Difference between revisions of "Schwarzschild black hole"

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(Problem 19.)
(Problem 19.)
Line 562: Line 562:
 
\frac{1}{h}\cdot\gamma\beta\sqrt{h}\cdot hk^{0}=
 
\frac{1}{h}\cdot\gamma\beta\sqrt{h}\cdot hk^{0}=
 
\sqrt{h}\gamma k^{0}(1\pm\beta).\]
 
\sqrt{h}\gamma k^{0}(1\pm\beta).\]
Choosing the sign ``$+$'' here, which corresponds to the emitter moving towards the horizon and light travaling outwards, we obtain the relativistic Dopper effect  
+
Choosing the sign "$+$" here, which corresponds to the emitter moving towards the horizon and light travaling outwards, we obtain the relativistic Dopper effect  
 
\begin{equation}\label{RelatDoppler-radial}
 
\begin{equation}\label{RelatDoppler-radial}
 
\omega_{\beta}=
 
\omega_{\beta}=

Revision as of 18:20, 17 June 2012


The spherically symmetric solution of Einstein's equations in vacuum for the spacetime metric has the form \cite{Schw} \begin{align}\label{Schw} ds^{2}=h(r)\,dt^2-h^{-1}(r)\,dr^2-r^2 d\Omega^{2}, &\qquad\mbox{where}\quad h(r)=1-\frac{r_g}{r};\quad r_{g}=\frac{2GM}{c^{2}};\\ d\Omega^{2}=d\theta^{2}+\sin^{2}\theta\, d\varphi^{2}&\;\text{-- metric of unit sphere.}\nonumber \end{align} The Birkhoff's theorem (1923) \cite{Birkhoff,Jebsen} states, that this solution is unique up to coordinate transformations. The quantity $r_g$ is called the Schwarzschild radius, or gravitational radius, $M$ is the mass of the central body or black hole.

Simple problems

Problem 1.

Find the interval of local time (proper time of stationary observer) at a point $(r,\theta,\varphi)$ in terms of coordinate time $t$, and show that $t$ is the proper time of an observer at infinity. What happens when $r\to r_{g}$?

Problem 2.

What is the physical distance between two points with coordinates $(r_{1},\theta,\varphi)$ and $(r_{2},\theta,\varphi)$? Between $(r,\theta,\varphi_{1})$ and $(r,\theta,\varphi_{2})$? How do these distances behave in the limit $r_{1},r\to r_{g}$?

Problem 3.

What would be the answers to the previous two questions for $r<r_g$ and why*? Why the Schwarzschild metric cannot be imagined as a system of "welded" rigid rods in $r<r_g$, as it can be in the external region?


Problem 4.

Calculate the acceleration of a test particle with zero velocity.

Problem 5.

Show that Schwarzschild metric is a solution of Einstein's equation in vacuum.

Symmetries and integrals of motion

For background on Killing vectors see problems K1, K2, K3 of chapter 2.

Problem 6.

What integral of motion arises due to existance of a timelike Killing vector? Express it through the physical velocity of the particle.

Problem 7.

Derive the Killing vectors for a sphere in Cartesian coordinate system; in spherical coordinates.

Problem 8.

Verify that in coordinates $(t,r,\theta,\varphi)$ vectors \[ \begin{array}{l} \Omega^{\mu}=(1,0,0,0),\\ R^{\mu}=(0,0,0,1),\\ S^{\mu}=(0,0,\cos\varphi,-\cot\theta\sin\varphi),\\ T^{\mu}=(0,0,-\sin\varphi,-\cot\theta\cos\varphi) \end{array}\] are the Killing vectors of the Schwarzschild metric.

Problem 9.

Show that existence of Killing vectors $S^\mu$ and $T^\mu$ leads to motion of particles in a plane.

Problem 10.

Show that the particles' motion in the plane is stable.

Problem 11.

Write down explicitly the conserved quantities $p_{\mu}\Omega^{\mu}$ and $p_{\mu}R^{\mu}$ for movement in the plane $\theta=\pi/2$.

Problem 12.

What is the work needed to pull a particle from the horizon to infinity? Will a black hole's mass change if we drop a particle with zero initial velocity from immediate proximity of the horizon?

Radial motion

Consider a particle's radial motion: $\dot{\varphi}=\dot{\theta}=0$. In this problem one is especially interested in asymptotes of all functions as $r\to r_{g}$.

Let us set $c=1$ here and henceforth measure time in the units of length, so that $x^{0}=t$, $\beta=v$, etc., and introduce the notation \[h(r)\equiv g_{00}(r)=-\frac{1}{g_{11}(r)}= 1-\frac{r_{g}}{r} \underset{r\to r_g+0}{\longrightarrow}+0.\]

Problem 13.

Derive the equation for null geodesics (worldlines of massless particles).

Problem 14.

Use energy conservation to derive $v(r)$, $\dot{r}(r)=dr/dt$, $r(t)$ for a massive particle. Initial conditions: $g_{00}|_{\dot{r}=0}=h_{0}$ (the limiting case $h_{0}\to 1$ is especially interesting and simple).

Problem 15.

Show that the equation of radial motion in terms of proper time of the particle is the same as in the non-relativistic Newtonian theory. Calculate the proper time of the fall from $r=r_0$ to the center. Derive the first correction in $r_{g}/r$ to the Newtonian result. Initial velocity is zero.

Problem 16.

Derive the equations of radial motion in the ultra-relativistic limit.

Problem 17.

A particle (observer) falling into a black hole is emitting photons, which are detected on the same radial line far away from the horizon (i.e. the photons travel from emitter to detector radially). Find $r$, $v$ and $\dot{r}$ as functions of the signal's detection time in the limit $r\to r_g$.

Blackness of black holes

A source radiates photons of frequency $\omega_i$, its radial coordinate at the time of emission is $r=r_{em}$. Find the frequency of photons registered by a detector situated at $r=r_{det}$ on the same radial line in different situations described below. By stationary observers here, we mean stationary in the static Schwarzschild metric; "radius" is the radial coordinate $r$.

Problem 18.

The source and detector are stationary.

Problem 19.

The source is falling freely without initial velocity from radius $r_0$; it flies by the stationary detector at the moment of emission.

Problem 20.

problem formulation

Problem 21.

problem formulation

Orbital motion, effective potential

Due to high symmetry of the Schwarzschild metric, a particle's worldline is completely determined by the normalizing condition $u^{\mu}u_{\mu}=\epsilon$, where $\epsilon=1$ for a massive particle and $\epsilon=0$ for a massless one, plus two conservation laws---of energy and angular momentum.

Problem 22.

problem formulation

Problem 23.

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Problem 24.

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Problem 25.

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Problem 26.

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Miscellaneous problems

Problem 27.

problem formulation

Problem 28.

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Problem 29.

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Problem 30.

problem formulation

Solving Einstein's equations for a spherically symmetric metric of general form in vacuum (energy-momentum tensor equals to zero), one can reduce the metric to \[ds^2=f(t)\Big(1-\frac{C}{r}\Big)dt^2 -\Big(1-\frac{C}{r}\Big)^{-1}dr^2-r^2 d\Omega^2,\] where $C$ is some integration constant, and $f(t)$ an arbitrary function of time $t$.

Problem 31.

problem formulation

Problem 32.

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Problem 33.

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Different coordinates, maximal extension

We saw that a particle's proper time of reaching the singularity is finite. However, the Schwarzschild metric has a (removable) coordinate singularity at $r=r_{g}$. In order to eliminate it and analyze the casual structure of the full solution, it is convenient to use other coordinate frames. Everywhere below we transform the coordinates $r$ and $t$, while leaving the angular part unchanged.

Problem 34.

problem formulation

Problem 35.

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Problem 36.

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Problem 37.

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Problem 38.

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Problem 39.

problem formulation