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 metric was discovered by Hans Reissner, Hermann Weyl, Gunnar Nordström and George Barker Jeffery.

The metric

See also: Derivation of the Reissner–Nordström metric

In spherical coordinates (t, r, θ, φ), the line element for the Reissner–Nordström metric is

${\rm {d}}s^{2}=\left(1-{\frac {r_{\rm {s}}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}\right)c^{2}\,{\rm {d}}t^{2}-\left(1-{\frac {r_{\rm {s}}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}\right)^{-1}{\rm {d}}r^{2}-r^{2}\,{\rm {d}}\Omega ^{2},$

where c is the speed of light, t is the time coordinate (measured by a stationary clock at infinity), r is the radial coordinate, dΩ² is a 2-sphere defined by

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

r_s is the Schwarzschild radius of the body given by

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

and r_Q is a characteristic length scale given by

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

Here 1/4πε₀ is Coulomb force constant K.

The total mass of the central body and its irreducible mass are related by

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

The difference between M and M_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 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 that 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 a 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. As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component g^rr diverges; that is, where

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

This equation has two solutions:

$r_{\pm }={\frac {1}{2}}\left(r_{\rm {s}}\pm {\sqrt {r_{\rm {s}}^{2}-4r_{\rm {Q}}^{2}}}\right).$

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 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). 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. Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic potential is

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

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 θ dφ in the electromagnetic potential.

Gravitational time dilation

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

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

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

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

Christoffel symbols

The Christoffel symbols

$\Gamma _{jk}^{i}=\sum _{s=0}^{3}\ {\frac {g^{is}}{2}}\left({\frac {\partial g_{j}}{\partial x^{k}}}+{\frac {\partial g_{sk}}{\partial x^{j}}}-{\frac {\partial g_{jk}}{\partial x^{s}}}\right)$

with the indicies

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

give the nonvanishing expressions

$\Gamma _{10}^{0}={\frac {Mr+Q^{2}}{r\left(r(r-2M)-Q^{2}\right)}}$

$\Gamma _{00}^{1}={\frac {\left(Mr+Q^{2}\right)\left(r(2M-r)+Q^{2}\right)}{r^{5}}}$

$\Gamma _{11}^{1}={\frac {Mr+Q^{2}}{2Mr^{2}+Q^{2}r-r^{3}}}$

$\Gamma _{22}^{1}=2M-{\frac {Q^{2}}{r}}+r$

$\Gamma _{33}^{1}={\frac {\sin ^{2}\theta \left(r(r-2M)-Q^{2}\right)}{r}}$

$\Gamma _{21}^{2}=r^{-1}$

$\Gamma _{33}^{2}=-\sin \theta \cos \theta$

$\Gamma _{31}^{3}=r^{-1}$

$\Gamma _{32}^{3}=\cot \theta$

Given the Christoffel symbols, one can compute the geodesics of a test-particle.

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 Ω instead of θ and 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

${\ddot {x}}^{i}=-\sum _{=0}^{3}\ \sum _{k=0}^{3}\ \Gamma _{jk}^{i}\ {{\dot {x}}^{j}}\ {{\dot {x}}^{k}}+q\ {F^{ik}}\ {{\dot {x}}_{k}}$

which gives

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

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

${\ddot {\Omega }}=-{\frac {2\ {\dot {\Omega }}\ {\dot {r}}}{r}}$

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

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

The first derivatives ${\dot {x}}^{i}$ and the contravariant components of the local 3-velocity $v^{i}$ are related by

${\dot {x}}^{i}={\frac {v^{i}}{\sqrt {(1-v^{2})\ |g_{ii}|}}}.$

which gives the initial conditions

${\dot {r}}={\frac {v_{\parallel }{\sqrt {r\ (r-2M)-Q^{2}}}}{r{\sqrt {(1-v^{2})}}}}$

${\dot {\Omega }}={\frac {v_{\perp }}{r{\sqrt {(1-v^{2})}}}}$

The specific orbital energy

$E={\frac {\sqrt {Q^{2}+(r-2)r}}{r{\sqrt {1-v^{2}}}}}$

and the specific relative angular momentum

$L={\frac {v_{\perp }\ r}{\sqrt {1-v^{2}}}}$

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

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

Alternative formulation of metric

The metric can alternatively be expressed like this:

$g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!$

$f={\frac {G}{r^{2}}}\left[2Mr-Q^{2}\right]$

$\mathbf {k} =(k_{x},k_{y},k_{z})=\left({\frac {x}{r}},{\frac {y}{r}},{\frac {z}{r}}\right)$

$k_{0}=1.\!$

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 η is the Minkowski tensor.

See also

• Black hole electron

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. Retrieved 27 April 2013.

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.