Schwarzschild black hole

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The spherically symmetric solution of Einstein's equations in vacuum for the spacetime metric has the form$^{*}$ \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) 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.

$^{*}$ K. Schwarzschild, On the gravitational field of a mass point according to Einstein's theory, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl., p.189 (1916) arXiv:physics/9905030v1.

$^{**}$ G.D. Birkhoff, Relativity and Modern Physics, p.253, Harvard University Press, Cambridge (1923); J.T. Jebsen, "Ark. Mat. Ast. Fys." (Stockholm) 15, nr.18 (1921), see also arXiv:physics/0508163v2.


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.

The source is freely falling the same way, while the detector is stationary at $r_{det}>r_{em}$.

Problem 21.

The source is falling freely and emitting continuously photons with constant frequency, the detector is stationary far away from the horizon $r_{det}\gg r_{g}$. How does the detected light's intensity depend on $r_{em}$ at the moment of emission? On the proper time of detector?

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.

Show that the ratio of specific energy to specific angular momentum of a particle equals to $r_{g}/b$, where $b$ is the impact parameter at infinity (for unbounded motion).

Problem 23.

Derive the geodesics' equations; bring the equation for $r(\lambda)$ to the form \[\frac{1}{2}\Big(\frac{dr}{d\lambda}\Big)^{2} +V_{\epsilon}(r)=\varepsilon,\] where $V_{\epsilon}(r)$ is a function conventionally termed as effective potential.

Problem 24.

Plot and investigate the function $V(r)$. Find the radii of circular orbits and analyze their stability; find the regions of parameters $(E,L)$ corresponding to bound and unbound motion, fall into the black hole. Consider the cases of a) massless, b) massive particles.

Problem 25.

Derive the gravitational capture cross-section for a massless particle; the first correction to it for a massive particle ultra-relativistic at infinity. Find the cross-section for a non-relativistic particle to the first order in $v^2/c^2$.

Problem 26.

Find the minimal radius of stable circular orbit and its parameters. What is the maximum gravitational binding energy of a particle in the Schwarzschild spacetime?

Miscellaneous problems

Problem 27.

Gravitational lensing is the effect of deflection of a light beam's (photon's) trajectory in the gravitational field. Derive the deflection of a photon's trajectory in Schwarzschild metric in the limit $L/r_{g}\gg 1$. Show that it is twice the value for a massive particle with velocity close to $c$ in the Newtonian theory.

Problem 28.

Show that the $4$-acceleration of a stationary particle in the Schwarzschild metric can be presented in the form \[a_{\mu}=-\partial_{\mu}\Phi,\quad \text{where}\quad \Phi=\ln \sqrt{g_{00}} =\tfrac{1}{2}\ln g_{00}\] is some generalization of the Newtonian gravitational potential.

Problem 29.

Let us reformulate the problem in a coordinate-independent manner. Suppose we have an arbitrary stationary metric with timelike Killing vector $\xi^\mu$, and we denote the $4$-velocity of a stationary observer by $u^{\mu}=\xi^{\mu}/V$. What is the $4$-force per unit mass that we need to apply to a test particle in order to make it stay stationary? Show in coordinate-independent way that the answer coincides with $\partial_{\mu}\Phi$ (up to the sign), and rewrite $\Phi$ in coordinate-independent form.

Problem 30.

Surface gravity $\kappa$ of the Schwarzschild horizon can be defined as acceleration of a stationary particle at the horizon, measured in the proper time of a stationary observer at infinity. Find $\kappa$.

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.

Suppose all the matter is distributed around the center of symmetry, and its energy-momentum tensor is spherically symmetric, so that the form of $g_{\mu\nu}$ written above is correct. Show that the solution in the exterior region is reduced to the Schwarzschild metric and find the relation between $C$ and the system's mass $M$.

Problem 32.

Let there be a spherically symmetric void $r<r_{0}$ in the spherically symmetric matter distribution. Show that spacetime in the void is flat.

Problem 33.

Let the matter distribution be spherically symmetric and filling regions $r<r_{0}$ and $r_{1}<r<r_{2}$ ($r_{0}<r_{1}$). Can one affirm, that the solution in the layer of empty space $r_{0}<r<r_{1}$ is also the Schwarzschild metric?

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.

Make coordinate transformation in the Schwarzschild metric near the horizon $(r-r_{g})\ll r_{g}$ by using physical distance to the horizon as a new radial coordinate instead of $r$, and show that in the new coordinates it reduces near the horizon to the Rindler metric.

Problem 35.

Derive the Schwarzschild metric in coordinates $t$ and $r^\star=r+r_{g}\ln|r-r_g|$. How do the null geodesics falling to the center look like in $(t,r^\star)$? What range of values of $r^\star$ corresponds to the region $r>r_g$?

Problem 36.

Rewrite the metric in coordinates $r$ and $u=t-r^\star$, find the equations of null geodesics and the value of $g=det(g_{\mu\nu})$ at $r=r_{g}$. Likewise in coordinates $r$ and $v=t+r^\star$; in coordinates $(u,v)$. The coordinate frames $(v,r)$ and $(u,r)$ are called the ingoing and outgoing Eddington-Finkelstein coordinates.

Problem 37.

Rewrite the Schwarzschild metric in coordinates $(u',v')$ and in the Kruskal coordinates $(T,R)$ (Kruskal solution), defined as follows: \[v'=e^{v/2r_g},\quad u'=-e^{-u/2r_g};\qquad T=\frac{u'+v'}{2},\quad R=\frac{v'-u'}{2}.\] What are the equations of null geodesics, surfaces $r=const$ and $t=const$, of the horizon $r=r_{g}$, singularity $r=0$, in the coordinates $(T,R)$? What is the range space of $(T,R)$? Which regions in the Schwarzschild coordinates do the regions $\{\text{I}:\;R>|T|\}$, $\{\text{II}:\;T>|R|\}$, $\{\text{III}:\;R<-|T|\}$ and $\{\text{IV}:\;T<-|R|\}$ correspond to? Which of them are casually connected and which are not? What is the geometry of the spacelike slice $T=const$ and how does it evolve with time $T$?

Problem 38.

Pass to coordinates \[v''=\arctan\frac{v'}{\sqrt{r_g}},\quad u''=\arctan\frac{u'}{\sqrt{r_g}}\] and draw the spacetime diagram of the Kruskal solution in them.

Problem 39.

The Kruskal solution describes an eternal black hole. Suppose, for simplicity, that some black hole is formed as a result of radial collapse of a spherically symmetric shell of massless particles. What part of the Kruskal solution will be realized, and what will not be? What is the casual structure of the resulting spacetime?