Kerr–Newman metric

The Kerr–Newman metric is a solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding a charged, rotating mass. This solution has not been especially useful for describing astrophysical phenomena, because observed astronomical objects do not possess an appreciable net electric charge. The solution has instead been of primarily theoretical and mathematical interest. (It is assumed that the cosmological constant equals zero which is near enough to the truth.)

History

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 $g_{\mu \nu }\!$ 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.

Four related solutions may be summarized by the following table:

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

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

Mathematical form

The Kerr–Newman metric describes the geometry of spacetime in the vicinity of a rotating mass M with charge Q. The formula for this metric depends upon what coordinates or coordinate conditions are selected. One way to express this metric is by writing down its line element in a particular set of spherical coordinates, also called Boyer–Lindquist coordinates:

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

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

$\alpha ={\frac {J}{Mc}}\,,$

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

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

have been introduced for brevity. Here r_s is the Schwarzschild radius (in meters) of the massive body, which is related to its mass M by

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

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

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

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

Component form

The components of the Kerr–Newman metric can be read off after a simple algebraic re-arrangement:

${\begin{aligned}c^{2}d\tau ^{2}&={\frac {(\Delta -\alpha ^{2}\sin ^{2}\theta )}{\rho ^{2}}}\;c^{2}\;dt^{2}-\left({\frac {\rho ^{2}}{\Delta }}\right)dr^{2}\\&-\rho ^{2}d\theta ^{2}+(\alpha ^{2}\Delta \sin ^{2}\theta -r^{4}-2r^{2}\alpha ^{2}-\alpha ^{4}){\frac {\sin ^{2}\theta \;d\phi ^{2}}{\rho ^{2}}}\\&-(\Delta -r^{2}-\alpha ^{2}){\frac {2\alpha \sin ^{2}\theta \;c\;dt\;d\phi }{\rho ^{2}}}\end{aligned}}$

Alternative (Kerr–Schild) formulation

The Kerr–Newman metric can be expressed in "Kerr–Schild" form, using a particular set of Cartesian coordinates as follows. These solutions were proposed by Kerr and Schild in 1965.

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

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

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

$k_{0}=1.\!$

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 tensor, and a is a constant rotational parameter of the spinning object. It is understood that the vector ${\vec {a}}$ is directed along the positive z-axis. The quantity r is not the radius, but rather is implicitly defined like this:

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

Notice that the quantity r becomes the usual radius

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

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:

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

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

$A_{\mu }={\frac {Q}{R}}k_{\mu }$

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.

Special cases and generalizations

The Kerr–Newman metric is a generalization of other exact solutions in general relativity:

• Kerr metric if the charge Q is zero.
• Reissner–Nordström metric if the angular momentum J (or a) is zero.
• Schwarzschild metric if the charge Q and the angular momentum J (or a) are zero.
• Minkowski space if the mass M, the charge Q, and the rotation parameter a are all zero. Also, if gravity is intended to be removed, Minkowski space arises if the gravitational constant G is zero (with electric and magnetic fields more complicated than simply the fields of a charged magnetic dipole).

The Kerr–Newman solution (with cosmological constant equal to zero) is also a special case of more general exact solutions of the Einstein–Maxwell Equations.

Some aspects 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 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.

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 been detected empirically yet.

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 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 stability issues. 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 an important electric charge.

The Kerr–Newman metric defines a black hole with an event horizon only when the following relation is satisfied:

$a^{2}+Q^{2}\leq M^{2}.$

An electron's a and 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. 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.

The Russian theorist Alexander Burinskii wrote in 2007: "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". The Burinskii paper 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.

The electromagnetic fields

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.

$A_{\mu }=\left(-\phi ,A_{x},A_{y},A_{z}\right)\,$

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

${\vec {E}}=-{\vec {\nabla }}\phi \,$

${\vec {B}}={\vec {\nabla }}\times {\vec {A}}\,$

Using the Kerr-Newman formula for the four-potential in the Kerr-Schild form yields the following concise complex formula for the fields:

${\vec {E}}+i{\vec {B}}=-{\vec {\nabla }}\Omega \,$

$\Omega ={\frac {Q}{\sqrt {({\vec {R}}-i{\vec {a}})^{2}}}}\,$

The quantity omega ($\Omega$) 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.

Bibliography

• Wald, Robert M. (1984). General Relativity. Chicago: The University of Chicago Press. pp. 312–324. ISBN 0-226-87032-4.

External links

• Scholarpedia article on the subject co-authored by Ezra T. Newman himself
• SR Made Easy, chapter 11: Charged and Rotating Black Holes and Their Thermodynamics