# Reissner-Nordström Metric

## The Reissner-Nordström Metric

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.

$F_{\mu\nu} = \begin{bmatrix} 0 & E_r/c & 0 & 0 \\ -E_r/c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

$F^{\mu\nu} = \begin{bmatrix} 0 & -E_r/c & 0 & 0 \\ E_r/c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

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.

$\kappa T_{tt} = A\frac{\Phi}{r^4}$ $\kappa T_{rr} = -B\frac{\Phi}{r^4}$ $\kappa T_{\theta\theta} = \frac{\Phi}{r^2}$ $\kappa T_{\phi\phi} = \kappa T_{\theta\theta}\sin^2\theta$

### Field Equations

The field equations are calculated in the same way as for the Schwarzschild interior metric.

$-\frac{A}{r^2B}\bigg(1 - r\frac{B'}{B} - B\bigg) = A\frac{\Phi}{r^4}$ $\frac{1}{r^2}\bigg(1 + r\frac{A'}{A} - B\bigg) = -B\frac{\Phi}{r^4}$ $\frac{r^2}{2B}\bigg(\frac{A''}{A} + \frac{A'}{rA} - \frac{B'}{rB} - \frac{A'(AB)'}{2A^2B}\bigg) = \frac{\Phi}{r^2}$

Rearrange.

$1 - r\frac{B'}{B} - B = -B\frac{\Phi}{r^2}\qquad{(1)}$ $1 + r\frac{A'}{A} - B = -B\frac{\Phi}{r^2}\qquad{(2)}$ $\frac{A''}{A} + \frac{A'}{rA} - \frac{B'}{rB} - \frac{A'(AB)'}{2A^2B} = \frac{2B\Phi}{r^4}\qquad{(3)}$

### Solve the Equations

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 Reissner-Nordström Solution

The metric is now defined.

$ds^2 = c^2d \tau^2 = \bigg(1 - \frac{r_s}{r} + \frac{\Phi}{r^2}\bigg)c^2dt^2 - \bigg(1 - \frac{r_s}{r} + \frac{\Phi}{r^2}\bigg)^{-1}dr^2 - r^2d\theta^2 - r^2 \sin^2\theta d\phi^2$