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| The [[Schwarzschild solution]] is one of the simplest and most useful solutions of the
| | == mainly taking with it a variety of treasures related. ' == |
| [[Einstein field equations]] (see [[general relativity]]). It describes [[spacetime]] in the vicinity of a non-rotating massive spherically-symmetric object. It is worthwhile deriving this metric in some detail; the following is a reasonably rigorous derivation that is not always seen in the textbooks.
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| == Assumptions and notation == | | By a number of strong, strong like a disciple. 'Baba Ta said,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_55.htm ゴルフ オークリー サングラス],' In the course of training,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_48.htm オークリー サングラス 人気ランキング], the Mo Yun vine can become strong and constantly evolving. Mo Yun vine evolutionary course there are many, mainly taking with it a variety of treasures related. '<br><br>'if placed on Earth,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_55.htm ゴルフ オークリー サングラス], can only absorb cosmic energy,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_70.htm オークリー サングラス 登山], the top seven or eight bands will star a line of strength.'<br><br>'But if you have a myriad of wood grain to it taking ya,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_5.htm 偏光サングラス オークリー], it can grow to a hundred percent 'cosmic'.' Baba Ta eyes shine, 'the owner of more than half of his wealth Mo Yun almost all cultivated vine,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_14.htm オークリー ゴルフ サングラス], that Mo Yun vine ...... finally grow up as 'immortal' masters rely Mo Yun vine but several escaped the catastrophe. just one last time, the enemy is too strong ...... Mo Yun vine also killed. '<br><br>'More than half of Fortune?' Luo Feng can not imagine,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_45.htm 自転車 サングラス オークリー].<br><br>a strong minion has nine immortal masterpiece strong, more than half of his wealth is how amazing. Arguably,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_45.htm オークリー サングラス ジョウボーン], the ship 'meteorite ink asterisk |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.3dfocus.com.cn/bbs/home.php?mod=space&uid=103345 http://www.3dfocus.com.cn/bbs/home.php?mod=space&uid=103345]</li> |
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| | <li>[http://happy-marron.com/cgi/diary/nicky/nicky.cgi http://happy-marron.com/cgi/diary/nicky/nicky.cgi]</li> |
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| | <li>[http://nucold.com/cgi-bin/guestbook/guestbook.cgi http://nucold.com/cgi-bin/guestbook/guestbook.cgi]</li> |
| | |
| | </ul> |
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| Working in a [[coordinate chart]] with coordinates <math> \left(r, \theta, \phi, t \right)</math> labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum. For the purposes of this article, these assumptions may be stated as follows (see the relevant links for precise definitions):
| | == then discard a pulse.' == |
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| (1) A [[spherically symmetric spacetime]] is one in which all metric components are unchanged under any rotation-reversal <math>\theta \rightarrow - \theta</math> or <math>\phi \rightarrow - \phi</math>.
| | Beast Warrior of the word of God, a spiritual teacher of the beast read the word of God. '<br><br>'both at the same practice, and more to help me study 'a beast of God'.' Luo Feng illegal channels.<br><br>vast sky.<br><br>Chaos Santo back to Luo Feng,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_56.htm オークリー偏光サングラス], standing void in a long time.<br><br>'teacher,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_12.htm スポーツサングラス オークリー].' Luofeng Gong Jing.<br><br>'decision?' Chaos Santo voice calm,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_7.htm オークリー サングラス 偏光].<br><br>'decision.' Luo Feng eyes very bright, 'disciples decided to Musha a pulse,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_30.htm オークリーサングラス カスタム], a pulse of all spiritual practice! really wrong if the disciples in the future, then discard a pulse.'<br><br>'Since you have decided, I will not say more, I will see you ...... Warrior, spiritual reading teacher fellow,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_26.htm オークリー サングラス 偏光レンズ], can go a step further.' Santo turned chaotic,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_16.htm オークリーサングラス一覧], but also looking at Luo Feng.<br><br>Luo Feng looked at the teacher.<br><br>eyes relative,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_15.htm オークリーサングラス ランキング], Luo Feng was not the slightest cowardice.<br><br>'good.' Chaos Santo nodded, 'Nothing wrong,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_24.htm オークリー サングラス フロッグスキン], but must adhere to the original mind, heart to go into.' |
| | | 相关的主题文章： |
| (2) A [[static spacetime]] is one in which all metric components are independent of the time coordinate <math>t</math> (so that <math>\frac {\part g_{\mu \nu}}{\part t}=0</math>) and the geometry of the spacetime is unchanged under a time-reversal <math>t \rightarrow -t</math>.
| | <ul> |
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| (3) A [[Einstein field equation|vacuum solution]] is one that satisfies the equation <math>T_{ab}=0</math>. From the [[Einstein field equations]] (with zero [[cosmological constant]]), this implies that <math>R_{ab}=0</math> (after contracting <math> R_{ab}-\frac{R}{2} g_{ab}=0</math> and putting <math>R = 0</math>).
| | <li>[http://www.hanban.com/plus/feedback.php?aid=314 http://www.hanban.com/plus/feedback.php?aid=314]</li> |
| | | |
| (4) [[Metric signature]] used here is <math>(-,+,+,+)</math>.
| | <li>[http://www.novaclassicrock.nl/cgi-bin/guestbook/guestbook.cgi http://www.novaclassicrock.nl/cgi-bin/guestbook/guestbook.cgi]</li> |
| | | |
| == Diagonalising the metric ==
| | <li>[http://www2u.biglobe.ne.jp/~ida2/cgi-bin/honey/honey.cgi http://www2u.biglobe.ne.jp/~ida2/cgi-bin/honey/honey.cgi]</li> |
| | | |
| The first simplification to be made is to diagonalise the metric. Under the [[coordinate transformation]], <math>(r, \theta, \phi, t) \rightarrow (r, \theta, \phi, -t)</math>, all metric components should remain the same. The metric components <math>g_{\mu 4}</math> (<math>\mu \ne 4</math>) change under this transformation as:
| | </ul> |
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| :<math>g_{\mu 4}'=\frac{\part x^{\alpha}}{\part x^{'\mu}} \frac{\part x^{\beta}}{\part x^{'4}} g_{\alpha \beta}= -g_{\mu 4}</math> (<math>\mu \ne 4</math>)
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| But, as we expect <math>g'_{\mu 4}= g_{\mu 4}</math> (metric components remain the same), this means that:
| |
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| :<math>g_{\mu 4}=\, 0</math> (<math>\mu \ne 4</math>) | |
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| Similarly, the coordinate transformations <math>(r, \theta, \phi, t) \rightarrow (r, \theta, -\phi, t)</math> and <math>(r, \theta, \phi, t) \rightarrow (r, -\theta, \phi, t)</math> respectively give:
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| :<math>g_{\mu 3}=\, 0</math> (<math>\mu \ne 3</math>)
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| :<math>g_{\mu 2}=\, 0</math> (<math>\mu \ne 2</math>)
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| Putting all these together gives:
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| :<math>g_{\mu \nu }=\, 0 </math> (<math> \mu \ne \nu </math>)
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| and hence the metric must be of the form:
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| :<math>ds^2=\, g_{11}\,d r^2 + g_{22} \,d \theta ^2 + g_{33} \,d \phi ^2 + g_{44} \,dt ^2</math>
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| where the four metric components are independent of the time coordinate <math>t</math> (by the static assumption).
| |
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| == Simplifying the components ==
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| On each [[hypersurface]] of constant <math>t</math>, constant <math>\theta</math> and constant <math>\phi</math> (i.e., on each radial line), <math>g_{11}</math> should only depend on <math>r</math> (by spherical symmetry). Hence <math>g_{11}</math> is a function of a single variable:
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| :<math>g_{11}=A\left(r\right)</math> | |
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| A similar argument applied to <math>g_{44}</math> shows that:
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| :<math>g_{44}=B\left(r\right)</math>
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| On the hypersurfaces of constant <math>t</math> and constant <math>r</math>, it is required that the metric be that of a 2-sphere:
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| :<math>dl^2=r_{0}^2 (d \theta^2 + \sin^2 \theta\, d \phi^2)</math>
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| Choosing one of these hypersurfaces (the one with radius <math>r_{0}</math>, say), the metric components restricted to this hypersurface (which we denote by <math>\tilde{g}_{22}</math> and <math>\tilde{g}_{33}</math>) should be unchanged under rotations through <math>\theta</math> and <math>\phi</math> (again, by spherical symmetry). Comparing the forms of the metric on this hypersurface gives:
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| :<math>\tilde{g}_{22}\left(d \theta^2 + \frac{\tilde{g}_{33}}{\tilde{g}_{22}} \,d \phi^2 \right) = r_{0}^2 (d \theta^2 + \sin^2 \theta \,d \phi^2)</math>
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| which immediately yields:
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| :<math>\tilde{g}_{22}=r_{0}^2</math> and <math>\tilde{g}_{33}=r_{0}^2 \sin ^2 \theta</math>
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| But this is required to hold on each hypersurface; hence,
| |
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| :<math>g_{22}=\, r^2</math> and <math>g_{33}=\, r^2 \sin^2 \theta</math>
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| Thus, the metric can be put in the form:
| |
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| :<math>ds^2=A\left(r\right)dr^2+r^2\,d \theta^2+r^2 \sin^2 \theta \,d \phi^2 + B\left(r\right) dt^2</math> | |
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| with <math>A</math> and <math>B</math> as yet undetermined functions of <math>r</math>. Note that if <math>A</math> or <math>B</math> is equal to zero at some point, the metric would be [[Mathematical singularity|singular]] at that point.
| |
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| == Calculating the Christoffel symbols ==
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| Using the metric above, we find the [[Christoffel symbols]], where the indices are <math>(0,1,2,3)=(r,\theta,\phi,t)</math>. The sign <math>'</math> denotes a total derivative of a function.
| |
| : <math>\Gamma^0_{ik} = \begin{bmatrix}
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| A'/\left( 2A \right) & 0 & 0 & 0\\
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| 0 & -r/A & 0 & 0\\
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| 0 & 0 & -r \sin^2 \theta /A & 0\\
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| 0 & 0 & 0 & -B'/\left( 2A \right) \end{bmatrix}</math>
| |
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| : <math>\Gamma^1_{ik} = \begin{bmatrix} | |
| 0 & 1/r & 0 & 0\\
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| 1/r & 0 & 0 & 0\\
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| 0 & 0 & -\sin\theta\cos\theta & 0\\
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| 0 & 0 & 0 & 0 \end{bmatrix}</math>
| |
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| : <math>\Gamma^2_{ik} = \begin{bmatrix}
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| 0 & 0 & 1/r & 0\\
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| 0 & 0 & \cot\theta & 0\\
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| 1/r & \cot\theta & 0 & 0 \\
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| 0 & 0 & 0 & 0 \end{bmatrix}</math>
| |
| | |
| : <math>\Gamma^3_{ik} = \begin{bmatrix}
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| 0 & 0 & 0 & B'/\left( 2B \right)\\
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| 0 & 0 & 0 & 0\\
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| 0 & 0 & 0 & 0 \\
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| B'/\left( 2B \right) & 0 & 0 & 0\end{bmatrix}</math>
| |
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| == Using the field equations to find <math>A(r)</math> and <math>B(r)</math> ==
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| To determine <math>A</math> and <math>B</math>, the [[Einstein's field equation|vacuum field equations]] are employed:
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| :<math>R_{ab}=\, 0</math>
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| Only four of these equations are nontrivial and upon simplification become:
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| <math>4 \dot{A} B^2 - 2 r \ddot{B} AB + r \dot{A} \dot{B}B + r \dot{B} ^2 A=0</math>
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| <math>r \dot{A}B + 2 A^2 B - 2AB - r \dot{B} A=0</math>
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| <math> - 2 r \ddot{B} AB + r \dot{A} \dot{B}B + r \dot{B} ^2 A - 4\dot{B} AB=0</math>
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| (The fourth equation is just <math>\sin^2 \theta</math> times the second equation)
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| where the dot means the ''r'' derivative of the functions.
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| Subtracting the first and third equations produces:
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| <math>\dot{A}B +A \dot{B}=0 \Rightarrow A(r)B(r) =K</math>
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| where <math>K</math> is a non-zero real constant. Substituting <math>A(r)B(r) \, =K</math> into the second equation and tidying up gives:
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| <math>r \dot{A} =A(1-A)</math> | |
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| which has general solution:
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| <math>A(r)=\left(1+\frac{1}{Sr}\right)^{-1}</math>
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| for some non-zero real constant <math>S</math>. Hence, the metric for a static, spherically symmetric vacuum solution is now of the form:
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| <math>ds^2=\left(1+\frac{1}{S r}\right)^{-1}dr^2+r^2(d \theta^2 + \sin^2 \theta d \phi^2)+K \left(1+\frac{1}{S r}\right)dt^2</math>
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| Note that the spacetime represented by the above metric is [[asymptotically flat]], i.e. as <math>r \rightarrow \infty</math>, the metric approaches that of the [[Minkowski metric]] and the spacetime manifold resembles that of [[Minkowski space]].
| |
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| == Using the Weak-Field Approximation to find <math>K</math> and <math>S</math> ==
| |
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| The geodesics of the metric (obtained where <math>ds</math> is extremised) must, in some limit (e.g., toward infinite speed of light), agree with the solutions of Newtonian motion (e.g., obtained by [[Lagrange equations]]). (The metric must also limit to [[Minkowski space]] when the mass it represents vanishes.)
| |
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| <math>0=\delta\int\frac{ds}{dt}dt=\delta\int(KE+PE_g)dt</math> | |
| | |
| (where <math>KE</math> is the kinetic energy and <math>PE_g</math> is the Potential Energy due to gravity) The constants <math>K</math> and <math>S</math> are fully determined by some variant of this approach; from the [[weak-field approximation]] one arrives at the result:
| |
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| <math>g_{44}=K\left(1 +\frac{1}{Sr}\right) \approx -c^2+\frac{2Gm}{r} = -c^2 \left(1-\frac{2Gm}{c^2 r} \right)</math>
| |
| | |
| where <math>G</math> is the [[gravitational constant]], <math>m</math> is the mass of the gravitational source and <math>c</math> is the speed of light. It is found that:
| |
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| <math>K=\, -c^2</math> and <math>\frac{1}{S}=-\frac{2Gm}{c^2}</math>
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| Hence:
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| <math>A(r)=\left(1-\frac{2Gm}{c^2 r}\right)^{-1}</math> and <math>B(r)=-c^2 \left(1-\frac{2Gm}{c^2 r}\right)</math>
| |
| | |
| So, the Schwarzschild metric may finally be written in the form:
| |
| | |
| <math>ds^2=\left(1-\frac{2Gm}{c^2 r}\right)^{-1}dr^2+r^2(d \theta^2 +\sin^2 \theta d \phi^2)-c^2 \left(1-\frac{2Gm}{c^2 r}\right)dt^2</math>
| |
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| Note that:
| |
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| <math>\frac{2Gm}{c^2}=r_s</math>
| |
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| is the definition of the [[Schwarzschild radius]] for an object of mass <math>m</math>, so the Schwarzschild metric may be rewritten in the alternative form:
| |
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| <math>ds^2=\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2(d\theta^2 +\sin^2\theta d\phi^2)-c^2\left(1-\frac{r_s}{r}\right)dt^2</math>
| |
| | |
| which shows that the metric becomes singular approaching the [[event horizon]] (that is, <math>r \rightarrow r_s</math>). The metric singularity is not a physical one (although there is a real physical singularity at <math>r=0</math>), as can be shown by using a suitable coordinate transformation (e.g. the [[Kruskal–Szekeres coordinates|Kruskal-Szekeres coordinate system]]).
| |
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| == Alternative form in isotropic coordinates ==
| |
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| The original formulation of the metric uses anisotropic coordinates in which the velocity of light is not the same in the radial and transverse directions. [[Arthur Eddington|A S Eddington]]<ref>A S Eddington, [http://books.google.com/books?id=Hhg0AAAAIAAJ&pg=PA93 "Mathematical Theory of Relativity"], Cambridge UP 1922 (2nd ed.1924, repr.1960), at [http://books.google.com/books?id=Hhg0AAAAIAAJ&pg=PA85 page 85] and [http://books.google.com/books?id=Hhg0AAAAIAAJ&pg=PA93 page 93]. Symbol usage in the Eddington source for interval s and time-like coordinate t has been converted for compatibility with the usage in the derivation above.</ref> gave alternative forms in [[isotropic coordinates]]. For isotropic spherical coordinates <math>r_1</math>, <math>\theta</math>, <math>\phi</math>, coordinates <math>\theta</math> and <math>\phi</math> are unchanged, and then (provided r >= 2Gm/c<sup>2</sup> <ref>H. A. Buchdahl, "Isotropic coordinates and Schwarzschild metric", International Journal of Theoretical Physics, Vol.24 (1985) pp.731-739.</ref>)
| |
| | |
| <math>r = r_1 \left(1+\frac{Gm}{2c^2 r_1}\right)^{2}</math> . . ., <math>dr = dr_1 \left(1-\frac{(Gm)^2}{4c^4 r_1^2}\right)</math> . . ., and
| |
| | |
| <math>\left(1-\frac{2Gm}{c^2 r}\right) = \left(1-\frac{Gm}{2c^2 r_1}\right)^{2}/\left(1+\frac{Gm}{2c^2 r_1}\right)^{2}</math> . . .
| |
| | |
| Then for isotropic rectangular coordinates <math>x</math>, <math>y</math>, <math>z</math>,
| |
| | |
| <math>x = r_1\, \sin(\theta)\, \cos(\phi) \dots,</math> <math>y = r_1\, \sin(\theta)\, \sin(\phi) \dots,</math> <math>z = r_1\, \cos(\theta) \dots</math>
| |
| | |
| The metric then becomes, in isotropic rectangular coordinates:
| |
| | |
| <math>ds^2= \left(1+\frac{Gm}{2c^2 r_1}\right)^{4}(dx^2+dy^2+dz^2) -c^2 dt^2 \left(1-\frac{Gm}{2c^2 r_1}\right)^{2}/\left(1+\frac{Gm}{2c^2 r_1}\right)^{2}</math> . . . | |
| | |
| == Dispensing with the static assumption - Birkhoff's theorem==
| |
| | |
| In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and [[Static spacetime|static]]. In fact, the static assumption is stronger than required, as [[Birkhoff's theorem (relativity)|Birkhoff's theorem]] states that any spherically symmetric vacuum solution of [[Einstein's field equations]] is [[Stationary spacetime|stationary]]; then one obtains the Schwarzschild solution. Birkhoff's theorem has the consequence that any pulsating star which remains spherically symmetric cannot generate [[gravitational wave]]s (as the region exterior to the star must remain static).
| |
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| == See also ==
| |
| * [[Kerr metric]]
| |
| * [[Reissner-Nordström metric]]
| |
| | |
| == References ==
| |
| <references />
| |
| | |
| [[Category:Exact solutions in general relativity]]
| |
| [[Category:Article proofs]]
| |
mainly taking with it a variety of treasures related. '
By a number of strong, strong like a disciple. 'Baba Ta said,ゴルフ オークリー サングラス,' In the course of training,オークリー サングラス 人気ランキング, the Mo Yun vine can become strong and constantly evolving. Mo Yun vine evolutionary course there are many, mainly taking with it a variety of treasures related. '
'if placed on Earth,ゴルフ オークリー サングラス, can only absorb cosmic energy,オークリー サングラス 登山, the top seven or eight bands will star a line of strength.'
'But if you have a myriad of wood grain to it taking ya,偏光サングラス オークリー, it can grow to a hundred percent 'cosmic'.' Baba Ta eyes shine, 'the owner of more than half of his wealth Mo Yun almost all cultivated vine,オークリー ゴルフ サングラス, that Mo Yun vine ...... finally grow up as 'immortal' masters rely Mo Yun vine but several escaped the catastrophe. just one last time, the enemy is too strong ...... Mo Yun vine also killed. '
'More than half of Fortune?' Luo Feng can not imagine,自転車 サングラス オークリー.
a strong minion has nine immortal masterpiece strong, more than half of his wealth is how amazing. Arguably,オークリー サングラス ジョウボーン, the ship 'meteorite ink asterisk
相关的主题文章：
then discard a pulse.'
Beast Warrior of the word of God, a spiritual teacher of the beast read the word of God. '
'both at the same practice, and more to help me study 'a beast of God'.' Luo Feng illegal channels.
vast sky.
Chaos Santo back to Luo Feng,オークリー偏光サングラス, standing void in a long time.
'teacher,スポーツサングラス オークリー.' Luofeng Gong Jing.
'decision?' Chaos Santo voice calm,オークリー サングラス 偏光.
'decision.' Luo Feng eyes very bright, 'disciples decided to Musha a pulse,オークリーサングラス カスタム, a pulse of all spiritual practice! really wrong if the disciples in the future, then discard a pulse.'
'Since you have decided, I will not say more, I will see you ...... Warrior, spiritual reading teacher fellow,オークリー サングラス 偏光レンズ, can go a step further.' Santo turned chaotic,オークリーサングラス一覧, but also looking at Luo Feng.
Luo Feng looked at the teacher.
eyes relative,オークリーサングラス ランキング, Luo Feng was not the slightest cowardice.
'good.' Chaos Santo nodded, 'Nothing wrong,オークリー サングラス フロッグスキン, but must adhere to the original mind, heart to go into.'
相关的主题文章：