaystyle M^2-(J^2/M^2{)}^2-Q^2=0}$ the solution is called "extremal"; in the super-extremal regime,, there is no event horizon and the interior singularity is observable as anaked singularity. That is, the Kerr–Newman metric defines a black hole with an event horizon only when the combined charge and angular momentum are sufficiently small compared to the mass.^[8]

An electron's angular momentumJ and chargeQ (suitably specified ingeometrized units) both exceed its massM, in which case the metric has no event horizon. Thus, there can be no such thing as ablack hole electron — only anaked spinning ring singularity.^[25] Such a metric has several seemingly unphysical properties, such as the ring's violation of thecosmic censorship hypothesis, and also appearance of causality-violatingclosed timelike curves in the immediate vicinity of the ring.^[11]

In an analysis of the Kerr–Newman solution for a rotating charged body,Brandon Carter remarked that the solutions predict agyromagnetic ratio equal to that predicted by the quantum mechanicalDirac equation that approximately matches experimental evidence for theelectron.^[11]: 1562 This lead to the application of the Kerr–Newman 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 Kerr–Newman parameters in the regime where there is no event horizon but there is a naked singularity.^{[26][27]: 2442} For example, for the electron the charge-to-mass ratio${\displaystyle Q/M\approx {10}^{21}}$ and the spin effects are both orders of magnitude larger than this limit.^[28]

If the Kerr–Newman potential is considered as a classical model for an electron (anelectron black hole), it predicts an electron having not only a magnetic dipole moment, but also other multipole moments, such as an electric quadrupole moment.^[8] No experimental measurements of such a quadrupole moment have been reported.^[28]

• Wald, Robert M. (1984).General Relativity. Chicago: The University of Chicago Press. pp. 312–324.ISBN 978-0-226-87032-8.

• Adamo, Tim; Newman, Ezra (2014)."Kerr–Newman metric".Scholarpedia.9 (10) 31791.arXiv:1410.6626.Bibcode:2014SchpJ...931791N.doi:10.4249/scholarpedia.31791. co-authored byEzra T. Newman
• SR Made Easy, chapter 11: Charged and Rotating Black Holes and Their Thermodynamics

• Exact solutions in general relativity
• Equations
• Metric tensors
• Gravitational singularities

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