Kerr-Newman metric - Citizendia

General relativity

$G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}\,$

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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. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the General relativity (GR is a Theory of Gravitation that was developed by Albert Einstein between 1907 and 1915 The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying Albert Einstein 's theory of General In General relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational The Gödel metric is an exact solution of the Einstein field equations in which the Stress-energy tensor contains two terms the first representing the In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body The Kasner metric is an exact solution to Einstein 's theory of General relativity. The Milne model was a special relativistic cosmological model proposed by Edward Arthur Milne. In General relativity, the pp-wave spacetimes, or pp-waves for short are an important family of Exact solutions of Einstein's field equation The ADM Formalism developed by Arnowitt, Deser and Misner is a Hamiltonian formulation of General relativity. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the 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. In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body

1 Mathematical form
2 Alternative mathematical form
3 Special cases
4 History
5 See also
6 References
7 Source(s)

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

$\begin{align} c^2 \mathrm d\tau^2 & = \left[ 1 - \frac{r_s r - r_Q^2}{\rho^2} \right] c^2 \mathrm d t^2 - \frac{\rho^2}{\Lambda^2} \mathrm d r^2 - \rho^2 \mathrm d\theta^2 \\ & - \left[ r^2 + \alpha^2 + \left( r_s r - r_Q^2 \right) \frac{\alpha^2}{\rho^2}\sin^2\theta \right] \sin^2 \theta \ \mathrm d\phi^2 \\ & + \left( r_s r - r_Q^2 \right) \frac{2\alpha\sin^2\theta}{\rho^2}\;c \mathrm d t\;\mathrm d\phi \end{align}$

where r_s is the Schwarzschild radius

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

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

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

where 1/4πε₀ is Coulomb's force constant. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form The length-scales α, ρ and Λ have been introduced for brevity

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

ρ 2 = r 2 + α 2 cos 2 θ

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

Alternative mathematical form

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

$ds^{2}=-\frac{\Lambda^{2}}{\rho^{2}}\left(dt-\alpha\sin^{2}\theta d\phi\right)^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left[\left(r^{2}+\alpha^{2}\right)d\phi-\alpha dt\right]^{2} +\frac{\rho^{2}}{\Lambda^{2}}dr^{2}+\rho^{2}d\theta^{2}$

$\Lambda^{2}\ \stackrel{\mathrm{def}}{=}\ r^{2}-2Mr+\alpha^{2}+Q^{2}$

$\rho^{2}\ \stackrel{\mathrm{def}}{=}\ r^{2}+ \alpha^{2}\cos^{2}\theta$

$\alpha\ \stackrel{\mathrm{def}}{=}\ \frac{J}{M}$

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 . A geometrized unit system or geometric unit system is a system of Natural units in which the base physical units are chosen so that the Speed of light . .

Kerr metric if the charge Q (or, equivalently, r_Q) is zero. In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body
Reissner-Nordström metric if the angular momentum J (or, equivalently, α) is zero. In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational
Schwarzschild metric if the charge Q and the angular momentum J are zero. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside
orthogonal metric for the oblate spheroidal coordinates in the non-relativistic limit where M (or, equivalently, r_s) goes to zero. Oblate spheroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional elliptic coordinate system

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

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

${x} = \sqrt {r^2 + \alpha^2} \sin\theta\cos\phi$

${y} = \sqrt {r^2 + \alpha^2} \sin\theta\sin\phi$

${z} = r \cos\theta \quad$

empty Minkowski space but in a usual spherical coordinate system if $M = Q = J = 0$ . 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. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial

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

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

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. In General relativity, a Spacetime is said to be stationary if it admits a global nowhere zero Timelike Killing vector field. The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the The electromagnetic field is a physical field produced by electrically charged objects. 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. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction.

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. Roy Patrick Kerr (born 1934 is a New Zealander Mathematician who is best known for discovering the Kerr vacuum, an exact solution to the In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body Ezra T Newman is an American physicist well known for his many contributions to General relativity. Year 1965 ( MCMLXV) was a Common year starting on Friday (link will display full calendar of the 1965 Gregorian calendar.

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. Ezra T Newman is an American physicist well known for his many contributions to General relativity. This solution is called the Kerr-Newman metric. It is a generalisation of the Kerr metric. In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body

References

^ Kerr, RP (1963). In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside Black hole#Major features of rotating black holes A rotating black hole is a Black hole that possesses Angular momentum. In General relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter Roy Patrick Kerr (born 1934 is a New Zealander Mathematician who is best known for discovering the Kerr vacuum, an exact solution to the "Gravitational field of a spinning mass as an example of algebraically special metrics". Physical Review Letters 11: 237–238. doi:10.1103/PhysRevLett.11.237. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Landau, LD; Lifshitz, EM (1975). Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968 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.
^ Boyer, RH; Lindquist RW (1967). "Maximal Analytic Extension of the Kerr Metric". J. Math. Phys. 8: 265–281. doi:10.1063/1.1705193. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

Wald, RM (1984). Robert Wald (b June 29, 1947) is a Physicist who specializes in General relativity and the Thermodynamics of Black holes General Relativity. Chicago: The University of Chicago Press, pp. 312–324. ISBN 978-0-226-87032-8.

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