Astronomy:Reissner–Nordström metric

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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 [math]\displaystyle{ (t, r, \theta, \varphi) }[/math], the Reissner–Nordström metric (i.e. the line element) is [math]\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, }[/math] where [math]\displaystyle{ c }[/math] is the speed of light, [math]\displaystyle{ \tau }[/math] is the proper time, [math]\displaystyle{ t }[/math] is the time coordinate (measured by a stationary clock at infinity), [math]\displaystyle{ r }[/math] is the radial coordinate, [math]\displaystyle{ (\theta, \varphi) }[/math] are the spherical angles, and [math]\displaystyle{ r_\text{s} }[/math] is the Schwarzschild radius of the body given by [math]\displaystyle{ r_\text{s} = \frac{2GM}{c^2}, }[/math] and [math]\displaystyle{ r_Q }[/math] is a characteristic length scale given by [math]\displaystyle{ r_Q^2 = \frac{Q^2 G}{4\pi\varepsilon_0 c^4}. }[/math] Here, [math]\displaystyle{ \varepsilon_0 }[/math] is the electric constant.

The total mass of the central body and its irreducible mass are related by[6][7] [math]\displaystyle{ M_{\rm irr}= \frac{c^2}{G} \sqrt{\frac{r_+^2}{2}} \ \to \ M=\frac{Q ^2}{16\pi\varepsilon_0 G M_{\rm irr}} + M_{\rm irr}. }[/math]

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

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

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

These concentric event horizons become degenerate for 2rQ = rs, which corresponds to an extremal black hole. Black holes with 2rQ > rs can not 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 [math]\displaystyle{ A_\alpha = (Q/r, 0, 0, 0). }[/math]

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  in the electromagnetic potential.[clarification needed]

Gravitational time dilation

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

Christoffel symbols

The Christoffel symbols [math]\displaystyle{ \Gamma^i_{j k} = \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) }[/math] with the indices [math]\displaystyle{ \{ 0, \ 1, \ 2, \ 3 \} \to \{ t, \ r, \ \theta, \ \varphi \} }[/math] give the nonvanishing expressions [math]\displaystyle{ \begin{align} \Gamma^t_{t r} & = \frac{M r-Q^2}{r ( Q^2 + r^2 - 2 M r ) } \\[6pt] \Gamma^r_{t t} & = \frac{(M r-Q^2) \left(r^2-2Mr+Q^2\right)}{r^5} \\[6pt] \Gamma^r_{r r} & = \frac{Q^2-M r}{Q^2 r-2 M r^2+r^3} \\[6pt] \Gamma^r_{\theta \theta} & = -\frac{r^2-2Mr+Q^2}{r} \\[6pt] \Gamma^r_{\varphi \varphi} & = -\frac{\sin ^2 \theta \left(r^2-2Mr+Q^2\right)}{r} \\[6pt] \Gamma^\theta_{\theta r} & = \frac{1}{r} \\[6pt] \Gamma^\theta_{\varphi \varphi} & = - \sin \theta \cos \theta \\[6pt] \Gamma^\varphi_{\varphi r} & = \frac{1}{r} \\[6pt] \Gamma^\varphi_{\varphi \theta} & = \cot \theta \end{align} }[/math]

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

Equations of motion

[13]

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 [math]\displaystyle{ \ddot x^i = - \sum_{j=0}^3 \ \sum_{k=0}^3 \ \Gamma^i_{j k} \ {\dot x^j} \ {\dot x^k} + q \ {F^{i k}} \ {\dot x_k} }[/math] which yields [math]\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} }[/math] [math]\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} }[/math] [math]\displaystyle{ \ddot \theta = -\frac{2 \ \dot\theta \ \dot{r}}{r} . }[/math]

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

Constants of the motion are provided by solutions [math]\displaystyle{ S (t,\dot t,r,\dot r,\theta,\dot\theta,\varphi,\dot\varphi) }[/math] to the partial differential equation[14] [math]\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} }[/math] after substitution of the second derivatives given above. The metric itself is a solution when written as a differential equation [math]\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 . }[/math]

The separable equation [math]\displaystyle{ \frac{\partial S}{\partial r}-\frac{2}{r}\dot\theta\frac{\partial S}{\partial \dot\theta}=0 }[/math] immediately yields the constant relativistic specific angular momentum [math]\displaystyle{ S_2=L=r^2\dot\theta; }[/math] a third constant obtained from [math]\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 }[/math] is the specific energy (energy per unit rest mass)[15] [math]\displaystyle{ S_3=E=\frac{\dot t(r^2-2Mr+Q^2)}{r^2} + \frac{qQ}{r} . }[/math]

Substituting [math]\displaystyle{ S_2 }[/math] and [math]\displaystyle{ S_3 }[/math] into [math]\displaystyle{ S_1 }[/math] yields the radial equation [math]\displaystyle{ c\int d\,\tau =\int \frac{r^2\,dr}{ \sqrt{r^4(E-1)+2Mr^3-(Q^2+L^2)r^2+2ML^2r-Q^2L^2 } } . }[/math]

Multiplying under the integral sign by [math]\displaystyle{ S_2 }[/math] yields the orbital equation [math]\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^2r-Q^2L^2 } }. }[/math]

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

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

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

Alternative formulation of metric

The metric can be expressed in Kerr–Schild form like this: [math]\displaystyle{ \begin{align} 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{align} }[/math]

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

Notes

  1. Reissner, H. (1916). "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie" (in de). Annalen der Physik 50 (9): 106–120. doi:10.1002/andp.19163550905. Bibcode1916AnP...355..106R. https://zenodo.org/record/1447315. 
  2. Weyl, H. (1917). "Zur Gravitationstheorie" (in de). Annalen der Physik 54 (18): 117–145. doi:10.1002/andp.19173591804. Bibcode1917AnP...359..117W. https://zenodo.org/record/1424330. 
  3. Nordström, G. (1918). "On the Energy of the Gravitational Field in Einstein's Theory". Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam 26: 1201–1208. Bibcode1918KNAB...20.1238N. 
  4. Jeffery, G. B. (1921). "The field of an electron on Einstein's theory of gravitation". Proc. Roy. Soc. Lond. A 99 (697): 123–134. doi:10.1098/rspa.1921.0028. Bibcode1921RSPSA..99..123J. 
  5. Big Think
  6. Thibault Damour: Black Holes: Energetics and Thermodynamics, S. 11 ff.
  7. Ashgar Quadir: The Reissner Nordström Repulsion
  8. Chandrasekhar, S. (1998). The Mathematical Theory of Black Holes (Reprinted ed.). Oxford University Press. p. 205. ISBN 0-19850370-9. http://www.oup.com/us/catalog/general/subject/Physics/Relativity/?view=usa&ci=9780198503705. Retrieved 13 May 2013. "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. Global Structure of the Kerr Family of Gravitational Fields, Physical Review, page 174
  11. Leonard Susskind: The Theoretical Minimum: Geodesics and Gravity, (General Relativity Lecture 4, timestamp: 34m18s)
  12. Eva Hackmann, Hongxiao Xu: Charged particle motion in Kerr–Newmann space-times
  13. Nordebo, Jonatan. "The Reissner-Nordström metric". https://www.diva-portal.org/smash/get/diva2:912393/FULLTEXT01.pdf. 
  14. Smith, Jr., B. R. (2009). "First order partial differential equations in classical dynamics". Am. J. Phys. 77 (12): 1147–1153. doi:10.1119/1.3223358. Bibcode2009AmJPh..77.1147S. 
  15. Misner, C. W. (1973). Gravitation. W. H. Freeman Co.. pp. 656–658. ISBN 0-7167-0344-0. 

References

External links