2.1. The black hole gravity - davidar/scholarpedia GitHub Wiki

The black hole gravitational field is described by three parameters: mass <math>M\ ,</math> angular momentum <math>J</math> and charge <math>Q\ .</math> It is convincingly argued that the astrophysical black holes relevant for accretion discs are uncharged, <math>Q = 0\ .</math> They are described by the stationary and axially symmetric Kerr geometry, with the metric <math>g_{\mu\nu}</math> given in the spherical Boyer-Lindquist coordinates <math>t, \phi, r, \theta</math> by the explicitly known functions of the radius <math>r</math> and the polar angle <math>\theta\ ,</math> which are listed in the table below. The table also gives the contravariant form of the metric, <math>g^{\mu\nu}\ ,</math> defined by <math>g^{\mu\beta}\,g_{\nu\beta} = \delta^{\mu}_{~\nu}\ .</math> It is defined, <math>\Delta = r^2 - 2Mr + a^2\ ,</math> <math>\Sigma = r^2 + a^2\cos^2\theta\ .</math> The signature <math>(+\,-\,-\,-)</math> is used.

The mass and angular momentum have been rescaled into the <math>c = G = 1</math> units, <math>M \rightarrow GM/c^2\ ,</math> <math>J \rightarrow a = J/c\ .</math> For a proper black hole solution it must be <math>\vert a \vert \le M\ ,</math> and the metric with <math>\vert a \vert > M</math> corresponds to a naked singularity. The Penrose cosmic censor hypothesis (unproved) states that there are no naked singularities in the Universe.

   <table cellpadding="0" cellspacing="1" border="background: #999999"></table> <tr style="background: #ffffff;&amp;amp;amp;#10;color: #000000"></tr> <td colspan="5" style="background:&amp;amp;amp;amp;#10;#ffffff" valign="middle" align="center">

<td width="540" colspan="4" style="middle" align="center"> <math>g^{\mu\nu}</math></td> </tr></td width="540"> <math>t</math> <td width="135" style="background: #ffffff" valign="center"> <math>\phi</math> </td></td width="135"> <math>r</math> <math>\theta</math> <td width="1" style="background: #999999" valign="center"></td></td width="1"> <math>t</math> <td width="135" style="background: #ffffff" valign="center"> <math>\phi</math> </td> <td valign="middle" align="center"> <math>r</math> </td> <td style="middle" align="center"> <math>\theta</math> </td> </tr></td width="135"> <math>t</math> <math>1 - 2\,M\,r/\Sigma</math> <math>4\,M\,a\,r\sin^2\theta/\Sigma</math> <math>0</math> <math>0</math>

<math>(r^2 + a^2)^2/\Sigma\,\Delta</math>
<math>- a^2\Delta\sin^2\theta/\Sigma\,\Delta</math> <math>2M\,\,a\,r/\Sigma\,\Delta</math> <math>0</math> <math>0</math> <math>\phi</math> <math>4\,M\,a\,r\sin^2\theta/\Sigma</math> <math>-(r^2 + a^2)\sin^2\theta</math>
<math>-2\,M\,a^2r\sin^4\theta/\Sigma</math> <math>0</math> <math>0</math>

<math>2M\,\,a\,r/\Sigma\,\Delta</math> <math>-\frac{\Delta-a^2\sin^2\theta}{\Delta\Sigma\sin^2\theta}</math> <math>0</math> <math>0</math> <math>r</math> <math>0</math> <math>0</math> <td width="135" style="background: #ffffff" valign="center"> <math>-\Sigma/\Delta</math> </td></td width="135"> <math>0</math>

<math>0</math> <math>0</math> <td width="135" style="background: #ffffff" valign="center"> <math>-\Delta/\Sigma</math> </td> <td valign="middle" align="center"> <math>0</math> </td> </tr></td width="135"> <math>\theta</math> <math>0</math> <math>0</math> <td width="135" style="background: #ffffff" valign="center"> <math>0</math> </td></td width="135"> <math>-\Sigma</math>

<td width="135" style="background: #ffffff" valign="center"> <math>0</math> </td></td width="135"> <math>0</math> <math>0</math> <td width="135" style="background: #ffffff" valign="center"> <math>-1/\Sigma</math> </td> </tr></table></td width="135"> In any stationary and axially symmetric spacetime, and in particular in the Kerr geometry, for matter rotating on circular orbits with four velocity <math>u^{\nu} = (u^t, u^{\phi})</math> it is <math>\Omega = u^{\phi}/u^t</math> and <math>j = - u_{\phi}/u_t\ ,</math> from which (and <math>u^{\nu}\,u_{\nu} = 1 </math>) it follows that,

<math> \Omega = -\frac{j\,g_{tt} + g_{t\phi}}{j\,g_{t\phi} + g_{\phi\phi}}, ~~~ j =-\frac{\Omega\,g_{\phi\phi} +

g_{t\phi}}{\Omega\,g_{t\phi} + g_{tt}}, ~~~ U_{eff} = -\frac{1}{2} \ln \left( g^{tt} - 2j\,g^{t\phi} + j^2\,g^{\phi \phi}\right). </math> <math>(3.2)</math></tr></table> From equations (3.1), (3.2) and <math>dR^2 = g_{rr}dr^2\ ,</math> <math>dZ^2 = g_{\theta\theta}d\theta^2\ ,</math> one derives that the Keplerian frequency <math>\Omega_K</math> and the two epicyclic frequencies (radial <math>\omega_R</math> and vertical <math>\omega_Z</math>) equal </tr></table> Here the dimensionless <math>{r_*}</math> and <math>\vert{a_*}\vert \le 1</math> are defined by <math>{r_*} = rc^2/GM\ ,</math> <math>{a_*} = Jc/GM^2\ .</math> In the strong gravity, i.e. for <math>{r_*} \sim 1\ ,</math> the three frequencies scale as <math>1/M\ .</math> The radial epicyclic oscillations, <math>\delta r(t) \sim \exp(- i\,\omega_R\,t)\ ,</math> become dynamically unstable at <math>r < ISCO\ ,</math> because there <math>\omega_R^2(r) < 0\ .</math> Stable circular Keplerian orbits exist only with radii greater than the radius of ISCO (the innermost stable circular orbit radius). All Keplerian orbits closer to the black hole than ISCO are unstable: without an extra support by non-gravitational forces (i.e. pressure or magnetic field) matter cannot stay there orbiting freely, but instead it must fall down into the black hole. This strong-field property of Einstein's gravity, absent in Newton's theory, is the most important physical effect in the black hole accretion disc physics.

<math> \Omega_K = \frac{c^3}{GM}\left( {r_}^{3/2} + {a_}\right)^{-1}, ~~~ \omega_R^2 = \Omega_K^2 \left( 1 - 6{r_}^{-1} + 8{a_}\,{r_}^{-3/2} - 3{a_}^2\,{r_}^2 \right), ~~~ \omega_Z^2 = \Omega_K^2 \left( 1 - 4{a_}\,{r_}^{-3/2} + 3{_a}^2\,{r_*}^{-2} \right). </math>	<math>(3.3)</math>

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2.1. The black hole gravity - davidar/scholarpedia GitHub Wiki

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