Reissner–Nordström metric

In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations that 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,<ref>Template:Cite journal</ref> Hermann Weyl,<ref>Template:Cite journal</ref> Gunnar Nordström<ref>Template:Cite journal</ref> and George Barker Jeffery<ref>Template:Cite journal</ref> independently.<ref>Template:Cite web</ref>

Metric

In spherical coordinates Template:Tmath, the Reissner–Nordström metric (i.e. the line element) is

<math>

ds^2</math> <math>= c^2\, d\tau^2</math> <math> = \left( 1 - \frac{r_\text{s}}{r} + \frac{r_{\rm Q}^2}{r^2} \right) c^2\, dt^2</math> <math> -\left( 1 - \frac{r_\text{s}}{r} + \frac{r_Q^2}{r^2} \right)^{-1} \, dr^2</math> <math> - ~ r^2 \, d\theta^2</math> <math> - ~ r^2\sin^2\theta \, d\varphi^2 ,</math> where

• <math>c</math> is the speed of light
• <math>\tau</math> is the proper time
• <math>t</math> is the time coordinate (measured by a stationary clock at infinity).
• <math>r</math> is the radial coordinate
• <math>(\theta, \varphi)</math> are the spherical angles
• <math>r_\text{s}</math> is the Schwarzschild radius of the body given by <math>\textstyle r_\text{s} = \frac{2GM}{c^2}</math>
• <math>r_Q</math> is a characteristic length scale given by <math>\textstyle r_Q^2 = \frac{Q^2 G}{4\pi\varepsilon_0 c^4}</math>
• <math>\varepsilon_0</math> is the electric constant.

The total mass of the central body and its irreducible mass are related by<ref>Thibault Damour: Black Holes: Energetics and Thermodynamics, S. 11 ff.</ref><ref name="quadir">Template:Cite journal</ref>

<math>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>M</math> and <math>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>Q</math> (or equivalently, the length scale Template:Tmath) 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>r_\text{s}/r</math> goes to zero. In the limit that both <math>r_Q/r</math> and <math>r_\text{s}/r</math> go to zero, the metric becomes the Minkowski metric for special relativity.

In practice, the ratio <math>r_\text{s}/r</math> is often extremely small. For example, the Schwarzschild radius of the Earth is roughly Template:Val. 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.<ref>Template:Cite book</ref> As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component <math>g_{rr}</math> diverges; that is, where <math display=block> 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 display=block>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 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).<ref name="hamilton">Andrew Hamilton: The Reissner Nordström Geometry (Casa Colorado)</ref> 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.<ref>Template:Cite journal</ref> Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic potential is <math display=block>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 dφ in the electromagnetic potential.Template:Clarify

Gravitational time dilation

The gravitational time dilation in the vicinity of the central body is given by <math display=block>\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 display=block>v_{\rm esc}=\frac{\sqrt{\gamma^2-1}}{\gamma}. </math>

Christoffel symbols

The Christoffel symbols <math display=block> \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 display=block>\{ 0, \ 1, \ 2, \ 3 \} \to \{ t, \ r, \ \theta, \ \varphi \}</math> give the nonvanishing expressions <math display=block> \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}{r (Q^2 -2 M r+r^2)} \\[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.<ref>Leonard Susskind: The Theoretical Minimum: Geodesics and Gravity, (General Relativity Lecture 4, timestamp: 34m18s)</ref><ref>Template:Cite journal</ref>

Tetrad form

Instead of working in the holonomic basis, one can perform efficient calculations with a tetrad.<ref>Template:Cite book</ref> Let <math> {\bf e}_I = e_{\mu I} </math> be a set of one-forms with internal Minkowski index Template:Tmath, such that Template:Tmath. The Reissner metric can be described by the tetrad

<math> {\bf e}_0 = G^{1/2} \, dt </math>

<math> {\bf e}_1 = G^{-1/2} \, dr </math>

<math> {\bf e}_2 = r \, d\theta </math>

<math> {\bf e}_3 = r \sin \theta \, d\varphi </math>

where Template:Tmath. The parallel transport of the tetrad is captured by the connection one-forms Template:Tmath. These have only 24 independent components compared to the 40 components of Template:Tmath. The connections can be solved for by inspection from Cartan's equation Template:Tmath, where the left hand side is the exterior derivative of the tetrad, and the right hand side is a wedge product.

<math> \boldsymbol \omega_{10} = \frac12 \partial_r G \, dt</math>

<math> \boldsymbol \omega_{20} = \boldsymbol \omega_{30} = 0</math>

<math> \boldsymbol \omega_{21} = - G^{1/2} \, d\theta</math>

<math> \boldsymbol \omega_{31} = - \sin \theta \, G^{1/2} \, d \varphi</math>

<math> \boldsymbol \omega_{32} = - \cos \theta \, d\varphi</math>

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

Equations of motion

<ref name="Nordebo">Template:Cite web</ref>

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 display=block> \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 display=block>\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 display=block>\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 display=block>\ddot \theta = -\frac{2 \ \dot\theta \ \dot{r}}{r} .</math>

All total derivatives are with respect to proper time Template:Tmath.

Constants of the motion are provided by solutions <math>S (t,\dot t,r,\dot r,\theta,\dot\theta,\varphi,\dot\varphi) </math> to the partial differential equation<ref>Template:Cite journal</ref> <math display=block> 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 display=block>

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 display=block> \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 display=block> S_2=L=r^2\dot\theta; </math> a third constant obtained from <math display=block> \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)<ref>Template:Cite book</ref> <math display=block>S_3=E=\frac{\dot t(r^2-2Mr+Q^2)}{r^2} + \frac{qQ}{r} .</math>

Substituting <math>S_2</math> and <math>S_3</math> into <math>S_1</math> yields the radial equation <math display=block>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>S_2</math> yields the orbital equation <math display=block>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 display=block>\gamma= \frac{q \ Q \ r^3 + E \ r^4}{r^2 \ (r^2-2 r+Q^2)} .</math>

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

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

• Black hole electron

References

• Template:Cite book
• Template:Cite book

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.