Kerr-Newman metric

The Kerr-Newman metric is a solution of Einstein's general relativity field equation that describes the spacetime geometry in the region surrounding a charged, rotating mass. Like the Kerr metric, the interior solution exists mathematically and satisfies Einstein's field equations, but is probably not representative of the actual metric of a physical black hole due to stability issues.

Mathematical form

The Kerr-Newman metric^[1][2] describes the geometry of spacetime in the vicinity of a mass M rotating with angular momentum J and charge Q

Failed to parse (Missing texvc executable; please see math/README to configure.): c^{2} d\tau^{2} = \left( 1 - \frac{r_{s} r - r_{Q}^{2}}{\rho^{2}} \right) c^{2} dt^{2} - \frac{\rho^{2}}{\Lambda^{2}} dr^{2} - \rho^{2} d\theta^{2} -

Failed to parse (Missing texvc executable; please see math/README to configure.): \left[ r^{2} + \alpha^{2} + \left( r_{s} r - r_{Q}^{2} \right) \frac{\alpha^{2}}{\rho^{2}} \sin^{2} \theta \right] \sin^{2} \theta \ d\phi^{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): + \left( r_{s} r - r_{Q}^{2} \right) \frac{2\alpha \sin^{2} \theta }{\rho^{2}} c dt d\phi

where r_s is the Schwarzschild radius

Failed to parse (Missing texvc executable; please see math/README to configure.): r_{s} = \frac{2GM}{c^{2}}

and the length-scale r_Q corresponds to the electrical charge Q

Failed to parse (Missing texvc executable; please see math/README to configure.): r_{Q}^{2} = \frac{Q^{2}G}{4\pi\epsilon_{0} c^{4}}

where 1/4πε₀ is Coulomb's force constant. The length-scales α, ρ and Λ have been introduced for brevity

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = \frac{J}{Mc}

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta

Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda^{2} = r^{2} - r_{s} r + \alpha^{2} + r_{Q}^{2}

Alternative mathematical form

The Kerr-Newman metric can also be written in geometrized units

Failed to parse (Missing texvc executable; please see math/README to configure.): ds^{2}=-\frac{\Lambda^{2}}{\rho^{2}}\left(dt-\alpha\sin^{2}\theta d\phi\right)^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left[\left(r^{2}+\alpha^{2}\right)d\phi-\alpha dt\right]^{2} +\frac{\rho^{2}}{\Lambda^{2}}dr^{2}+\rho^{2}d\theta^{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda^{2}\ \stackrel{\mathrm{def}}{=}\ r^{2}-2Mr+\alpha^{2}+Q^{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho^{2}\ \stackrel{\mathrm{def}}{=}\ r^{2}+ \alpha^{2}\cos^{2}\theta

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha\ \stackrel{\mathrm{def}}{=}\ \frac{J}{M}

where

M is the mass of the black hole

J is the angular momentum of the black hole

Q is the charge of the black hole

Special cases

The Kerr-Newman metric becomes the ...

• Kerr metric if the charge Q (or, equivalently, r_Q) is zero.
• Reissner-Nordström metric if the angular momentum J (or, equivalently, α) is zero.
• Schwarzschild metric if the charge Q and the angular momentum J are zero.
• orthogonal metric for the oblate spheroidal coordinates in the non-relativistic limit where M (or, equivalently, r_s) goes to zero.

Failed to parse (Missing texvc executable; please see math/README to configure.): c^{2} d\tau^{2} = c^{2} dt^{2} - \frac{\rho^{2}}{r^{2} + \alpha^{2}} dr^{2} - \rho^{2} d\theta^{2} - \left( r^{2} + \alpha^{2} \right) \sin^{2}\theta d\phi^{2}

which are equivalent to the Boyer-Lindquist coordinates^[3]

Failed to parse (Missing texvc executable; please see math/README to configure.): {x} = \sqrt {r^2 + \alpha^2} \sin\theta\cos\phi

Failed to parse (Missing texvc executable; please see math/README to configure.): {y} = \sqrt {r^2 + \alpha^2} \sin\theta\sin\phi

Failed to parse (Missing texvc executable; please see math/README to configure.): {z} = r \cos\theta \quad

• empty Minkowski space but in a usual spherical coordinate system if Failed to parse (Missing texvc executable; please see math/README to configure.): M = Q = J = 0

.

As for the Kerr metric, the Kerr-Newman metric defines a black hole only when

Failed to parse (Missing texvc executable; please see math/README to configure.): a^2 + Q^2 \leq M^2.

Newman's result represents the most general stationary, axisymmetric asymptotically flat solution of Einstein's equations in the presence of an electromagnetic field in four dimensions. Since the matter content of the solution reduces to an electromagnetic field, it is referred as an electrovac solution of Einstein's equations. 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 solution is named after Roy Kerr, discoverer of the uncharged rotating solution named after him (see Kerr metric) and Ezra T. Newman, co-discoverer of the charged solution in 1965.

History

In 1965, Ezra Newman found the axi-symmetric solution for Einstein's field equation for a black hole which is both rotating and electrically charged. This solution is called the Kerr-Newman metric. It is a generalisation of the Kerr metric.

See also

• Kerr metric
• Reissner-Nordström metric
• Schwarzschild metric
• Rotating black hole
• Exact solutions in general relativity
• BKL singularity

References

1. ^ Kerr, RP (1963). "Gravitational field of a spinning mass as an example of algebraically special metrics". Physical Review Letters 11: 237–238. 
2. ^ Landau, LD; Lifshitz, EM (1975). The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2), revised 4th English ed., New York: Pergamon Press, pp. 321–330. ISBN 978-0-08-018176-9. 
3. ^ Boyer, RH; Lindquist RW (1967). "Maximal Analytic Extension of the Kerr Metric". J. Math. Phys. 8: 265–281. 

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

Source(s)

• Kerr-Newman Black Holecs:Kerrova-Newmanova metrika

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