Black Holes

Abstract

Spacetime outside black holes with and without rotation—i.e. the Kerr and Schwarzschild spacetimes—is studied. By considering the motion of free particles in the Kerr spacetime we find an exact expression for the angular velocity of the inertial dragging . Hawking radiation from a non-rotating black hole is also studied.

Exercises

• 10.1. A spaceship falling into a black hole

  • (a) In this problem we will consider a spaceship (A) falling radially from the Earth (neglecting the gravitational field of the earth) into a Schwarzschild black hole with mass the mass of the Sun, $M = M_{\text{Sun}}$ at the position of the Sun, 150 million km from the Earth.

    What is the Schwarzschild radius of the black hole?

    Find the equations of motion of the spaceship in curvature coordinates r and t, using the proper time $\tau$ as time parameter.

    Solve the equations of motion with the initial condition that the space ship falls from rest at the Earth at proper time $\tau = 0$.

    When (in terms of $\tau$) does the spaceship reach the Schwarzschild radius? And the singularity?

  • (b) The spaceship (A) has radio contact with a stationary space station (B) at $r_{B} = 1$ light years. The radio signals are sent with intervals $\Delta T$ and with frequency $\omega$ from both A and B. The receivers at A and B receive signals with frequency $\omega_{A}$ and $\omega_{B}$, respectively. Find $\omega_{A}$ and $\omega_{B}$ as a function of the position of the spaceship. Hint: Perform the calculation in two steps. At first find the change in frequency between two stationary inertial systems in the points $r_{A}$ (the position of the spaceship) and $r_{B}$. Then calculate the change in frequency due to a transfer into an inertial system with the velocity of the spaceship.

• 10.2. Kinematics in the Kerr spacetime

  A Kerr black hole is an electrically neutral, rotating black hole. When spacetime outside a Kerr black hole is described in Boyer–Lindquist coordinates , the line element is the following,

  $$ {\text{d}}s^{2} = - e^{2\nu } {\text{d}}t^{2} + e^{2\mu } {\text{d}}r^{2} + e^{2\lambda } {\text{d}}\theta^{2} + e^{2\psi } ({\text{d}}\phi - \omega {\text{d}}t)^{2} , $$

  where

  $$ \begin{aligned} & e^{2\nu } = \frac{{\rho^{2} \Delta }}{{\varSigma^{2} }},e^{2\mu } = \frac{{\rho^{2} }}{\Delta },e^{2\lambda } = \rho^{2} , \\ & e^{2\psi } = \left( {\frac{{\varSigma^{2} }}{{\rho^{2} }}} \right)\sin^{2} \theta ,\omega = - \frac{{g_{t\phi } }}{{g_{\phi \phi } }} = \frac{2Mar}{{\varSigma^{2} }}, \\ & \rho^{2} = r^{2} + a^{2} \cos^{2} \theta , \, \Delta = r^{2} + a^{2} - 2Mr, \, \varSigma^{2} = (r^{2} + a^{2} )^{2} - a^{2} \Delta \sin^{2} \theta . \, \\ \end{aligned} $$

  Here M is the mass of the hole and a its spin per unit mass.

  • (a) Consider light moving in negative and positive direction of $\phi$ in the equatorial plane, $\theta = \pi /2$. What is the coordinate velocity $c_{\phi } = d\phi /dt$ of light?

    We now want to investigate the Sagnac effect in the Kerr space. An emitter–receiver is attached to a point in the BL coordinate system. Light signals with the frequency $\nu$ are sent by means of mirrors in both directions along the circle $r = r_{0}$, $\theta = \pi /2$. Find the travel time difference of light travelling in opposite directions, when the signals reach the receiver.

  • (b) A ZAMO is an observer with vanishing angular momentum. In the following a ZAMO in the Kerr spacetime will be describing particles with fixed r- and $\theta$-coordinates. Introduce an orthonormal basis $(\vec{e}_{{\hat{t}\prime }} ,\vec{e}_{{\hat{r}\prime }} ,\vec{e}_{{\hat{\theta }\prime }} ,\vec{e}_{{\hat{\phi }\prime }} )$, where $\vec{e}_{{\hat{t}\prime }}$ is the 4-velocity of a ZAMO. The dual basis 1-forms are

    $$ \begin{aligned} & \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{{\hat{t}^{{\prime }} }} = e^{\nu } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{t} ,\quad \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{{\hat{r}^{{\prime }} }} = e^{\mu } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{r} , \\ & \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{{\hat{\theta }^{{\prime }} }} = e^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{\theta } ,\quad \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{{\hat{\phi }^{{\prime }} }} = e^{\psi } (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{\phi } - \omega \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega }^{t} ). \\ \end{aligned} $$

    (10.51)

    Show that

    $$ \begin{aligned} & \vec{e}_{{\hat{t}^{{\prime }} }} = e^{ - \nu } (\vec{e}_{t} + \omega \vec{e}_{\phi } ),\quad \vec{e}_{{\hat{r}^{{\prime }} }} = e^{ - \mu } \vec{e}_{r} , \\ & \vec{e}_{{\hat{\theta }^{{\prime }} }} \, = e^{ - \lambda } \vec{e}_{\theta } ,\quad \vec{e}_{{\hat{\phi }^{{\prime }} }} = e^{ - \psi } \vec{e}_{\phi } , \\ \end{aligned} $$

    (10.52)

    where $(\vec{e}_{t} ,\vec{e}_{r} ,\vec{e}_{\theta } ,\vec{e}_{\phi } )$ are the coordinate basis vectors in the BL coordinate system.

    Given a particle with 4-velocity components

    $$ U^{\mu } = \left( {\dot{t},\,\,\,\dot{\phi }} \right) = \dot{t}\left( {1,\,\,\,\Omega } \right),\,\,\,\,\,\,\Omega = \frac{{{\text{d}}\phi }}{{{\text{d}}t}} $$

    (10.53)

    in the Boyer–Lindquist coordinate system .

    Show that the physical velocity of the particle, measured by a ZAMO, is

    $$ v^{{\hat{\phi }^{{\prime }} }} = e^{\psi - \nu } (\Omega - \omega ). $$

    (10.54)

    What is the velocity $v_{0}^{{\hat{\phi }^{{\prime }} }}$ of a fixed coordinate point measured by a ZAMO?

  • (c) Introduce an orthonormal basis field given by the expressions

    $$ \vec{e}_{{\hat{0}}} = ( - g_{00} )^{ - 1/2} \vec{e}_{0} ,\quad \vec{e}_{{\hat{i}}} = (\gamma_{ii} )^{ - 1/2} [\vec{e}_{i} - (g_{i0} /g_{00} )\vec{e}_{0} ], $$

    (10.55)

    where

    $$ \gamma_{ii} = g_{ii} - g_{i0}^{2} /g_{00} . $$

    Show that

    $$ \begin{aligned} & \vec{e}_{{\hat{t}}} = \hat{\gamma }e^{ - \nu } \vec{e}_{t} ,\quad \vec{e}_{{\hat{r}}} = e^{ - \mu } \vec{e}_{r} , \\ & \vec{e}_{{\hat{\theta }}} = e^{ - \lambda } \vec{e}_{r} ,\quad \vec{e}_{{\hat{\phi }}} = \hat{\gamma }^{ - 1} e^{ - \psi } \vec{e}_{\phi } + \hat{\gamma }e^{ - \nu } v_{0}^{{\hat{\phi }^{{\prime }} }} \vec{e}_{t} , \\ \end{aligned} $$

    (10.56)

    where $\hat{\gamma } = (1 - (v_{0}^{{\hat{\phi }^{{\prime }} }} )^{2} )^{ - 1/2}$. Find the dual basis 1-forms.

• 10.3. A gravitomagnetic clock effect

  This problem is concerned with the difference of proper time shown by two clocks moving freely in opposite directions in the equatorial plane of the Kerr spacetime outside a rotating body. The clocks move along a path with $r = {\text{constant}}$ and $\theta = \pi /2$.

  • (a) Show that in this case the radial geodesic equation reduces to

    $$ {\varGamma}_{tt}^{r} {\text{d}}t^{2} + 2{\varGamma }_{\phi t}^{r} {\text{d}}\phi {\text{d}}t + {\varGamma }_{\phi \phi }^{r} {\text{d}}\phi^{2} = 0. $$

    (10.57)

    Calculate the Christoffel symbols and show that the equation takes the form

    $$ \left( {\frac{{{\text{d}}t}}{{{\text{d}}\phi }}} \right)^{2} - 2a\frac{{{\text{d}}t}}{{{\text{d}}\phi }} + a^{2} - \frac{{r^{3} }}{M} = 0, $$

    (10.58)

    where M is the mass of the rotating body and a its angular momentum per unit mass, $a = J/M$.

  • (b) Show that the time difference for one closed orbit in $(\phi \to \phi + 2\pi )$ the direct and the retrograde direction is $t_{ + } - t_{ - } \approx 4\pi a = 4\pi J/M$, or in S.I. units,

    $$ t_{ + } - t_{ - } = 4\pi a = 4\pi J/mc^{2} . $$

    (10.59)

    Estimate this time difference for clocks in satellites moving in the equatorial plane of the Earth. (The mass of the Earth is $m = 6\,\cdot\,10^{26}$ kg and its angular momentum $J = 10^{34} {\mkern 1mu} {\text{kg}}\;{\text{m}}^{2} /{\text{s}}$.)