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Orion1
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A non-rotating [itex]J = 0[/itex] and charge neutral [itex]Q = 0[/itex] spherically symmetric metric is defined by the Schwarzschild metric:
[tex]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)[/tex]
The next metric form for a non-rotating [itex]J = 0[/itex] and charged [itex]Q \neq 0[/itex] spherically symmetric metric is defined as:
[tex]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)[/tex]
Which reduces directly to the Schwarzschild metric for [itex]Q = 0[/itex].
Wikipedia said:In the limit that the charge [itex]Q[/itex] (or equivalently, the length-scale [itex]r_Q[/itex]) goes to zero, one recovers the Schwarzschild metric.
However, the formal definition for a non-rotating [itex]J = 0[/itex] and charged [itex]Q \neq 0[/itex] spherically symmetric metric is the Reissner–Nordström metric:
[tex]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}[/tex]
Where the solid angle is defined as:
[tex]d \Omega^2 = d \theta^2 + \sin^2 \theta d \phi^2[/tex]
The Reissner–Nordström metric:
[tex]\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)}[/tex]
Reference:
http://en.wikipedia.org/wiki/Schwarzschild_metric"
http://en.wikipedia.org/wiki/Reissner-Nordström_black_hole"
http://en.wikipedia.org/wiki/Solid_angle"
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