Reissner-Nordström metriği: Revizyonlar arasındaki fark
| [kontrol edilmemiş revizyon] | [kontrol edilmemiş revizyon] |
İçerik silindi İçerik eklendi
Ramazantunc (mesaj | katkılar) Değişiklik özeti yok |
Ramazantunc (mesaj | katkılar) Değişiklik özeti yok |
||
18. satır:
| [[Kerr–Newman metric|Kerr–Newman]]
|}
where ''Q'' represents the body's [[electric charge]] and ''J'' represents its spin [[angular momentum]].
==The metric==
In [[spherical coordinates]] (''t'', ''r'', θ, φ), the [[line element]] for the Reissner–Nordström metric is
:<math>
ds^2 =
\left( 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2} \right) c^2\, dt^2 - \frac{1}{1 - r_\mathrm{S}/r + r_Q^2/r^2}\, dr^2 - r^2\, d\theta^2 - r^2 \sin^2 \theta \, d\phi^2,</math>
where ''c'' is the [[speed of light]], ''t'' is the time coordinate (measured by a stationary clock at infinity), ''r'' is the radial coordinate, ''r''<sub>S</sub> is the [[Schwarzschild radius]] of the body given by
:<math>
r_{s} = \frac{2GM}{c^2},
</math>
and ''r<sub>Q</sub>'' is a characteristic length scale given by
:<math>
r_{Q}^{2} = \frac{Q^2 G}{4\pi\varepsilon_{0} c^4}.
</math>
Here 1/4πε<sub>0</sub> is [[Coulomb's law|Coulomb force constant]].<ref name="landau_1975" >Landau 1975.</ref>
In the limit that the charge ''Q'' (or equivalently, the length-scale ''r''<sub>''Q''</sub>) goes to zero, one recovers the [[Schwarzschild metric]]. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio ''r''<sub>S</sub>/''r'' goes to zero. In that limit that both ''r<sub>Q</sub>''/''r'' and ''r''<sub>S</sub>/''r'' go to zero, the metric becomes the [[Minkowski metric]] for [[special relativity]].
In practice, the ratio ''r''<sub>S</sub>/''r'' is often extremely small. For example, the Schwarzschild radius of the [[Earth]] is roughly 9 [[millimeter|mm]] (3/8 [[inch]]), whereas a [[satellite]] in a [[geosynchronous orbit]] has a radius ''r'' that is roughly four billion times larger, at 42,164 [[kilometer|km]] (26,200 [[mile]]s). Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio only becomes large close to [[black hole]]s and other ultra-dense objects such as [[neutron star]]s.
==Charged black holes==
Although charged black holes with ''r<sub>Q</sub>'' ≪ ''r''<sub>S</sub> are similar to the [[Schwarzschild black hole]], they have two horizons: the [[event horizon]] and an internal [[Cauchy horizon]].<ref>{{cite book |last=Chandrasekhar |first=S. |authorlink=Subrahmanyan Chandrasekhar |title=The Mathematical Theory of Black Holes |year=1998 |publisher=Oxford University Press |isbn=0-19850370-9 |edition=Reprinted |url=http://www.oup.com/us/catalog/general/subject/Physics/Relativity/?view=usa&ci=9780198503705 |accessdate=13 May 2013 |page=205 |quote=And finally, the fact that the Reissner-Nordström solution has two horizons, an external event horizon and an internal 'Cauchy horizon,' provides a convenient bridge to the study of the Kerr solution in the subsequent chapters.}}</ref> As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component ''g<sup>rr</sup>'' diverges; that is, where
:<math> 0 = 1/g^{rr} = 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2}.</math>
This equation has two solutions:
:<math>
r_\pm = \frac{1}{2}\left(r_{s} \pm \sqrt{r_{s}^2 - 4r_{Q}^2}\right).
</math>
These concentric [[event horizon]]s become [[Degenerate energy level|degenerate]] for 2''r<sub>Q</sub>'' = ''r''<sub>S</sub>, which corresponds to an [[extremal black hole]]. Black holes with 2''r<sub>Q</sub>'' > ''r''<sub>S</sub> are believed not to exist in nature because they would contain a [[naked singularity]]; their appearance would contradict [[Roger Penrose]]'s [[cosmic censorship hypothesis]] which is generally believed to be true.{{citation needed|date=January 2013}} Theories with [[supersymmetry]] usually guarantee that such "superextremal" black holes cannot exist.
| |||