Reissner–Nordström metric explained

In physics and astronomy, the Reissner - Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. The analogous solution for a charged, rotating body is given by the Kerr–Newman metric.

The metric was discovered between 1916 and 1921 by Hans Reissner,^[1] Hermann Weyl,^[2] Gunnar Nordström^[3] and George Barker Jeffery^[4] independently.^[5]

The metric

(t, r, \theta, \varphi) , the Reissner - Nordström metric (i.e. the line element) isds ^{2=c} ^{2} d\tau ^{2} = \left( 1 - (r _{s})/(r) + (r ^{2} _{\rm Q})/(r ^{2}) \right) c ^{2} dt ^{2} - \left( 1 - (r _{s})/(r) + (r ^{2} _{Q})/(r ^{2}) \right) ^{-1} dr ^{2} - r ^{2} d\theta ^{2} - r ^{2\sin} ^{2\theta} d\varphi ^{2,}

• Where

c is the speed of light.\tau is the proper time.t is the time coordinate (measured by a stationary clock at infinity).r is the radial coordinate.(\theta, \varphi) are the spherical angles.r _{s} is the Schwarzschild radius of the body given byr _{s} = (2 GM)/(c ^{2}) , .r _{Q} is a characteristic length scale given byr ^{2} _{Q} = (Q ^{2} G)/(4 \pi\varepsilon _{0} c ^{4}) . \varepsilon _{0} is the electric constant.

The total mass of the central body and its irreducible mass are related by^{[6] [7]}

M _{\rm} = (c ^{2})/(\sqrt{ G) (r ^{2} _{+})/(2) } \ \to \ M=\frac + M_.

The difference between

M and M _{\rm} is due to the equivalence of mass and energy, which makes the electric field energy also contribute to the total mass.

In the limit that the charge

Q (or equivalently, the length scale r _{Q} ) goes to zero, one recovers the Schwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio r _{s/r} goes to zero. In the limit that both r _{Q/r} and r _{s/r} go to zero, the metric becomes the Minkowski metric for special relativity.

In practice, the ratio

r _{s/r} is often extremely small. For example, the Schwarzschild radius of the Earth is roughly 9 mm (3/8 inch), whereas a satellite in a geosynchronous orbit has an orbital radius r that is roughly four billion times larger, at 42,164 km (26,200 miles). Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio only becomes large close to black holes and other ultra-dense objects such as neutron stars.

Charged black holes

Although charged black holes with r_Q ≪ r_s are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon.^[8] As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component

g _{rr} diverges; that is, where$$1\; -\; \backslash frac\; +\; \backslash frac\; =\; -\backslash frac\; =\; 0.$$

This equation has two solutions:$$r\_\backslash pm\; =\; \backslash frac\backslash left(r\_\; \backslash pm\; \backslash sqrt\backslash right).$$

These concentric event horizons become degenerate for 2r_Q = r_s, which corresponds to an extremal black hole. Black holes with 2r_Q > r_s cannot exist in nature because if the charge is greater than the mass there can be no physical event horizon (the term under the square root becomes negative).^[9] Objects with a charge greater than their mass can exist in nature, but they can not collapse down to a black hole, and if they could, they would display a naked singularity.^[10] Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic potential is$$A\_\backslash alpha\; =\; (Q/r,\; 0,\; 0,\; 0).$$

If magnetic monopoles are included in the theory, then a generalization to include magnetic charge P is obtained by replacing Q² by Q² + P² in the metric and including the term P cos θ dφ in the electromagnetic potential.

Gravitational time dilation

The gravitational time dilation in the vicinity of the central body is given by$$\backslash gamma\; =\; \backslash sqrt$$

= \sqrt,which relates to the local radial escape velocity of a neutral particle$$v\_=\backslash frac.$$

Christoffel symbols

The Christoffel symbols$$\backslash Gamma^i\_\; =\; \backslash sum\_^3\; \backslash \; \backslash frac\; \backslash left(\backslash frac+\backslash frac-\backslash frac\backslash right)$$with the indices$$\backslash \; \backslash to\; \backslash$$give the nonvanishing expressions

Given the Christoffel symbols, one can compute the geodesics of a test-particle.^{[11] [12]}

Tetrad form

Instead of working in the holonomic basis, one can perform efficient calculations with a tetrad.^[13] Let

{\bf e} _{I} = e _{\mu} be a set of one-forms with internal Minkowski index I \in\{ 0 , 1 , 2 , 3 \} , such that η ^{IJ} e _{\mu} e _{\nu} = g _{\mu\nu} . The Reissner metric can be described by the tetrad{\bf e} _{0} = G ^{1/2} dt ,{\bf e} _{1} = G ^{-1/2} dr ,{\bf e} _{2} = r d\theta {\bf e} _{3} = r \sin \theta d\varphi

where

G(r) = 1 - r ^{-1} _{sr} + r ^{2 r} _{Q} ^{-2} . The parallel transport of the tetrad is captured by the connection one-forms \boldsymbol \omega _{IJ} = - \boldsymbol \omega _{JI} = \omega _{\mu} = e ^{\nu} _{I} \nabla _{\mu} e _{J\nu} . These have only 24 independent components compared to the 40 components of \Gamma ^{λ} _{\mu\nu} . The connections can be solved for by inspection from Cartan's equation d{\bf e} _{I} = {\bf e} ^{J} \wedge \boldsymbol \omega _{IJ} , where the left hand side is the exterior derivative of the tetrad, and the right hand side is a wedge product.\boldsymbol \omega _{10} = (12)/(\partial) _{r} G dt \boldsymbol \omega _{20} = \boldsymbol \omega _{30} = 0 \boldsymbol \omega _{21} = - G ^{1/2} d\theta \boldsymbol \omega _{31} = - \sin \theta G ^{1/2} d \varphi \boldsymbol \omega _{32} = - \cos \theta d\varphi {\bf R} _{IJ} = R _{\mu\nu} can be constructed as a collection of two-forms by the second Cartan equation {\bf R} _{IJ} = d \boldsymbol \omega _{IJ} + \boldsymbol \omega _{IK} \wedge \boldsymbol \omega ^{K{}} _{J,} which again makes use of the exterior derivative and wedge product. This approach is significantly faster than the traditional computation with \Gamma ^{λ} _{\mu\nu} ; note that there are only four nonzero \boldsymbol \omega _{IJ} compared with nine nonzero components of \Gamma ^{λ} _{\mu\nu} .

Equations of motion

^[14]

Because of the spherical symmetry of the metric, the coordinate system can always be aligned in a way that the motion of a test-particle is confined to a plane, so for brevity and without restriction of generality we use θ instead of φ. In dimensionless natural units of G = M = c = K = 1 the motion of an electrically charged particle with the charge q is given by$$\backslash ddot\; x^i\; =\; -\; \backslash sum\_^3\; \backslash \; \backslash sum\_^3\; \backslash \; \backslash Gamma^i\_\; \backslash \; \backslash \; +\; q\; \backslash \; \backslash$$which yields$$\backslash ddot\; t\; =\; \backslash frac\backslash dot\backslash dot+\backslash frac\; \backslash \; \backslash dot$$$$\backslash ddot\; r\; =\; \backslash frac+\backslash frac+\backslash frac\; +\; \backslash frac\; \backslash \; \backslash dot$$$$\backslash ddot\; \backslash theta\; =\; -\backslash frac\; .$$

All total derivatives are with respect to proper time

• a = (da)/(d\tau) .

Constants of the motion are provided by solutions

S (t, • • t,r, • r,\theta, • \theta,\varphi, \varphi) to the partial differential equation^[15] $$0=\backslash dot\; t\backslash dfrac+\backslash dot\; r\backslash frac+\backslash dot\backslash theta\backslash frac+\backslash ddot\; t\; \backslash frac\; +\backslash ddot\; r\; \backslash frac\; +\; \backslash ddot\backslash theta\; \backslash frac$$after substitution of the second derivatives given above. The metric itself is a solution when written as a differential equation$$S\_1=1\; =\; \backslash left(1\; -\; \backslash frac\; +\; \backslash frac\; \backslash right)\; c^2\backslash ,\; ^2\; -\backslash left(1\; -\; \backslash frac\; +\; \backslash frac\; \backslash right)^\; \backslash ,\; ^2\; -\; r^2\; \backslash ,\; ^2\; .$$

The separable equation$$\backslash frac-\backslash frac\backslash dot\backslash theta\backslash frac=0$$immediately yields the constant relativistic specific angular momentum$$S\_2=L=r^2\backslash dot\backslash theta;$$a third constant obtained from$$\backslash frac-\backslash frac\backslash dot\; t\backslash frac=0$$is the specific energy (energy per unit rest mass)^[16] $$S\_3=E=\backslash frac\; +\; \backslash frac\; .$$

Substituting

S _{2} and S _{3} into S _{1} yields the radial equation $$c\backslash int\; d\backslash ,\backslash tau\; =\backslash int\; \backslash frac\; .$$

Multiplying under the integral sign by

S _{2} yields the orbital equation$$c\backslash int\; Lr^2\backslash ,d\backslash theta\; =\backslash int\; \backslash frac.$$

The total time dilation between the test-particle and an observer at infinity is$$\backslash gamma=\; \backslash frac\; .$$

The first derivatives

• x ^{i} and the contravariant components of the local 3-velocity v ^{i} are related by$$\backslash dot\; x^i\; =\; \backslash frac,$$which gives the initial conditions$$\backslash dot\; r\; =\; \backslash frac$$$$\backslash dot\; \backslash theta\; =\; \backslash frac\; .$$

The specific orbital energy$$E=\backslash frac+\backslash frac$$and the specific relative angular momentum$$L=\backslash frac$$of the test-particle are conserved quantities of motion.

v _{\parallel} and v _{\perp} are the radial and transverse components of the local velocity-vector. The local velocity is therefore$$v\; =\; \backslash sqrt\; =\; \backslash sqrt.$$

Alternative formulation of metric

The metric can be expressed in Kerr–Schild form like this:

Notice that k is a unit vector. Here M is the constant mass of the object, Q is the constant charge of the object, and η is the Minkowski tensor.

See also

• Black hole electron

References

• Book: Adler . R. . Introduction to General Relativity . Bazin . M. . Schiffer . M. . 1965 . McGraw-Hill Book Company . 978-0-07-000420-7 . New York . 395–401.
• Book: Wald, Robert M. . Robert Wald . General Relativity . 1984 . The University of Chicago Press . 978-0-226-87032-8 . Chicago . 158, 312–324.

External links

• Spacetime diagrams including Finkelstein diagram and Penrose diagram, by Andrew J. S. Hamilton
• "Particle Moving Around Two Extreme Black Holes" by Enrique Zeleny, The Wolfram Demonstrations Project.

Notes and References

1. Reissner . H. . 1916 . Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie . Annalen der Physik . en . 355 . 9 . 106–120 . 1916AnP...355..106R . 10.1002/andp.19163550905 . 0003-3804.
2. Weyl . Hermann . 1917 . Zur Gravitationstheorie . Annalen der Physik . en . 359 . 18 . 117–145 . 1917AnP...359..117W . 10.1002/andp.19173591804 . 0003-3804.
3. Nordström . G. . 1918 . On the Energy of the Gravitational Field in Einstein's Theory . Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings . 20 . 2 . 1238–1245 . 1918KNAB...20.1238N.
4. Jeffery . G. B. . 1921 . The field of an electron on Einstein's theory of gravitation . Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character . en . 99 . 697 . 123–134 . 1921RSPSA..99..123J . 10.1098/rspa.1921.0028 . 0950-1207 . free.
5. Web site: Siegel . Ethan . 2021-10-13 . Surprise: the Big Bang isn't the beginning of the universe anymore . 2024-09-03 . Big Think . en-US.
6. [Thibault Damour]
7. Qadir . Asghar . December 1983 . Reissner-Nordstrom repulsion . Physics Letters A . en . 99 . 9 . 419–420 . 1983PhLA...99..419Q . 10.1016/0375-9601(83)90946-5.
8. Book: Chandrasekhar, Subrahmanyan . Subrahmanyan Chandrasekhar . The mathematical theory of black holes . 2009 . Clarendon Press . 978-0-19-850370-5 . Reprinted . Oxford classic texts in the physical sciences . Oxford . 205 . And finally, the fact that the Reissner–Nordström solution has two horizons, an external event horizon and an internal 'Cauchy horizon,' provides a convenient bridge to the study of the Kerr solution in the subsequent chapters..
9. Andrew Hamilton: The Reissner Nordström Geometry (Casa Colorado)
10. Carter . Brandon . Brandon Carter . 25 October 1968 . Global Structure of the Kerr Family of Gravitational Fields . Physical Review . en . 174 . 5 . 1559–1571 . 10.1103/PhysRev.174.1559 . 0031-899X.
11. [Leonard Susskind]
12. Hackmann . Eva . Xu . Hongxiao . 2013 . Charged particle motion in Kerr-Newmann space-times . Physical Review D . en . 87 . 12 . 124030 . 10.1103/PhysRevD.87.124030 . 1304.2142 . 1550-7998.
13. Book: Wald, Robert M. . General relativity . 2009 . Univ. of Chicago Press . 978-0-226-87033-5 . Repr. . Chicago.
14. Web site: Nordebo . Jonatan . The Reissner-Nordström metric . diva-portal . 8 April 2021.
15. Smith . B. R. . December 2009 . First-order partial differential equations in classical dynamics . American Journal of Physics . en . 77 . 12 . 1147–1153 . 2009AmJPh..77.1147S . 10.1119/1.3223358 . 0002-9505.
16. Book: Misner . Charles W. . Gravitation . Thorne . Kip S. . Wheeler . John Archibald . Kaiser . David . 2017 . Princeton University Press . 978-0-691-17779-3 . Princeton, N.J . 656–658 . on1006427790 . etal.

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