Schwarzschild black hole - Revision history

Igor at 10:02, 2 May 2013

2013-05-02T10:02:59Z

Igor at 10:01, 2 May 2013

2013-05-02T10:01:26Z

Igor at 20:12, 19 November 2012

2012-11-19T20:12:39Z

Igor: /* Miscellaneous problems */

2012-11-19T20:10:08Z

‎Miscellaneous problems

Igor at 17:24, 13 October 2012

2012-10-13T17:24:22Z

Igor at 20:53, 24 July 2012

2012-07-24T20:53:43Z

Igor at 20:51, 24 July 2012

2012-07-24T20:51:35Z

Igor at 19:38, 24 July 2012

2012-07-24T19:38:29Z

Igor: /* Blackness of black holes */

2012-07-24T19:17:42Z

‎Blackness of black holes

Igor: /* Problem 6: energy */

2012-07-24T19:12:47Z

‎Problem 6: energy

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 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 is the translation of the original paper at [http://arxiv.org/abs/physics/9905030v1 arXiv:physics/9905030v1]; please disregard the abstract).
+$^{*}$ 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 [http://arxiv.org/abs/physics/9905030v1 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);

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 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) [http://arxiv.org/abs/physics/9905030v1 arXiv:physics/9905030v1].
+$^{*}$ 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 is the translation of the original paper at [http://arxiv.org/abs/physics/9905030v1 arXiv:physics/9905030v1]; please disregard the abstract).
 $^{**}$ G.D. Birkhoff, Relativity and Modern Physics, p.253, Harvard University Press, Cambridge (1923);

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-=== Problem 34: derivation of Schwarzschild ===
+=== Problem 34: Einstein equations for spherically symmetric case ===
-Consider a static, spherically symmetric spacetime, described by a metric
+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.
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-=== Problem 34: Rindler metric ===
+=== 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.
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-=== Problem 35: tortoise coordinate ===
+=== 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 $r>r_g$?
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-=== Problem 36: introducing null coordinates ===
+=== 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.
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-=== Problem 37: Kruskal-Sekeres metric ===
+=== 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
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 <div id="BlackHole51"></div>
-=== Problem 38: Penrose diagram ===
+=== Problem 39: Penrose diagram ===
 Pass to coordinates
 \[v''=\arctan\frac{v'}{\sqrt{r_g}},\quad
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 <div id="BlackHole52"></div>
-=== Problem 39: more realistic collapse ===
+=== 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?
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      <p style="text-align: left;">
 With such matter distribution the solution in region $r>r_{2}$ is Schwarzschild, as the constant $C$ and function $f(t)$ are fixed in the usual way. In region $r_{0}<r<r_{1}$, however, we do not have the freedom to choose a new time coordinate, as it will be determined by the smooth sewing-up of the metric at the boundaries (in region $r_{1}<r<r_{2}$ the solution is quite different, of course). Therefore $f(t)\neq 1$ in the inner region.	</p>
    </div>
 </div>

← Older revision		Revision as of 17:24, 13 October 2012
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	[[Category:Black Holes\|2]]		[[Category:Black Holes\|2]]
−	~~=Schwarzshild black hole=~~

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

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 [[Category:Black Holes|2]]
 =Schwarzshild black hole=
 The spherically symmetric solution of Einstein's equations in vacuum for the spacetime metric has the form$^{*}$
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 J.T. Jebsen, "Ark. Mat. Ast. Fys." (Stockholm) 15, nr.18 (1921), see also [http://arxiv.org/abs/physics/0508163 arXiv:physics/0508163v2].
+__TOC__
 ==Simple problems==

← Older revision		Revision as of 20:51, 24 July 2012
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	[[Category:Black Holes\|2]]		[[Category:Black Holes\|2]]
		+	=Schwarzshild black hole=
		+	__TOC__

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

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 	&\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
+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.
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      <p style="text-align: left;">
-The first integral of motion is, up to a multiplier, the energy (see \ref{EnergyStat}):
+The first integral of motion is, up to a multiplier, the energy (see Eq. [[Technical_warm-up#EnergyStat|EnergyStat]]):
 \begin{equation}\label{SchwInt-E1}
 	 E=\Omega_{\mu}u^{\mu}=
 	g_{00}\frac{dx^{0}}{d\lambda}=h\frac{dt}{d\lambda}=
 	\left(1-\frac{r_g}{r}\right)\frac{dt}{d\lambda},\end{equation}
-where $\lambda$ is the geodesic parameter: for a massive particle we can always choose the natural parametrization  $d\lambda=ds=\gamma^{-1}\sqrt{g_{00}}\,dt$ (see (\ref{IntervalStaticCase})) so that $E$ is energy per unit mass; recovering the multipliers, for true energy we obtain
+where $\lambda$ is the geodesic parameter: for a massive particle we can always choose the natural parametrization  $d\lambda=ds=\gamma^{-1}\sqrt{g_{00}}\,dt$ (see Eq. [[Technical_warm-up#IntervalStaticCase|IntervalStaticCase]]) so that $E$ is energy per unit mass; recovering the multipliers, for true energy we obtain
 \begin{equation}\label{SchwInt-E2}
 	\varepsilon_{m}\equiv mcE=
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 \[ds^{2}=l^{2}d\omega^{2}-dl^{2}-r^{2}(l)d\Omega^{2},
 	\qquad\mbox{where}\quad\omega=\frac{ct}{2r_{g}}.\]
-Comparing with the Rindler metric (\ref{Rindler}), we see that $l\equiv\rho=\frac{c^2}{a}$, and $\tau=t\frac{l}{2r_g}$. Thus the static Schwarzschild metric near the horizon has the same form as Minkowskii metric for a uniformly accelerated observer with acceleration $a=\frac{c^2}{l}$, which tends to infinity at the horizon. Then using the inverse transformation, from Rindler to Minkowskii, we can turn the Schwarzschild metric near the horizon into the flat one, thus showing explicitly that $r=r_g$ is just a removable coordinate singularity.</p>
+Comparing with the [[Technical_warm-up#Rindler|Rindler metric]], we see that $l\equiv\rho=\frac{c^2}{a}$, and $\tau=t\frac{l}{2r_g}$. Thus the static Schwarzschild metric near the horizon has the same form as Minkowskii metric for a uniformly accelerated observer with acceleration $a=\frac{c^2}{l}$, which tends to infinity at the horizon. Then using the inverse transformation, from Rindler to Minkowskii, we can turn the Schwarzschild metric near the horizon into the flat one, thus showing explicitly that $r=r_g$ is just a removable coordinate singularity.</p>
    </div>
 </div>
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 	=\frac{4r_{g}^3}{r}e^{-r/r_{g}}
 	(dT^2-dR^2)-r^{2}d\Omega^2.\]
-Equations of null geodegics: $u=const$ and $v=const$, or in terms of $(T,R)$: $T=\pm R+const$. The surface  $r=const$ is mapped to a hyperbola $T^2-R^2=-2(r-r_g)e^{r/r_g}$; singularity to a hyperbola $T^2-R^2=2r_g$ with \emph{two} branches also; the horizon to \emph{two} straight lines $T=\pm R$. A surface $t=const$ is mapped to straight line $T=R\tanh\frac{t}{r_g}$; at $t\to\pm\infty$ it coincides with the horizon. Every point $(T,R)$ here represents a two-sphere.</p>
+Equations of null geodegics: $u=const$ and $v=const$, or in terms of $(T,R)$: $T=\pm R+const$. The surface  $r=const$ is mapped to a hyperbola $T^2-R^2=-2(r-r_g)e^{r/r_g}$; singularity to a hyperbola $T^2-R^2=2r_g$ with ''two'' branches also; the horizon to \emph{two} straight lines $T=\pm R$. A surface $t=const$ is mapped to straight line $T=R\tanh\frac{t}{r_g}$; at $t\to\pm\infty$ it coincides with the horizon. Every point $(T,R)$ here represents a two-sphere.</p>
 [[File:BHfig-KruskalCE.png|center|thumb|400px|The Kuskal diagram in coordinates $(T,R)$ and the mapping onto it of the coordinate grid $(t,r)$. Singularity is two hyperbola branches, in the past and future, while the horizon is two straight lines $T=\pm R$. Null geodesics are also straight lines $T\pm R=const$]]

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 === 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.
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-=== Problem 20: adding distance ===
+=== Problem 20: adding the two effects ===
 The source is freely falling the same way, while the detector is stationary at $r_{det}>r_{em}$.
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 === Problem 21: intensity ===
-=== Problem 21: registered 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?
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 For background on Killing vectors see problems [[Equations of General Relativity#equ_oto-kill1|K1]], [[Equations of General Relativity#equ_oto-kill2|K2]], [[Equations of General Relativity#equ_oto-kill3|K3]] of chapter 2.
 <div id="BlackHole19"></div>
-=== Problem 6: energy ===
+=== 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.
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 If a Killing vector field is timelike we can use the coordinate frame in which $K^\mu$ is the unitary vector of the time coordinate (see problem [[Equations of General Relativity#equ_oto-kill2|K2]] of chapter 2.) $K^{\mu}=(1,0,0,0)$, while the spacelike basis vectors are orthogonal to it. Then the integral of motion is energy in the corresponding (stationary) frame $p_{\mu}K^{\mu}=p_{0}\equiv\varepsilon/c$.
-Using the result (\ref{u0}), which holds for arbitrary gravitational field, we obtain the expression for energy, which is the integral of motion in a stationary metric
+Using the result [[Technical_warm-up#u0|u0]], which holds for arbitrary gravitational field, we obtain the expression for energy, which is the integral of motion in a stationary metric
 \begin{equation}\label{EnergyStat}
 	\varepsilon=mc^{2}u_{0}=