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 and charge Q.
The metric was discovered between 1916 and 1921 by Hans Reissner, Hermann Weyl, Gunnar Nordström, and George Barker Jeffery independently.
Define the metric
The metric can defined in the same way as for the Schwarzschild outer metric.
\[ds^2 = A(r)dt^2 - B(r)dr^2 - r^2d\theta^2 - r^2 \sin^2\theta d\phi^2\]
Where \(A\) and \(B\) are unknown functions of \(r\). The interior metric must agree with the outer metric at \(r = r_g\).
The metric and its inverse are now defined to be:
\[ g_{\alpha\beta} =
\begin{bmatrix}
A & 0 & 0 & 0\\
0 & -B & 0 & 0\\
0 & 0 & -r^2 & 0\\
0 & 0 & 0 & -r^2 \sin^2 \theta
\end{bmatrix}\]
\[ g^{\alpha\beta} =
\begin{bmatrix}
\frac{1}{A} & 0 & 0 & 0\\
0 & -\frac{1}{B} & 0 & 0\\
0 & 0 & -\frac{1}{r^2} & 0\\
0 & 0 & 0 & -\frac{1}{r^2 \sin^2 \theta}
\end{bmatrix}\]
The Stress-Energy Tensor
The stress-energy tensor is derived from the electromagnetic tensor. In this case there is no magnetic field and the electric field only has a radial component.
In spherical polar coordinates.
\[F_{\mu\nu} =
\begin{bmatrix}
0 & E_r/c & 0 & 0 \\
-E_r/c & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}\]
\[F^{\mu\nu} =
\begin{bmatrix}
0 & -E_r/c & 0 & 0 \\
E_r/c & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}\]
\[E_r = \frac{Q}{4\pi\epsilon_0r^2}\]
The stress-energy tensor is:
\[T_{\alpha\beta} = \frac{1}{\mu_0}\bigg(\frac{1}{4}g_{\alpha\beta}F_{\mu\nu}F^{\mu\nu} - g_{\beta\nu}F_{\alpha\mu}F^{\nu\mu}\bigg)\]
\[T_{tt} = -\frac{1}{2\mu_0}AF_{01}F^{01}\]
\[T_{rr} = \frac{1}{2\mu_0}BF_{01}F^{01}\]
\[T_{\theta\theta} = -\frac{1}{2\mu_0}r^2F_{01}F^{01}\]
\[T_{\phi\phi} = T_{\theta\theta}\sin^2\theta\]
Extract the common term, multiply by the Einstein constant and simplify. Given \(\epsilon_0\mu_0 = c^{-2}\).
\[\frac{\kappa}{2\mu_0}F_{01}F^{01} = -\frac{8\pi G}{2\mu_0c^4}\frac{E_r^2}{c^2} = - \frac{4\pi G}{\mu_0c^6} \bigg(\frac{Q}{4\pi\epsilon_0r^2}\bigg)^2 = -\frac{GQ^2}{4\pi\epsilon_0c^4r^4}\]
Define a constant \(\Phi\).
\[\Phi = \frac{Q^2}{4\pi\epsilon_0c^4}\]
\[\kappa T_{tt} = A\frac{\Phi}{r^4}\]
\[\kappa T_{rr} = -B\frac{\Phi}{r^4}\]
\[\kappa T_{\theta\theta} = \frac{\Phi}{r^2}\]
\[\kappa T_{\phi\phi} = \kappa T_{\theta\theta}\sin^2\theta\]
Field Equations
The field equations are calculated in the same way as for the Schwarzschild interior metric.
\[-\frac{A}{r^2B}\bigg(1 - r\frac{B'}{B} - B\bigg) = A\frac{\Phi}{r^4}\]
\[\frac{1}{r^2}\bigg(1 + r\frac{A'}{A} - B\bigg) = -B\frac{\Phi}{r^4}\]
\[\frac{r^2}{2B}\bigg(\frac{A''}{A} + \frac{A'}{rA} - \frac{B'}{rB} - \frac{A'(AB)'}{2A^2B}\bigg) = \frac{\Phi}{r^2}\]
\[1 - r\frac{B'}{B} - B = -B\frac{\Phi}{r^2}\]
\[1 + r\frac{A'}{A} - B = -B\frac{\Phi}{r^2}\]
\[\frac{A''}{A} + \frac{A'}{rA} - \frac{B'}{rB} - \frac{A'(AB)'}{2A^2B} = \frac{2B\Phi}{r^4}\]
Solve the Equations
\[\frac{A'}{A} + \frac{B'}{B} = \frac{A'B + AB'}{AB} = \frac{(AB)'}{AB} = 0\]
Where \(k_1\) is a constant of integration. As the metric must conform to the Minkowski metric \(k_1 = c^2\), \(AB = c^2\).
\[\frac{1}{B} - \frac{rB'}{B^2} = \frac{d}{dr}\bigg(\frac{r}{B}\bigg) = 1 - \frac{\Phi}{r^2}\]
\[\frac{r}{B} = r + \frac{\Phi}{r} + k_2\]
Where \(k_2\) is a constant of integration. This constant is a function of the mass of the body. As with the Schwarzschild outer solution \(k_2 = -r_s\).
The Reissner-Nordström Solution
The metric is now defined.
\[ds^2 = c^2d \tau^2 = \bigg(1 - \frac{r_s}{r} + \frac{\Phi}{r^2}\bigg)c^2dt^2 - \bigg(1 - \frac{r_s}{r} + \frac{\Phi}{r^2}\bigg)^{-1}dr^2 - r^2d\theta^2 -
r^2 \sin^2\theta d\phi^2\]
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