Difference between revisions of "Schwarzschild black hole"
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==Blackness of black holes== | ==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 $r$. | |
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| + | <div id="BlackHole31"></div> | ||
| + | === Problem 18. === | ||
| + | problem formulation | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"> | ||
| + | problem solution </p> | ||
| + | </div> | ||
| + | </div> | ||
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| + | <div id="BlackHole32"></div> | ||
| + | === Problem 19. === | ||
| + | problem formulation | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"> | ||
| + | problem solution </p> | ||
| + | </div> | ||
| + | </div> | ||
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| + | <div id="BlackHole33"></div> | ||
| + | === Problem 20. === | ||
| + | problem formulation | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"> | ||
| + | problem solution </p> | ||
| + | </div> | ||
| + | </div> | ||
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| + | <div id="BlackHole34"></div> | ||
| + | === Problem 21. === | ||
| + | problem formulation | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"> | ||
| + | problem solution </p> | ||
| + | </div> | ||
| + | </div> | ||
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==Orbital motion, effective potential== | ==Orbital motion, effective potential== | ||
Revision as of 08:56, 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.
Contents
Simple problems
Problem 1.
problem formulation
problem solution
Problem 2.
problem formulation
problem solution
Problem 3.
problem formulation
problem solution
Problem 4.
problem formulation
problem solution
Problem 5.
problem formulation
problem solution
Symmetries and integrals of motion
Problem 6.
problem formulation
problem solution
Problem 7.
problem formulation
problem solution
Problem 8.
problem formulation
problem solution
Problem 9.
problem formulation
problem solution
Problem 10.
problem formulation
problem solution
Problem 11.
problem formulation
problem solution
Problem 12.
problem formulation
problem solution
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 solution
Problem 14.
problem formulation
problem solution
Problem 15.
problem formulation
problem solution
Problem 16.
problem formulation
problem solution
Problem 17.
problem formulation
problem solution
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 solution
Problem 19.
problem formulation
problem solution
Problem 20.
problem formulation
problem solution
Problem 21.
problem formulation
problem solution
