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 ${\displaystyle (t,r,\theta ,\varphi )}$, the Reissner–Nordström metric (i.e. the line element) is

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, and ${\displaystyle r_{\text{s}}}$ is the Schwarzschild radius of the body given by and ${\displaystyle r_{Q}}$ is a characteristic length scale given by Here, ${\displaystyle \varepsilon _{0}}$ is the electric constant.

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

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

This equation has two solutions:

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).^[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

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

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

Christoffel symbols[]

The Christoffel symbols

with the indices give the nonvanishing expressions

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

which yields

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^[14]

after substitution of the second derivatives given above. The metric itself is a solution when written as a differential equation

The separable equation

immediately yields the constant relativistic specific angular momentum a third constant obtained from is the specific energy (energy per unit rest mass)^[15]

Substituting ${\displaystyle S_{2}}$ and ${\displaystyle S_{3}}$ into ${\displaystyle S_{1}}$ yields the radial equation

Multiplying under the integral sign by ${\displaystyle S_{2}}$ yields the orbital equation

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

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

which gives the initial conditions

The specific orbital energy

and the specific relative angular momentum 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

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

Notes[]

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