Difference between revisions of "Schwarzschild black hole"

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==Different coordinates, maximal extension==
 
==Different coordinates, maximal extension==
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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.
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=== Problem 34. ===
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=== Problem 35. ===
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Revision as of 09:31, 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.

problem formulation

Problem 2.

problem formulation

Problem 3.

problem formulation

Problem 4.

problem formulation

Problem 5.

problem formulation

Symmetries and integrals of motion

Problem 6.

problem formulation

Problem 7.

problem formulation

Problem 8.

problem formulation

Problem 9.

problem formulation

Problem 10.

problem formulation

Problem 11.

problem formulation

Problem 12.

problem formulation

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 $r\to r_{g}$.

Problem 13.

problem formulation

Problem 14.

problem formulation

Problem 15.

problem formulation

Problem 16.

problem formulation

Problem 17.

problem formulation

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.

problem formulation

Problem 19.

problem formulation

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.

problem formulation

Problem 24.

problem formulation

Problem 25.

problem formulation

Problem 26.

problem formulation

Miscellaneous problems

Problem 27.

problem formulation

Problem 28.

problem formulation

Problem 29.

problem formulation

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.

problem formulation

Problem 33.

problem formulation

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.

problem formulation

Problem 36.

problem formulation

Problem 37.

problem formulation

Problem 38.

problem formulation

Problem 39.

problem formulation

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