Difference between revisions of "Kerr black hole"

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(Created page with "==General axially symmetric metric== ==Limiting cases== ==Horizons and singularity== ==Stationary limit== ==Ergosphere and the Penrose process== ==Integrals of ...")
 
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Kerr solution is the solution of Einstein's equations in vacuum that describes a rotating black hole (or the metric outside of a rotating axially symmetric body) \cite{Kerr63}. In the Boyer-Lindquist coordinates \cite{BoyerLindquist67} it takes the form
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\begin{align}\label{Kerr}
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&&ds^2=\bigg(1-\frac{2\mu r}{\rho^2}\bigg)dt^2
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+\frac{4\mu a \,r\sin^{2}\theta}{\rho^2}
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\;dt\,d\varphi
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-\frac{\rho^2}{\Delta}\;dr^2-\rho^2\, d\theta^2
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+\qquad\nonumber\\
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&&-\bigg(
 +
r^2+a^2+\frac{2\mu r\,a^2 \,\sin^{2}\theta}{\rho^2}
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\bigg) \sin^2 \theta\;d\varphi^2;\\
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\label{Kerr-RhoDelta}
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&&\text{where}\quad
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\rho^2=r^2+a^2 \cos^2 \theta,\qquad
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\Delta=r^2-2\mu r+a^2.
 +
\end{align}
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Here $\mu$ is the black hole's mass, $J$ its angular momentum, $a=J/\mu$; $t$ and $\varphi$ are time and usual azimuth angle, while $r$ and $\theta$ are some coordinates that become the other two coordinates of the spherical coordinate system at $r\to\infty$.
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==General axially symmetric metric==
 
==General axially symmetric metric==
    
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A number of properties of the Kerr solution can be understood qualitatively without use of its specific form. In this problem we consider the axially symmetric metric of quite general kind
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\begin{equation}\label{AxiSimmMetric}
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ds^2=A dt^2-B(d\varphi-\omega dt)^{2}-
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C\,dr^2-D\,d\theta^{2},\end{equation}
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where functions $A,B,C,D,\omega$ depend only on $r$ and $\theta$.
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<div id="BlackHole53"></div>
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=== Problem 1. ===
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Find the components of metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$.
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<div class="NavFrame collapsed">
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   <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">
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The metric is:
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\begin{equation}\label{AxiSimmMetricmatrix}
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g_{\mu\nu}=\begin{pmatrix}
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A-\omega^2 B&0&0&\omega B\\
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0&-C&0&0\\
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0&0&-D&0\\
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\omega B&0&0&-B
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\end{pmatrix}.
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\end{equation}
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Taking into account the structure of $g_{\mu\nu}$, for the inverse matrix we get
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\begin{align*}
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&g^{rr}=\frac{1}{g_{rr}};\quad
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g^{\theta\theta}=\frac{1}{g_{\theta\theta}};\\
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&g^{tt}=
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\frac{g_{\varphi\varphi}}{|G|};\quad
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g^{\varphi\varphi}=\frac{g^{tt}}{|G|};
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\quad g^{t\varphi}
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=-\frac{g_{t\varphi}}{|G|};\\
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&\text{where}\quad
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G=g_{tt}g_{\varphi\varphi}-g_{t\varphi}^{2}.
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\end{align*}
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Using the explicit expression for  $g_{\mu\nu}$, we see that $G=-AB$ and thus finally
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\begin{equation}\label{AxiSimmMetricInvmatrix}
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g^{\mu\nu}=\begin{pmatrix}
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1/A&0&0&\omega/A\\
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0&-1/C&0&0\\
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0&0&-1/D&0\\
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\omega/A&0&0&\frac{\omega^2 B-A}{AB}
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\end{pmatrix}
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\end{equation} </p>
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  </div>
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</div>
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<div id="BlackHole54"></div>
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=== Problem 2. ===
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Write down the integrals of motion corresponding to Killing vectors $\partial_t$ and $\partial_\varphi$.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">
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A particle's integrals of motion are
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\begin{equation}\label{AxiSimm-Integrals}
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\bm{u}\cdot\bm{\xi}_t=u_{t};
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\quad
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\bm{u}\cdot\bm{\xi}_{\varphi}=u_{\varphi}.\end{equation}
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Energy and angular momentum are defined the same way as in the Schwarzshild case
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\[E=mc^{2}u_{t};\qquad L=-m u_\varphi.\] </p>
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  </div>
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</div>
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<div id="BlackHole55"></div>
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=== Problem 3. ===
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Find the coordinate angular velocity $\Omega=\tfrac{d\varphi}{dt}$ of a particle with zero angular momentum $u_{\mu}(\partial_{\varphi})^{\mu}=0$.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">
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For a particle moving in the axially symmetric field
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\begin{align*}
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u^{t}=&g^{t\mu}u_{\mu}=
 +
g^{tt}u_{t}+g^{t\varphi}u_{\varphi};\\
 +
u^{\varphi}=&g^{\varphi\mu}u_{\mu}=
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g^{\varphi t}u_{t}+g^{\varphi\varphi}u_{\varphi}.
 +
\end{align*}
 +
 
 +
Then for a particle with zero angular momentum (ZAMO)  $u_{\varphi}=0$ we get
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\[u^{t}=g^{tt}u_{t};
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\quad u^{\varphi}=g^{t\varphi}u_{t},\]
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and therefore its angular velocity is
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\[\frac{d\varphi}{dt}
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=\frac{d\varphi/ds}{dt/ds}=
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\frac{u^{\varphi}}{u^t}=
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\frac{g^{t\varphi}u_t}{g_{tt}u_t}=
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\frac{\omega/A}{1/A}=\omega(r,\theta).\]
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Now we see the physical meaning of the quantity $\omega(r,\theta)$. </p>
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  </div>
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</div>
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<div id="BlackHole55plus"></div>
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=== Problem 4. ===
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Calculate $A,B,C,D,\omega$ for the Kerr metric.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;">
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Let us introduce notation
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\[\Sigma^{2}=(r^2+a^2)^{2}-a^{2}\Delta \sin^{2}\theta,\]
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so that for the Kerr metric $g_{\varphi\varphi}=-\tfrac{\Sigma^2}{\rho^2}\sin^{2}\theta$. After some straightforward calculations then we obtain
 +
\begin{equation}\label{Kerr-ABCD}
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A=\frac{\Delta \rho^2}{\Sigma^2},\quad
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B=\frac{\Sigma^2}{\rho^2}\sin^2 \theta,\quad
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C=\frac{\rho^2}{\Delta},\quad
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D=\rho^2,\quad
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\omega=\frac{2\mu ra}{\Sigma^2}.
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\end{equation} </p>
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  </div>
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</div>
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 +
 
 
==Limiting cases==
 
==Limiting cases==
 
    
 
    

Revision as of 22:27, 18 July 2012

Kerr solution is the solution of Einstein's equations in vacuum that describes a rotating black hole (or the metric outside of a rotating axially symmetric body) \cite{Kerr63}. In the Boyer-Lindquist coordinates \cite{BoyerLindquist67} it takes the form \begin{align}\label{Kerr} &&ds^2=\bigg(1-\frac{2\mu r}{\rho^2}\bigg)dt^2 +\frac{4\mu a \,r\sin^{2}\theta}{\rho^2} \;dt\,d\varphi -\frac{\rho^2}{\Delta}\;dr^2-\rho^2\, d\theta^2 +\qquad\nonumber\\ &&-\bigg( r^2+a^2+\frac{2\mu r\,a^2 \,\sin^{2}\theta}{\rho^2} \bigg) \sin^2 \theta\;d\varphi^2;\\ \label{Kerr-RhoDelta} &&\text{where}\quad \rho^2=r^2+a^2 \cos^2 \theta,\qquad \Delta=r^2-2\mu r+a^2. \end{align} Here $\mu$ is the black hole's mass, $J$ its angular momentum, $a=J/\mu$; $t$ and $\varphi$ are time and usual azimuth angle, while $r$ and $\theta$ are some coordinates that become the other two coordinates of the spherical coordinate system at $r\to\infty$.

General axially symmetric metric

A number of properties of the Kerr solution can be understood qualitatively without use of its specific form. In this problem we consider the axially symmetric metric of quite general kind \begin{equation}\label{AxiSimmMetric} ds^2=A dt^2-B(d\varphi-\omega dt)^{2}- C\,dr^2-D\,d\theta^{2},\end{equation} where functions $A,B,C,D,\omega$ depend only on $r$ and $\theta$.

Problem 1.

Find the components of metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$.

Problem 2.

Write down the integrals of motion corresponding to Killing vectors $\partial_t$ and $\partial_\varphi$.

Problem 3.

Find the coordinate angular velocity $\Omega=\tfrac{d\varphi}{dt}$ of a particle with zero angular momentum $u_{\mu}(\partial_{\varphi})^{\mu}=0$.

Problem 4.

Calculate $A,B,C,D,\omega$ for the Kerr metric.


Limiting cases

Horizons and singularity

Stationary limit

Ergosphere and the Penrose process

Integrals of motion

The laws of mechanics of black holes

Particles' motion in the equatorial plane