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{is the 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) (there's a translation of the original paper at arXiv:physics/9905030v1; please disregard the abstract/foreword, which is incorrect).

$^{**}$ 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: local time

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: measuring distances

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: the inner region

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: acceleration

Calculate the acceleration of a test particle with zero velocity.

Problem 5: Schwarzschild is a vacuum solution

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: timelike Killing vector

What integral of motion arises due to existence of a timelike Killing vector? Express it through the physical velocity of the particle.

Problem 7: Killing vectors of a sphere

Derive the Killing vectors for a sphere in Cartesian coordinate system; in spherical coordinates.

Problem 8: spherical symmetry of Schwarzshild

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: planar motion

Show that existence of Killing vectors $S^\mu$ and $T^\mu$ leads to motion of particles in a plane.

Problem 10: stability of planar motion

Show that the particles' motion in the plane is stable.

Problem 11: remaining integrals of motion

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: work and mass

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: null geodesics

Derive the equation for null geodesics (worldlines of massless particles).

Problem 14: geodesic motion of massive particle

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: radial motion in terms of proper time

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: ultra-relativistic limit

Derive the equations of radial motion in the ultra-relativistic limit.

Problem 17: communication from near the black hole

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: stationary source and detector

The source and detector are stationary.

Problem 19: free-falling source

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: adding the two effects

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

Problem 21: intensity

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: impact parameter

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: geodesic equations and effective potential

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: bound and unbound motion

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: gravitational cross-section

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: innermost stable circular orbit

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

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: generalization of Newtonian potential

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: coordinate-invariant reformulation

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

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: uniqueness in exterior region

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: solution in a spherically symmetric void

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: shells

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?

Problem 34: Einstein equations for spherically symmetric case

Consider a static, spherically symmetric spacetime, described by metric \[ds^{2} =-f(r)dt^{2} +f^{-1} (r)dr^{2} +r^{2} d\Omega ^{2}, \] and write the Einstein's equations for it.

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 35: Rindler metric

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 36: tortoise coordinate

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 37: introducing null coordinates

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 38: Kruskal-Sekeres metric

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 39: Penrose diagram

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 40: more realistic collapse

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?