Difference between revisions of "Kerr black hole"
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| − | === Problem 1 | + | === Problem 1: preliminary algebra === |
Find the components of metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. | Find the components of metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. | ||
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| − | === Problem 2 | + | === Problem 2: integrals of motion === |
Write down the integrals of motion corresponding to Killing vectors $\partial_t$ and $\partial_\varphi$. | Write down the integrals of motion corresponding to Killing vectors $\partial_t$ and $\partial_\varphi$. | ||
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A particle's integrals of motion are | A particle's integrals of motion are | ||
\begin{equation}\label{AxiSimm-Integrals} | \begin{equation}\label{AxiSimm-Integrals} | ||
| − | \ | + | \mathbf{u}\cdot\mathbf{\xi}_t=u_{t}; |
\quad | \quad | ||
| − | \ | + | \mathbf{u}\cdot\mathbf{\xi}_{\varphi}=u_{\varphi}.\end{equation} |
Energy and angular momentum are defined the same way as in the Schwarzshild case | Energy and angular momentum are defined the same way as in the Schwarzshild case | ||
\[E=mc^{2}u_{t};\qquad L=-m u_\varphi.\] </p> | \[E=mc^{2}u_{t};\qquad L=-m u_\varphi.\] </p> | ||
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| − | === Problem 3 | + | === Problem 3: Zero Angular Momentum Observer (particle) === |
Find the coordinate angular velocity $\Omega=\tfrac{d\varphi}{dt}$ of a particle with zero angular momentum $u_{\mu}(\partial_{\varphi})^{\mu}=0$. | 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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| − | === Problem 4 | + | === Problem 4: rewriting metric again === |
Calculate $A,B,C,D,\omega$ for the Kerr metric. | Calculate $A,B,C,D,\omega$ for the Kerr metric. | ||
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==Limiting cases== | ==Limiting cases== | ||
Revision as of 22:31, 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 $r\to\infty$.
Contents
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: preliminary algebra
Find the components of metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$.
The metric is: \begin{equation}\label{AxiSimmMetricmatrix} g_{\mu\nu}=\begin{pmatrix} A-\omega^2 B&0&0&\omega B\\ 0&-C&0&0\\ 0&0&-D&0\\ \omega B&0&0&-B \end{pmatrix}. \end{equation} Taking into account the structure of $g_{\mu\nu}$, for the inverse matrix we get \begin{align*} &g^{rr}=\frac{1}{g_{rr}};\quad g^{\theta\theta}=\frac{1}{g_{\theta\theta}};\\ &g^{tt}= \frac{g_{\varphi\varphi}}{|G|};\quad g^{\varphi\varphi}=\frac{g^{tt}}{|G|}; \quad g^{t\varphi} =-\frac{g_{t\varphi}}{|G|};\\ &\text{where}\quad G=g_{tt}g_{\varphi\varphi}-g_{t\varphi}^{2}. \end{align*} Using the explicit expression for $g_{\mu\nu}$, we see that $G=-AB$ and thus finally \begin{equation}\label{AxiSimmMetricInvmatrix} g^{\mu\nu}=\begin{pmatrix} 1/A&0&0&\omega/A\\ 0&-1/C&0&0\\ 0&0&-1/D&0\\ \omega/A&0&0&\frac{\omega^2 B-A}{AB} \end{pmatrix} \end{equation}
Problem 2: integrals of motion
Write down the integrals of motion corresponding to Killing vectors $\partial_t$ and $\partial_\varphi$.
A particle's integrals of motion are \begin{equation}\label{AxiSimm-Integrals} \mathbf{u}\cdot\mathbf{\xi}_t=u_{t}; \quad \mathbf{u}\cdot\mathbf{\xi}_{\varphi}=u_{\varphi}.\end{equation} Energy and angular momentum are defined the same way as in the Schwarzshild case \[E=mc^{2}u_{t};\qquad L=-m u_\varphi.\]
Problem 3: Zero Angular Momentum Observer (particle)
Find the coordinate angular velocity $\Omega=\tfrac{d\varphi}{dt}$ of a particle with zero angular momentum $u_{\mu}(\partial_{\varphi})^{\mu}=0$.
For a particle moving in the axially symmetric field \begin{align*} u^{t}=&g^{t\mu}u_{\mu}= g^{tt}u_{t}+g^{t\varphi}u_{\varphi};\\ u^{\varphi}=&g^{\varphi\mu}u_{\mu}= 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 \[u^{t}=g^{tt}u_{t}; \quad u^{\varphi}=g^{t\varphi}u_{t},\] and therefore its angular velocity is \[\frac{d\varphi}{dt} =\frac{d\varphi/ds}{dt/ds}= \frac{u^{\varphi}}{u^t}= \frac{g^{t\varphi}u_t}{g_{tt}u_t}= \frac{\omega/A}{1/A}=\omega(r,\theta).\] Now we see the physical meaning of the quantity $\omega(r,\theta)$.
Problem 4: rewriting metric again
Calculate $A,B,C,D,\omega$ for the Kerr metric.
Let us introduce notation \[\Sigma^{2}=(r^2+a^2)^{2}-a^{2}\Delta \sin^{2}\theta,\] 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} A=\frac{\Delta \rho^2}{\Sigma^2},\quad B=\frac{\Sigma^2}{\rho^2}\sin^2 \theta,\quad C=\frac{\rho^2}{\Delta},\quad D=\rho^2,\quad \omega=\frac{2\mu ra}{\Sigma^2}. \end{equation}
