Reissner–Nordström metric

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

In spherical coordinates ${\displaystyle (t,r,\theta ,\varphi )}$, the Reissner–Nordström metric (i.e. the line element) is ${\displaystyle ds^{2}=c^{2}\,d\tau ^{2}=\left(1-{\frac {r_{\text{s}}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}\right)c^{2}\,dt^{2}-\left(1-{\frac {r_{\text{s}}}{r}}+{\frac {r_{Q}^{2}}{r^{2}}}\right)^{-1}\,dr^{2}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\varphi ^{2},}$

• Where ${\displaystyle c}$ is the speed of light.
• ${\displaystyle \tau }$ is the proper time.
• ${\displaystyle t}$ is the time coordinate (measured by a stationary clock at infinity).
• ${\displaystyle r}$ is the radial coordinate.
• ${\displaystyle (\theta ,\varphi )}$ are the spherical angles.
• ${\displaystyle r_{\text{s}}}$ is the Schwarzschild radius of the body given by

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

• ${\displaystyle r_{Q}}$ is a characteristic length scale given by

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

• ${\displaystyle \varepsilon _{0}}$ is the electric constant.

The total mass of the central body and its irreducible mass are related by^[6][7] ${\displaystyle M_{\rm {irr}}={\frac {c^{2}}{G}}{\sqrt {\frac {r_{+}^{2}}{2}}}\ \to \ M={\frac {Q^{2}}{16\pi \varepsilon _{0}GM_{\rm {irr}}}}+M_{\rm {irr}}.}$

The difference between ${\displaystyle M}$ and ${\displaystyle M_{\rm {irr}}}$ 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 ${\displaystyle Q}$ (or equivalently, the length scale ${\displaystyle r_{Q}}$) goes to zero, one recovers the Schwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio ${\displaystyle r_{\text{s}}/r}$ goes to zero. In the limit that both ${\displaystyle r_{Q}/r}$ and ${\displaystyle r_{\text{s}}/r}$ go to zero, the metric becomes the Minkowski metric for special relativity.

In practice, the ratio ${\displaystyle r_{\text{s}}/r}$ 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 ${\displaystyle r}$ 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 ${\displaystyle g_{rr}}$ diverges; that is, where $${\displaystyle 1-{\frac {r_{\rm {s}}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}=-{\frac {1}{g_{rr}}}=0.}$$

This equation has two solutions: $${\displaystyle r_{\pm }={\frac {1}{2}}\left(r_{\rm {s}}\pm {\sqrt {r_{\rm {s}}^{2}-4r_{\rm {Q}}^{2}}}\right).}$$

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 $${\displaystyle A_{\alpha }=(Q/r,0,0,0).}$$

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.^{[clarification needed]}

Gravitational time dilation

The gravitational time dilation in the vicinity of the central body is given by $${\displaystyle \gamma ={\sqrt {|g^{tt}|}}={\sqrt {\frac {r^{2}}{Q^{2}+(r-2M)r}}},}$$ which relates to the local radial escape velocity of a neutral particle $${\displaystyle v_{\rm {esc}}={\frac {\sqrt {\gamma ^{2}-1}}{\gamma }}.}$$

Christoffel symbols

The Christoffel symbols $${\displaystyle \Gamma _{jk}^{i}=\sum _{s=0}^{3}\ {\frac {g^{is}}{2}}\left({\frac {\partial g_{js}}{\partial x^{k}}}+{\frac {\partial g_{sk}}{\partial x^{j}}}-{\frac {\partial g_{jk}}{\partial x^{s}}}\right)}$$ with the indices $${\displaystyle \{0,\ 1,\ 2,\ 3\}\to \{t,\ r,\ \theta ,\ \varphi \}}$$ give the nonvanishing expressions $${\displaystyle {\begin{aligned}\Gamma _{tr}^{t}&={\frac {Mr-Q^{2}}{r(Q^{2}+r^{2}-2Mr)}}\\[6pt]\Gamma _{tt}^{r}&={\frac {(Mr-Q^{2})\left(r^{2}-2Mr+Q^{2}\right)}{r^{5}}}\\[6pt]\Gamma _{rr}^{r}&={\frac {Q^{2}-Mr}{r(Q^{2}-2Mr+r^{2})}}\\[6pt]\Gamma _{\theta \theta }^{r}&=-{\frac {r^{2}-2Mr+Q^{2}}{r}}\\[6pt]\Gamma _{\varphi \varphi }^{r}&=-{\frac {\sin ^{2}\theta \left(r^{2}-2Mr+Q^{2}\right)}{r}}\\[6pt]\Gamma _{\theta r}^{\theta }&={\frac {1}{r}}\\[6pt]\Gamma _{\varphi \varphi }^{\theta }&=-\sin \theta \cos \theta \\[6pt]\Gamma _{\varphi r}^{\varphi }&={\frac {1}{r}}\\[6pt]\Gamma _{\varphi \theta }^{\varphi }&=\cot \theta \end{aligned}}}$$

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 ${\displaystyle {\bf {e}}_{I}=e_{\mu I}}$ be a set of one-forms with internal Minkowski index ${\displaystyle I\in \{0,1,2,3\}}$, such that ${\displaystyle \eta ^{IJ}e_{\mu I}e_{\nu J}=g_{\mu \nu }}$. The Reissner metric can be described by the tetrad

${\displaystyle {\bf {e}}_{0}=G^{1/2}\,dt}$,

${\displaystyle {\bf {e}}_{1}=G^{-1/2}\,dr}$,

${\displaystyle {\bf {e}}_{2}=r\,d\theta }$

${\displaystyle {\bf {e}}_{3}=r\sin \theta \,d\varphi }$

where ${\displaystyle G(r)=1-r_{s}r^{-1}+r_{Q}^{2}r^{-2}}$. The parallel transport of the tetrad is captured by the connection one-forms ${\displaystyle {\boldsymbol {\omega }}_{IJ}=-{\boldsymbol {\omega }}_{JI}=\omega _{\mu IJ}=e_{I}^{\nu }\nabla _{\mu }e_{J\nu }}$. These have only 24 independent components compared to the 40 components of ${\displaystyle \Gamma _{\mu \nu }^{\lambda }}$. The connections can be solved for by inspection from Cartan's equation ${\displaystyle d{\bf {e}}_{I}={\bf {e}}^{J}\wedge {\boldsymbol {\omega }}_{IJ}}$, where the left hand side is the exterior derivative of the tetrad, and the right hand side is a wedge product.

${\displaystyle {\boldsymbol {\omega }}_{10}={\frac {1}{2}}\partial _{r}G\,dt}$

${\displaystyle {\boldsymbol {\omega }}_{20}={\boldsymbol {\omega }}_{30}=0}$

${\displaystyle {\boldsymbol {\omega }}_{21}=-G^{1/2}\,d\theta }$

${\displaystyle {\boldsymbol {\omega }}_{31}=-\sin \theta G^{1/2}d\varphi }$

${\displaystyle {\boldsymbol {\omega }}_{32}=-\cos \theta \,d\varphi }$

The Riemann tensor ${\displaystyle {\bf {R}}_{IJ}=R_{\mu \nu IJ}}$ can be constructed as a collection of two-forms by the second Cartan equation ${\displaystyle {\bf {R}}_{IJ}=d{\boldsymbol {\omega }}_{IJ}+{\boldsymbol {\omega }}_{IK}\wedge {\boldsymbol {\omega }}^{K}{}_{J},}$ which again makes use of the exterior derivative and wedge product. This approach is significantly faster than the traditional computation with ${\displaystyle \Gamma _{\mu \nu }^{\lambda }}$; note that there are only four nonzero ${\displaystyle {\boldsymbol {\omega }}_{IJ}}$ compared with nine nonzero components of ${\displaystyle \Gamma _{\mu \nu }^{\lambda }}$.

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 $${\displaystyle {\ddot {x}}^{i}=-\sum _{j=0}^{3}\ \sum _{k=0}^{3}\ \Gamma _{jk}^{i}\ {{\dot {x}}^{j}}\ {{\dot {x}}^{k}}+q\ {F^{ik}}\ {{\dot {x}}_{k}}}$$ which yields $${\displaystyle {\ddot {t}}={\frac {\ 2(Q^{2}-Mr)}{r(r^{2}-2Mr+Q^{2})}}{\dot {r}}{\dot {t}}+{\frac {qQ}{(r^{2}-2mr+Q^{2})}}\ {\dot {r}}}$$ $${\displaystyle {\ddot {r}}={\frac {(r^{2}-2Mr+Q^{2})(Q^{2}-Mr)\ {\dot {t}}^{2}}{r^{5}}}+{\frac {(Mr-Q^{2}){\dot {r}}^{2}}{r(r^{2}-2Mr+Q^{2})}}+{\frac {(r^{2}-2Mr+Q^{2})\ {\dot {\theta }}^{2}}{r}}+{\frac {qQ(r^{2}-2mr+Q^{2})}{r^{4}}}\ {\dot {t}}}$$ $${\displaystyle {\ddot {\theta }}=-{\frac {2\ {\dot {\theta }}\ {\dot {r}}}{r}}.}$$

All total derivatives are with respect to proper time ${\displaystyle {\dot {a}}={\frac {da}{d\tau }}}$.

Constants of the motion are provided by solutions ${\displaystyle S(t,{\dot {t}},r,{\dot {r}},\theta ,{\dot {\theta }},\varphi ,{\dot {\varphi }})}$ to the partial differential equation^[15] $${\displaystyle 0={\dot {t}}{\dfrac {\partial S}{\partial t}}+{\dot {r}}{\frac {\partial S}{\partial r}}+{\dot {\theta }}{\frac {\partial S}{\partial \theta }}+{\ddot {t}}{\frac {\partial S}{\partial {\dot {t}}}}+{\ddot {r}}{\frac {\partial S}{\partial {\dot {r}}}}+{\ddot {\theta }}{\frac {\partial S}{\partial {\dot {\theta }}}}}$$ after substitution of the second derivatives given above. The metric itself is a solution when written as a differential equation $${\displaystyle S_{1}=1=\left(1-{\frac {r_{s}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}\right)c^{2}\,{\dot {t}}^{2}-\left(1-{\frac {r_{s}}{r}}+{\frac {r_{Q}^{2}}{r^{2}}}\right)^{-1}\,{\dot {r}}^{2}-r^{2}\,{\dot {\theta }}^{2}.}$$

The separable equation $${\displaystyle {\frac {\partial S}{\partial r}}-{\frac {2}{r}}{\dot {\theta }}{\frac {\partial S}{\partial {\dot {\theta }}}}=0}$$ immediately yields the constant relativistic specific angular momentum $${\displaystyle S_{2}=L=r^{2}{\dot {\theta }};}$$ a third constant obtained from $${\displaystyle {\frac {\partial S}{\partial r}}-{\frac {2(Mr-Q^{2})}{r(r^{2}-2Mr+Q^{2})}}{\dot {t}}{\frac {\partial S}{\partial {\dot {t}}}}=0}$$ is the specific energy (energy per unit rest mass)^[16] $${\displaystyle S_{3}=E={\frac {{\dot {t}}(r^{2}-2Mr+Q^{2})}{r^{2}}}+{\frac {qQ}{r}}.}$$

Substituting ${\displaystyle S_{2}}$ and ${\displaystyle S_{3}}$ into ${\displaystyle S_{1}}$ yields the radial equation $${\displaystyle c\int d\,\tau =\int {\frac {r^{2}\,dr}{\sqrt {r^{4}(E-1)+2Mr^{3}-(Q^{2}+L^{2})r^{2}+2ML^{2}r-Q^{2}L^{2}}}}.}$$

Multiplying under the integral sign by ${\displaystyle S_{2}}$ yields the orbital equation $${\displaystyle c\int Lr^{2}\,d\theta =\int {\frac {L\,dr}{\sqrt {r^{4}(E-1)+2Mr^{3}-(Q^{2}+L^{2})r^{2}+2ML^{2}r-Q^{2}L^{2}}}}.}$$

The total time dilation between the test-particle and an observer at infinity is $${\displaystyle \gamma ={\frac {q\ Q\ r^{3}+E\ r^{4}}{r^{2}\ (r^{2}-2r+Q^{2})}}.}$$

The first derivatives ${\displaystyle {\dot {x}}^{i}}$ and the contravariant components of the local 3-velocity ${\displaystyle v^{i}}$ are related by $${\displaystyle {\dot {x}}^{i}={\frac {v^{i}}{\sqrt {(1-v^{2})\ |g_{ii}|}}},}$$ which gives the initial conditions $${\displaystyle {\dot {r}}={\frac {v_{\parallel }{\sqrt {r^{2}-2M+Q^{2}}}}{r{\sqrt {(1-v^{2})}}}}}$$ $${\displaystyle {\dot {\theta }}={\frac {v_{\perp }}{r{\sqrt {(1-v^{2})}}}}.}$$

The specific orbital energy $${\displaystyle E={\frac {\sqrt {Q^{2}-2rM+r^{2}}}{r{\sqrt {1-v^{2}}}}}+{\frac {qQ}{r}}}$$ and the specific relative angular momentum $${\displaystyle L={\frac {v_{\perp }\ r}{\sqrt {1-v^{2}}}}}$$ of the test-particle are conserved quantities of motion. ${\displaystyle v_{\parallel }}$ and ${\displaystyle v_{\perp }}$ are the radial and transverse components of the local velocity-vector. The local velocity is therefore $${\displaystyle v={\sqrt {v_{\perp }^{2}+v_{\parallel }^{2}}}={\sqrt {\frac {(E^{2}-1)r^{2}-Q^{2}-r^{2}+2rM}{E^{2}r^{2}}}}.}$$

Alternative formulation of metric

The metric can be expressed in Kerr–Schild form like this: $${\displaystyle {\begin{aligned}g_{\mu \nu }&=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\\[5pt]f&={\frac {G}{r^{2}}}\left[2Mr-Q^{2}\right]\\[5pt]\mathbf {k} &=(k_{x},k_{y},k_{z})=\left({\frac {x}{r}},{\frac {y}{r}},{\frac {z}{r}}\right)\\[5pt]k_{0}&=1.\end{aligned}}}$$

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

Notes

1. Reissner, H. (1916). "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie". Annalen der Physik. 355 (9): 106–120. Bibcode:1916AnP...355..106R. doi:10.1002/andp.19163550905. ISSN 0003-3804.
2. Weyl, Hermann (1917). "Zur Gravitationstheorie". Annalen der Physik. 359 (18): 117–145. Bibcode:1917AnP...359..117W. doi:10.1002/andp.19173591804. ISSN 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. Bibcode: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. 99 (697): 123–134. Bibcode:1921RSPSA..99..123J. doi:10.1098/rspa.1921.0028. ISSN 0950-1207.
5. Siegel, Ethan (2021-10-13). "Surprise: the Big Bang isn't the beginning of the universe anymore". Big Think. Retrieved 2024-09-03.
6. Thibault Damour: Black Holes: Energetics and Thermodynamics, S. 11 ff.
7. Qadir, Asghar (December 1983). "Reissner-Nordstrom repulsion". Physics Letters A. 99 (9): 419–420. Bibcode:1983PhLA...99..419Q. doi:10.1016/0375-9601(83)90946-5.
8. Chandrasekhar, Subrahmanyan (2009). The mathematical theory of black holes. Oxford classic texts in the physical sciences (Reprinted ed.). Oxford: Clarendon Press. p. 205. ISBN 978-0-19-850370-5. 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 (25 October 1968). "Global Structure of the Kerr Family of Gravitational Fields". Physical Review. 174 (5): 1559–1571. doi:10.1103/PhysRev.174.1559. ISSN 0031-899X.
11. Leonard Susskind: The Theoretical Minimum: Geodesics and Gravity, (General Relativity Lecture 4, timestamp: 34m18s)
12. Hackmann, Eva; Xu, Hongxiao (2013). "Charged particle motion in Kerr-Newmann space-times". Physical Review D. 87 (12): 124030. arXiv:1304.2142. doi:10.1103/PhysRevD.87.124030. ISSN 1550-7998.
13. Wald, Robert M. (2009). General relativity (Repr. ed.). Chicago: Univ. of Chicago Press. ISBN 978-0-226-87033-5.
14. Nordebo, Jonatan. "The Reissner-Nordström metric" (PDF). diva-portal. Retrieved 8 April 2021.
15. Smith, B. R. (December 2009). "First-order partial differential equations in classical dynamics". American Journal of Physics. 77 (12): 1147–1153. Bibcode:2009AmJPh..77.1147S. doi:10.1119/1.3223358. ISSN 0002-9505.
16. Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald; Kaiser, David; et al. (2017). Gravitation. Princeton, N.J: Princeton University Press. pp. 656–658. ISBN 978-0-691-17779-3. OCLC 1006427790.

References

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

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.