# Schwarzschild black hole

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.

## Contents

## Simple problems

### Problem 1.

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### Problem 2.

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### Problem 3.

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### Problem 4.

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### Problem 5.

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## Symmetries and integrals of motion

### Problem 6.

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### Problem 7.

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### Problem 8.

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### Problem 9.

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### Problem 10.

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### Problem 11.

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### Problem 12.

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