Difference between revisions of "Schwarzschild black hole"

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(Radial motion)
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==Radial motion==
 
==Radial motion==
    
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Consider a particle's radial motion: $\dot{\varphi}=\dot{\theta}=0$. In this problem one is especially interested in asymptotes of all functions as $r\to r_{g}$.
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=== Problem 13. ===
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=== Problem 14. ===
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problem formulation
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=== Problem 15. ===
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problem solution </p>
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=== Problem 16. ===
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=== Problem 17. ===
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problem formulation
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==Blackness of black holes==
 
==Blackness of black holes==
 
    
 
    

Revision as of 08:53, 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

Orbital motion, effective potential

Miscellaneous problems

Different coordinates, maximal extension