Difference between revisions of "Kerr black hole"

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==Horizons and singularity==
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Event horizon is a closed null surface. A null surface is a surface with null normal vector $n^\mu$:
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\[n^{\mu}n_{\mu}=0.\]
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This same notation means that $n^\mu$ belongs to the considered surface (which is not to be wondered at, as a null vector is always orthogonal to self). It can be shown further, that a null surface can be divided into a set of null geodesics. Thus the light cone touches it in each point: the future light cone turns out to be on one side of the surface and the past cone on the other side. This means that world lines of particles, directed in the future, can only cross the null surface in one direction, and the latter works as a one-way valve, -- "event horizon"
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<div id="label"></div>
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=== Problem 8: on null surfaces ===
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Show that if a surface is defined by equation $f(r)=0$, and on it $g^{rr}=0$, it is a null surface.
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  <div class="NavHead">solution</div>
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    <p style="text-align: left;">
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The normal vector $n_\mu$ to a surface $f(x)=0$ is directed along $\partial_{\mu}f$. It is null if
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\[g^{\mu\nu}(\partial_{\mu}f)(\partial_{\nu}f)=0.\]
  
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The normal to the surface $f(r)=0$ is directed along $\partial_{r}f$, i.e. $\partial_{\mu}f\sim \delta_{\mu}^{r}$ and the null condition takes the form
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\[0=g^{\mu\nu}\delta^{r}_{\mu}\delta^{r}_{\nu}
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=g^{rr}.\] </p>
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  </div>
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</div>
  
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<div id="label"></div>
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=== Problem 9: null surfaces in Kerr metric ===
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Find the surfaces $g^{rr}=0$ for the Kerr metric. Are they spheres?
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  <div class="NavHead">solution</div>
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    <p style="text-align: left;">
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Equations of surfaces, on which $g^{rr}$ terms to zero and $g_{rr}$ to infinity, are
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\begin{align}
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\Delta=0\quad\Leftrightarrow\quad
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r^2-2\mu r+a^2=0\quad\Leftrightarrow\quad
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r=r_{\pm},\quad\text{где}\nonumber\\
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\label{Kerr-Rhor+-}
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r_{\pm}\equiv\mu\pm\sqrt{\mu^2-a^2}.
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\end{align}
  
==Horizons and singularity==
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Although those are surfaces of constant $r$, their intrinsic metric is not spherical. Plugging $r=r_{\pm}$ into the spatial section $dt=0$ of the Kerr metric (\ref{Kerr}) ans using the relation $r_{\pm}^{2}+a^2=2\mu r_{\pm}$, which holds on the surfaces $r=r_\pm$, we obtain
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\[dl^{2}_{r=r_\pm}=
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\rho^{2}_{\pm}d\theta^{2}+
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\Big(\frac{2\mu r_\pm}{\rho_\pm}\Big)^2
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\sin^{2}\theta \,d\varphi^2
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=\rho_{\pm}^{2}
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\big(d\theta^2 +\sin^{2}\theta d\varphi^{2} \big)
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+2a^{2}(r^2 +a^2 \cos^{2}\tfrac{\theta}{2})
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\sin^{4}\theta\,d\varphi^{2}.\]
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The first terms is the metric of a sphere, while the second gives additional positive contribution to the distance measured along $\varphi$. Thus if we embedded such a surface into a three-dimensional Euclidean space, we'd get something similar to an oblate ellipsoid of rotation. </p>
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=== Problem 10: horizon area ===
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Calculate surface areas of the outer and inner horizons.
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The horizon's area is
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\begin{equation}\label{Kerr-HorizonSurface}
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S_{\pm}=\int\rho_\pm d\theta\cdot
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\frac{2\mu r_\pm}{\rho_\pm}\sin\theta d\varphi=
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2\mu r_{\pm}
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\int\limits_{0}^{\pi}\sin\theta d\theta
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\int\limits_{0}^{2\pi}d\varphi=
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8\pi \mu r_{\pm}=4\pi (r_{\pm}^{2}+a^2).
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\end{equation} </p>
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  </div>
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</div>
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<div id="label"></div>
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=== Problem 11: black holes and naked singularities ===
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What values of $a$ lead to existence of horizons?
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Solutions of $\Delta=0$ exist when
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\[a<m.\] </p>
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  </div>
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</div>
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On calculating curvature invariants, one can see they are regular on the horizons and diverge only at $\rho^2 \to 0$. Thus only the latter surface is a genuine singularity.
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=== Problem 12: surface $r=0$. ===
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Derive the internal metric of the surface $r=0$ in Kerr solution.
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  <div class="NavHead">solution</div>
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Let us consider the set of points $r=0$. Assuming $r=0$ and $dr=0$ in (\ref{Kerr}), from (\ref{Kerr-RhoDelta}) we obtain
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\[ \rho^2=a^2 \cos^2 \theta;\quad
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\Delta=a^2;\quad\Rightarrow\quad
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g_{\theta\theta}=-a^2,\quad
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g_{\varphi\varphi}=-a^{2}\sin^{2}\theta, \]
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so metric takes the form
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\begin{equation}\label{KerrR=0}
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ds^{2}_{r=0}=dt^2-a^{2}\cos^{2}\theta d\theta^2
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-a^{2}\sin^{2}\theta d\varphi^2=
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\Big\lwavy a\sin\theta=\eta\Big\rwavy=
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dt^{2}-\Big(d\eta^2+\eta^2 d\varphi^2 \Big).
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\end{equation}
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This is a flat disk of radius $a$, center $\eta=\theta=0$, with distance to the center measured by $\eta=a\sin\theta$. </p>
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=== Problem 13: circular singularity ===
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Show that the set of points $\rho=0$ is a circle. How it it situated relative to the inner horizon?
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  <div class="NavHead">solution</div>
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    <p style="text-align: left;">
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The boundary of the disk is a circle $\eta=a$, or in original variables
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\[\{r=0,\;\theta=\pi/2,\;\varphi\in[0,2\pi)\},\]
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which lies beyond the inner horizon. If $a=\mu$ (extremal Kerr black hole), then $r_{-}=0$ and it lies on the horizon. </p>
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  </div>
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</div>.
 
    
 
    
 
==Stationary limit==
 
==Stationary limit==

Revision as of 22:49, 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) . In the Boyer-Lindquist coordinates$^{**}$ 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 $r\to\infty$.

$^{*}$ R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), 237 (1963).

$^{**}$ R.H. Boyer, R.W. Lindquist. Maximal Analytic Extension of the Kerr Metric. J. Math. Phys 8, 265–281 (1967).

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 $g^{\mu\nu}$.

Problem 2: integrals of motion

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

Problem 3: Zero Angular Momentum Observer (particle)

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

Problem 4: some more simple algebra

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

Limiting cases

Problem 5: Schwarzshild limit

Show that in the limit $a\to 0$ the Kerr metric turns into Schwarzschild with $r_{g}=2\mu$.

Problem 6: Minkowski limit

Show that in the limit $\mu\to 0$ the Kerr metric describes Minkowski space with the spatial part in coordinates that are related to Cartesian as \begin{align*} &x=\sqrt{r^2+a^2}\;\sin\theta\cos\varphi, \nonumber\\ &y=\sqrt{r^2+a^2}\;\sin\theta\sin\varphi,\\ &z=r\;\cos\theta\nonumber,\\ &\text{where}\quad r\in[0,\infty),\quad \theta\in[0,\pi],\quad \varphi\in[0,2\pi).\nonumber \end{align*} Find equations of surfaces $r=const$ and $\theta=const$ in coordinates $(x,y,z)$. What is the surface $r=0$?

Problem 7: weak field rotation effect

Write the Kerr metric in the limit $a/r \to 0$ up to linear terms.

Horizons and singularity

Event horizon is a closed null surface. A null surface is a surface with null normal vector $n^\mu$: \[n^{\mu}n_{\mu}=0.\] This same notation means that $n^\mu$ belongs to the considered surface (which is not to be wondered at, as a null vector is always orthogonal to self). It can be shown further, that a null surface can be divided into a set of null geodesics. Thus the light cone touches it in each point: the future light cone turns out to be on one side of the surface and the past cone on the other side. This means that world lines of particles, directed in the future, can only cross the null surface in one direction, and the latter works as a one-way valve, -- "event horizon"

Problem 8: on null surfaces

Show that if a surface is defined by equation $f(r)=0$, and on it $g^{rr}=0$, it is a null surface.

Problem 9: null surfaces in Kerr metric

Find the surfaces $g^{rr}=0$ for the Kerr metric. Are they spheres?

Problem 10: horizon area

Calculate surface areas of the outer and inner horizons.

Problem 11: black holes and naked singularities

What values of $a$ lead to existence of horizons?

On calculating curvature invariants, one can see they are regular on the horizons and diverge only at $\rho^2 \to 0$. Thus only the latter surface is a genuine singularity.

Problem 12: surface $r=0$.

Derive the internal metric of the surface $r=0$ in Kerr solution.

Problem 13: circular singularity

Show that the set of points $\rho=0$ is a circle. How it it situated relative to the inner horizon?

.

Stationary limit

Ergosphere and the Penrose process

Integrals of motion

The laws of mechanics of black holes

Particles' motion in the equatorial plane