Schwarzschild and Reissner–Nordström metrics

[Orion1]

A non-rotating $J = 0$ and charge neutral $Q = 0$ spherically symmetric metric is defined by the Schwarzschild metric:
$$c^2 {d \tau}^{2} = \left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \frac{dr^2}{1 - \frac{r_s}{r}} - r^2 d\theta^2 - r^2 \sin^2 \theta \, d\phi^2 \right)$$

The next metric form for a non-rotating $J = 0$ and charged $Q \neq 0$ spherically symmetric metric is defined as:
$$c^2 {d \tau}^{2} = \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^2 dt^2 - \frac{dr^2}{1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}}} - r^2 d\theta^2 - r^2 \sin^2 \theta \, d\phi^2 \right)$$

Which reduces directly to the Schwarzschild metric for $Q = 0$.

In the limit that the charge $Q$ (or equivalently, the length-scale $r_Q$) goes to zero, one recovers the Schwarzschild metric.

However, the formal definition for a non-rotating $J = 0$ and charged $Q \neq 0$ spherically symmetric metric is the Reissner–Nordström metric:
$$c^2 {d \tau}^{2} = \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} - \frac{dr^{2}}{1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}}} - r^{2} d\Omega^{2}$$

Where the solid angle is defined as:
$$d \Omega^2 = d \theta^2 + \sin^2 \theta d \phi^2$$

The Reissner–Nordström metric:
$$\boxed{c^2 {d \tau}^{2} = \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^2 dt^2 - \frac{dr^2}{1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}}} - r^2 d\theta^2 - r^2 \sin^2 \theta \, d\phi^2 \right)}$$

Reference:
http://en.wikipedia.org/wiki/Schwarzschild_metric"
http://en.wikipedia.org/wiki/Reissner-Nordström_black_hole"
http://en.wikipedia.org/wiki/Solid_angle"

 

[George Jones]

$$d \Omega^2 = d \theta^2 + \sin^2 \theta d \phi^2$$

 

[cristo]

It might just be me but... what's the point of this thread?