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)}
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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?