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.
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 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.
\[ F_{\mu\nu} = \begin{bmatrix} 0 & E_r/c & 0 & 0 \\ -E_r/c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]
\[ F^{\mu\nu} = \begin{bmatrix} 0 & -E_r/c & 0 & 0 \\ E_r/c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]
Where:
\[E_r = \frac{Q}{4\pi\epsilon_0r^2}\]
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 \(\Phi\).
\[\Phi = \frac{Q^2}{4\pi\epsilon_0c^4}\]
Then.
\[\kappa T_{tt} = A\frac{\Phi}{r^4}\] \[\kappa T_{rr} = -B\frac{\Phi}{r^4}\] \[\kappa T_{\theta\theta} = \frac{\Phi}{r^2}\] \[\kappa T_{\phi\phi} = \kappa T_{\theta\theta}\sin^2\theta\]
The field equations are calculated in the same way as for the Schwarzschild interior metric.
\[-\frac{A}{r^2B}\bigg(1 - r\frac{B'}{B} - B\bigg) = A\frac{\Phi}{r^4}\] \[\frac{1}{r^2}\bigg(1 + r\frac{A'}{A} - B\bigg) = -B\frac{\Phi}{r^4}\] \[\frac{r^2}{2B}\bigg(\frac{A''}{A} + \frac{A'}{rA} - \frac{B'}{rB} - \frac{A'(AB)'}{2A^2B}\bigg) = \frac{\Phi}{r^2}\]
Rearrange.
\[1 - r\frac{B'}{B} - B = -B\frac{\Phi}{r^2}\](1) \[1 + r\frac{A'}{A} - B = -B\frac{\Phi}{r^2}\](2) \[\frac{A''}{A} + \frac{A'}{rA} - \frac{B'}{rB} - \frac{A'(AB)'}{2A^2B} = \frac{2B\Phi}{r^4}\](3)
Subtract ([1]) from ([2]) and divide by \(r\).
\[\frac{A'}{A} + \frac{B'}{B} = \frac{A'B + AB'}{AB} = \frac{(AB)'}{AB} = 0\]
Integrating gives.
\[AB = k_1\]
Where \(k_1\) is a constant of integration. As the metric must conform to the Minkowski metric \(k_1 = c^2\), \(AB = c^2\).
Divide ([1]) by \(B\) and rearrange.
\[\frac{1}{B} - \frac{rB'}{B^2} = \frac{d}{dr}\bigg(\frac{r}{B}\bigg) = 1 - \frac{\Phi}{r^2}\]
Integrate.
\[\frac{r}{B} = r + \frac{\Phi}{r} + k_2\]
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 metric is now defined.
\[ds^2 = c^2d \tau^2 = \bigg(1 - \frac{r_s}{r} + \frac{\Phi}{r^2}\bigg)c^2dt^2 - \bigg(1 - \frac{r_s}{r} + \frac{\Phi}{r^2}\bigg)^{-1}dr^2 - r^2d\theta^2 - r^2 \sin^2\theta d\phi^2\]