Kerr–Newman metric

In Dec 1963 Kerr and Schild found the Kerr–Schild metrics that gave all Einstein spaces that are exact linear perturbations of Minkowski space. In early 1964 Roy Kerr looked for all Einstein–Maxwell spaces with this same property. By Feb 1964 the special case where the Kerr–Schild spaces were charged (this includes 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 ${\displaystyle g_{\mu \nu }\!}$ 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:

where Q represents the body's electric charge and J represents its spin angular momentum.

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]

${\displaystyle J^{2}/M^{2}+Q^{2}\leq M^{2}.}$

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 and thus there can be no such thing as a black hole electron — only a naked spinning ring singularity.[10] Such a metric has several seemingly unphysical properties, such as the ring's violation of the cosmic censorship hypothesis, and also appearance of causality-violating closed timelike curves in the immediate vicinity of the ring.[11]

A 2007 paper by Russian theorist Alexander Burinskii describes an electron as a gravitationally confined ring singularity without an event horizon. It has some, but not all of the predicted properties of a black hole.[12] As Burinskii described it:

In this work we obtain an exact correspondence between the wave function of the Dirac equation and the spinor (twistorial) structure of the Kerr geometry. It allows us to assume that the Kerr–Newman geometry reflects the specific space–time structure of electron, and electron contains really the Kerr–Newman circular string of Compton size.[12]

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

• The Kerr metric as the charge Q goes to zero.
• The Reissner–Nordström metric as the angular momentum J (or a = J/M ) goes to zero.
• The Schwarzschild metric as both the charge Q and the angular momentum J (or a) are taken to zero.
• 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.

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.

One way to express this metric is by writing down its line element in a particular set of spherical coordinates,[13] also called Boyer–Lindquist coordinates:

${\displaystyle c^{2}d\tau ^{2}=-\left({\frac {dr^{2}}{\Delta }}+d\theta ^{2}\right)\rho ^{2}+\left(c\,dt-a\sin ^{2}\theta \,d\phi \right)^{2}{\frac {\Delta }{\rho ^{2}}}-\left(\left(r^{2}+a^{2}\right)d\phi -ac\,dt\right)^{2}{\frac {\sin ^{2}\theta }{\rho ^{2}}}}$

where the coordinates (r, θ, ϕ) are standard spherical coordinate system, and the length scales:

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

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

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

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

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

where G is the gravitational constant, and r_Q is a length scale corresponding to the electric charge Q of the mass

${\displaystyle r_{Q}^{2}={\frac {Q^{2}G}{4\pi \epsilon _{0}c^{4}}}}$

where 1/(4πε₀) is Coulomb's force constant.

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 metric is as follows.[16][17][18]

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}$

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

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

${\displaystyle k_{0}=1.\!}$

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 ${\displaystyle {\vec {a}}}$ is directed along the positive z-axis, i.e. ${\displaystyle {\vec {a}}=a{\hat {z}}}$. The quantity r is not the radius, but rather is implicitly defined like this:

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

Notice that the quantity r becomes the usual radius R

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

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:[16][19]

${\displaystyle A_{\mu }={\frac {Qr^{3}}{r^{4}+a^{2}z^{2}}}k_{\mu }}$

At large distances from the source (R ≫ a), these equations reduce to the Reissner–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.[1]

The total mass-equivalent M, which contains the electric field-energy and the rotational energy, and the irreducible mass M_irr are related by[22][23]

${\displaystyle M_{\rm {irr}}={\frac {1}{2}}{\sqrt {2M^{2}-r_{Q}^{2}c^{4}/G^{2}+2M{\sqrt {M^{2}-(r_{Q}^{2}+a^{2})c^{4}/G^{2}}}}}}$

which can be inverted to obtain

${\displaystyle M={\frac {4M_{\rm {irr}}^{2}+r_{Q}^{2}c^{4}/G^{2}}{2{\sqrt {4M_{\rm {irr}}^{2}-a^{2}c^{4}/G^{2}}}}}}$

In order to electrically charge and/or spin a neutral and static body, energy has to be applied to the system. Due to the mass–energy equivalence, 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,[24][25] the remaining mass–energy will always stay greater than or equal to M_irr.

Event horizons and ergospheres of a charged and spinning black hole in pseudospherical r,θ,φ and cartesian x,y,z coordinates.

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

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

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

${\displaystyle r_{\text{E}}^{\pm }={\frac {r_{\rm {s}}}{2}}\pm {\sqrt {{\frac {r_{\rm {s}}^{2}}{4}}-a^{2}\cos ^{2}\theta -r_{Q}^{2}}}.}$

Testparticle in orbit around a spinning and charged black hole (a/M = 0.9, Q/M = 0.4)

For brevity, we further use dimensionless natural units of ${\displaystyle G=M=c=K=1}$, with Coulomb's constant ${\displaystyle K}$, where ${\displaystyle a}$ reduces to ${\displaystyle Jc/G/M^{2}}$ and ${\displaystyle Q}$ to ${\displaystyle Q/M\ {\sqrt {K/G}}}$, and the equations of motion for a testparticle of charge ${\displaystyle q}$ become[26][27]

${\displaystyle {\dot {t}}}$${\displaystyle ={\frac {\csc ^{2}\theta \ ({L_{z}}(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 ))}{\Delta \rho ^{2}}}}$

${\displaystyle {\dot {r}}=\pm {\frac {\sqrt {((r^{2}+a^{2})\ E-a\ L_{z}-q\ Q\ r)^{2}-\Delta \ (C+r^{2})}}{\rho ^{2}}}}$

${\displaystyle {\dot {\theta }}=\pm {\frac {\sqrt {C-(a\cos \theta )^{2}-(a\ \sin ^{2}\theta \ E-L_{z})^{2}/\sin ^{2}\theta }}{\rho ^{2}}}}$

${\displaystyle {\dot {\phi }}={\frac {E\ (a\ \sin ^{2}\theta \ (r^{2}+a^{2})-a\ \sin ^{2}\theta \ \Delta )+L_{z}\ (\Delta -a^{2}\ \sin ^{2}\theta )-q\ Q\ r\ a\ \sin ^{2}\theta }{\rho ^{2}\ \Delta \ \sin ^{2}\theta }}}$

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

${\displaystyle C=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}(1-E^{2})+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)=a^{2}\ (1-E^{2})\ \sin ^{2}\delta +L_{z}^{2}\ \tan ^{2}\delta ={\rm {const.}}}$

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

Ray traced shadow of a spinning and charged black hole with the parameters a² + Q² = 1M². The left side of the black hole is rotating towards the observer.

${\displaystyle L_{z}={\frac {v^{\phi }\ {\bar {R}}}{\sqrt {1-v^{2}}}}={\rm {const.}}}$

and

${\displaystyle E={\sqrt {\frac {\Delta \ \rho ^{2}}{(1-v^{2})\ \chi }}}+\Omega \ L_{z}={\rm {const.}}}$

are also conserved quantities.

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

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

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

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

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

for the radial,

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

for the poloidial,

${\displaystyle v^{\phi }={\frac {L_{z}{\sqrt {1-v^{2}}}}{\bar {R}}}}$

for the axial and

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

for the total local velocity, where

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

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

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

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

${\displaystyle v_{\rm {esc}}={\frac {\sqrt {\varsigma ^{2}-1}}{\varsigma }}}$.

• 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 by Ezra T. Newman himself
• SR Made Easy, chapter 11: Charged and Rotating Black Holes and Their Thermodynamics