Reissner-Nordstrom Metric
Get Reissner%E2%80%93Nordstrom Metric essential facts below. View Videos or join the Reissner%E2%80%93Nordstrom Metric discussion. Add Reissner%E2%80%93Nordstrom Metric to your PopFlock.com topic list for future reference or share this resource on social media.
Reissner%E2%80%93Nordstrom 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]

## The metric

In spherical coordinates ${\displaystyle (t,r,\theta ,\phi )}$, the Reissner–Nordström metric (aka the line element) is

${\displaystyle g=\left(1-{\frac {r_{s}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}\right)c^{2}\,dt^{2}-\left(1-{\frac {r_{s}}{r}}+{\frac {r_{Q}^{2}}{r^{2}}}\right)^{-1}dr^{2}-r^{2}\,g_{\Omega },}$

where ${\displaystyle c}$ is the speed of light, ${\displaystyle t}$ is the time coordinate (measured by a stationary clock at infinity), ${\displaystyle r}$ is the radial coordinate, and ${\displaystyle g_{\Omega }}$ is the standard metric on the unit radius 2-sphere which if coordinatised by ${\displaystyle \Omega =(\theta ,\phi )}$ reads

${\displaystyle g_{\Omega }=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}$

${\displaystyle r_{s}}$ is the Schwarzschild radius of the body given by

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

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

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

Here ${\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}}$ is Coulomb force constant ${\displaystyle K}$.

The total mass of the central body and its irreducible mass are related by[5][6]

${\displaystyle M_{\rm {irr}}={\frac {c^{2}}{G}}{\sqrt {\frac {r_{+}^{2}}{2}}}\ \to \ M={\frac {Q^{2}K}{4GM_{\rm {irr}}}}+M_{\rm {irr}}}$.

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 ${\displaystyle r_{s}/r}$ goes to zero. In the limit that both ${\displaystyle r_{Q}/r}$ and ${\displaystyle r_{s}/r}$ go to zero, the metric becomes the Minkowski metric for special relativity.

In practice, the ratio ${\displaystyle r_{s}/r}$ 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 a 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 rQ rs are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon.[7] As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component grr diverges (is not g_{rr} divergent, or equivalently g^{rr}=0?); that is, where

${\displaystyle 0={\frac {1}{g^{rr}}}=1-{\frac {r_{\rm {s}}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}.}$

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 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[8] (the term under the square root becomes negative). 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.[9] Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

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

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 Pcos ? d? in the electromagnetic potential.[clarification needed]

## Gravitational time dilation

The gravitational time dilation in the vicinity of the central body is given by

${\displaystyle \varsigma ={\sqrt {|g^{tt}|}}={\sqrt {\frac {r^{2}}{Q^{2}+(r-2M)r}}}}$

which relates to the local radial escape-velocity of a neutral particle

${\displaystyle v_{\rm {esc}}={\frac {\sqrt {\varsigma ^{2}-1}}{\varsigma }}.}$

## 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

${\displaystyle \{0,\ 1,\ 2,\ 3\}\to \{t,\ r,\ \theta ,\ \phi \}}$

give the nonvanishing expressions

${\displaystyle \Gamma _{10}^{0}={\frac {Mr+Q^{2}}{r\left(r(r-2M)-Q^{2}\right)}}}$
${\displaystyle \Gamma _{00}^{1}={\frac {\left(Mr+Q^{2}\right)\left(r(2M-r)+Q^{2}\right)}{r^{5}}}}$
${\displaystyle \Gamma _{11}^{1}={\frac {Mr+Q^{2}}{2Mr^{2}+Q^{2}r-r^{3}}}}$
${\displaystyle \Gamma _{22}^{1}=2M-{\frac {Q^{2}}{r}}+r}$
${\displaystyle \Gamma _{33}^{1}={\frac {\sin ^{2}\theta \left(r(r-2M)-Q^{2}\right)}{r}}}$
${\displaystyle \Gamma _{21}^{2}=r^{-1}}$
${\displaystyle \Gamma _{33}^{2}=-\sin \theta \cos \theta }$
${\displaystyle \Gamma _{31}^{3}=r^{-1}}$
${\displaystyle \Gamma _{32}^{3}=\cot \theta }$

Given the Christoffel symbols, one can compute the geodesics of a test-particle.[10][11]

## Equations of motion

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 further use ? instead of ? and ?. 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 gives

${\displaystyle {\ddot {t}}={\frac {{\dot {r}}\ (q\ r\ Q+2(Q^{2}-r){\dot {t}})}{r((r-2)r+Q^{2})}}}$
${\displaystyle {\ddot {r}}={\frac {((r-2)\ r+Q^{2})(q\ r\ Q\ {\dot {t}}+r^{4}{\dot {\Omega }}^{2}+(Q^{2}-r)\ {\dot {t}}^{2})}{r^{5}}}+{\frac {(r-Q^{2}){\dot {r}}^{2}}{r\ ((r-2)\ r+Q^{2})}}}$
${\displaystyle {\ddot {\Omega }}=-{\frac {2\ {\dot {\Omega }}\ {\dot {r}}}{r}}}$

The total time dilation between the test-particle and an observer at infinity is

${\displaystyle {\dot {t}}={\frac {q\ Q\ r^{3}+E\ r^{4}}{r^{2}\ (r^{2}-2r+Q^{2})}}}$

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

${\displaystyle {\dot {r}}={\frac {v_{\parallel }{\sqrt {r\ (r-2M)-Q^{2}}}}{r{\sqrt {(1-v^{2})}}}}}$
${\displaystyle {\dot {\Omega }}={\frac {v_{\perp }}{r{\sqrt {(1-v^{2})}}}}}$
${\displaystyle E={\frac {\sqrt {Q^{2}+(r-2)r}}{r{\sqrt {1-v^{2}}}}}}$
${\displaystyle L={\frac {v_{\perp }\ r}{\sqrt {1-v^{2}}}}}$

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

${\displaystyle v={\sqrt {v_{\perp }^{2}+v_{\parallel }^{2}}}={\sqrt {\frac {E^{2}r^{2}-Q^{2}-r^{2}+2r}{E^{2}r^{2}}}}.}$

## Alternative formulation of metric

The metric can alternatively be expressed like this:

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}$
${\displaystyle f={\frac {G}{r^{2}}}\left[2Mr-Q^{2}\right]}$
${\displaystyle \mathbf {k} =(k_{x},k_{y},k_{z})=\left({\frac {x}{r}},{\frac {y}{r}},{\frac {z}{r}}\right)}$
${\displaystyle k_{0}=1.\!}$

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 ? is the Minkowski tensor.

## Notes

1. ^ Reissner, H. (1916). "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie". Annalen der Physik (in German). 50 (9): 106-120. Bibcode:1916AnP...355..106R. doi:10.1002/andp.19163550905.
2. ^ Weyl, H. (1917). "Zur Gravitationstheorie". Annalen der Physik (in German). 54 (18): 117-145. Bibcode:1917AnP...359..117W. doi:10.1002/andp.19173591804.
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.
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. Bibcode:1921RSPSA..99..123J. doi:10.1098/rspa.1921.0028.
5. ^
6. ^ Ashgar Quadir: The Reissner Nordström Repulsion
7. ^ Chandrasekhar, S. (1998). The Mathematical Theory of Black Holes (Reprinted ed.). Oxford University Press. p. 205. ISBN 0-19850370-9. Archived from the original on 29 April 2013. Retrieved 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.
8. ^ Andrew Hamilton: The Reissner Nordström Geometry Archived 2007-07-07 at the Wayback Machine (Casa Colorado)
9. ^ Carter, Brandon. Global Structure of the Kerr Family of Gravitational Fields, Physical Review, page 174
10. ^ Leonard Susskind: The Theoretical Minimum: Geodesics and Gravity, (General Relativity Lecture 4, timestamp: 34m18s)
11. ^ Eva Hackmann, Hongxiao Xu: Charged particle motion in Kerr-Newmann space-times