1. 개요
옙센-버크호프 정리(Jebsen-Birkhoff theorem)는 일반 상대론에서 진공, 구형 대칭인 해가 유일함을 설명하는 정리이다. 다시 말해, 그러한 해는 (다른 형태로 구해지더라도) 결국 슈바르츠실트 해와 똑같은 해이다. 따라서 다른 형태의 해를 굳이 고려할 필요가 없으며, 무엇보다 이러한 조건에서 중력장이 자동으로 정적인 형태가 되므로 구형 대칭 중력장에서는 중력 복사가 발생하지 않는다는 사실을 알 수 있다. 예를 들어, 단일 항성이 중력 붕괴를 하는 경우가 이에 해당한다. 노르웨이의 외르그 토프테 옙센(노르웨이어: Jørg Tofte Jebsen)이 1921년에 그리고 미국의 조지 데이비드 버크호프(George David Birkhoff)가 1923년에 각자 개별적으로 유도하였는데, 옙센의 발견은 나중에 알려져서 보통은 버크호프(버코프) 정리라고 알려져 있다. (허블 법칙 -> 허블-르메트르 법칙과 유사)[가][나][다][바][5][라][마]이것을 보이는 과정은 결국 아인슈타인 방정식을 푸는 것이다. 대신 슈바르츠실트 해와 다른 점은 여기에서는 각 함수를 [math(r)] 만이 아니라 (1)처럼 [math(r, \, t)]의 함수로 가정한다는 것인데, 요점은 결국 [math(t)]가 사라진다는 것이다. 리치 텐서의 각 성분은 (2)와 같이 계산된다.
[math( ds^2 = - A(r,t)dr^2 - 2B(r,t)dtdr -C(r,t)(d\theta^2 +\sin^2 \theta d\phi^2) +D(r,t)dt^2 \qquad \cdots~(1) )]
[math( \begin{pmatrix} R_{rr} = \dfrac{D}{2D} - \dfrac{D'}{4D}\left( \dfrac{A'}{A}+\dfrac{D'}{D} \right) - \dfrac{A'C'}{2AC}+\dfrac{C}{C} -\dfrac{C''}{2C^2} \\ \\
[math( \begin{pmatrix} R_{rr} = \dfrac{D}{2D} - \dfrac{D'}{4D}\left( \dfrac{A'}{A}+\dfrac{D'}{D} \right) - \dfrac{A'C'}{2AC}+\dfrac{C}{C} -\dfrac{C''}{2C^2} \\ \\
R_{\theta\theta} = \dfrac{C'}{4A}\left( -\dfrac{A'}{A} +\dfrac{D'}{D}\right)+\dfrac{C''}{2A}-1 \\ \\
R_{\phi\phi} = \sin^2\theta R_{\theta\theta} \\ \\
R_{tt} = -\dfrac{D''}{2A}+\dfrac{D'}{2A} \left( \dfrac{A'}{A} +\dfrac{D'}{D}\right) -\dfrac{D'C'}{2AC} \end{pmatrix} \qquad \cdots~(2) )]
R_{\phi\phi} = \sin^2\theta R_{\theta\theta} \\ \\
R_{tt} = -\dfrac{D''}{2A}+\dfrac{D'}{2A} \left( \dfrac{A'}{A} +\dfrac{D'}{D}\right) -\dfrac{D'C'}{2AC} \end{pmatrix} \qquad \cdots~(2) )]
2. 리치텐서
(2)를 정리하고[math( R_{11} = \dfrac{v' }{2v} - \dfrac{v'' }{4v^2} -\dfrac{v'\lambda' }{4v \lambda} - \dfrac{1}{r}\dfrac{\lambda'}{\lambda})]
[math( R_{22} = \dfrac{v' r}{2v \lambda} + \dfrac{1}{\lambda} -\dfrac{\lambda' r}{2\lambda\lambda} - 1 )]
[math( R_{33} = \left( \dfrac{rv' }{2v \lambda} + \dfrac{1}{\lambda} -\dfrac{r\lambda' }{2\lambda^2} - 1 \right) sin^2\theta = R_{22}\, sin^2\theta )]
[math( R_{44} = -\dfrac{v' }{2\lambda} + \dfrac{v' \lambda' }{4\lambda^2} +\dfrac{v'' }{4v \lambda} - \dfrac{1}{r}\dfrac{v'}{\lambda})]
일반적으로(generalized) 접근할수있는 대칭구형(symmetric sphere)에서 접평면(tangent plane)으로 다루어지는 리치텐서(Ricci tensor)들을 조사할 수 있다.
3. 슈바르츠실트 해
1921,1923년 에딩턴(A. S. EDDINGTON),1943년 리차드 톨먼(Richard Chace Tolman)등이 사용한 일반적인 슈바르츠실트 해를 얻는 과정으로 다루어지는 전형적인 아인슈타인 텐서의 표준 접근 경로[다]82.7[마][math( G_{11} = - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right))]
[math( G_{44} = + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
이와 비교해서 아래는 1921년 엡센의 슈바르츠실트 해에대한 리치텐서 단계에서의 빠른 접근 경로[가][바]
[math( g^{11}\left( R_{11} - \dfrac{1}{4}g_{11}R \right) -g^{44}\left( R_{44} - \dfrac{1}{4}g_{44}R \right) )]
옙센-버크호프 정리는 슈바르츠실트 해에 접근할때 [math(R_{33},R_{44})] 및 리치 스칼라 곡률 계산 생략이 가능하다.
이러한 옙센-버크호프 정리가 보여주는 빠른 경로에 대한 아이디어는 초창기 슈바르츠실트 솔루션(solution,해)을 직접 자신이 푼 카를 슈바르츠실트(Karl Schwarzschild)가 1916년에 <(직역)아인슈타인의 이론에 따른 질량 점의 중력장에 대해서 (Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie)>에서 보다 빠른 접근 경로를 기술한바있다.[바] 카를 슈바르츠실트는 그의 논문에서 특징된 다양체(manifold) 공간에서 측지선(geodesic line)을 따라 이동을 가정하고 얻은 중력장의 구성 요소들인 크리스토펠 심볼(symbol,기호)들 만으로 이를 기술했다.
다음은 슈바르츠실트(Schwarzschild)가 1916년에 솔루션(해)를 얻을때 사용한 크리스토펠 심볼(표기는 편미분)이다.[바]
[math( \Gamma _{11}^{1}=-{\dfrac {1}{2}}{\dfrac {1}{f_{1}}}{\dfrac {\partial f_{1}}{\partial x_{1}}},\quad \Gamma _{22}^{1}=+{\dfrac {1}{2}}{\dfrac {1}{f_{1}}}{\dfrac {\partial f_{2}}{\partial x_{1}}}{\dfrac {1}{1-x_{2}^{2}}}, \Gamma _{33}^{1}=+{\dfrac {1}{2}}{\dfrac {1}{f_{1}}}{\dfrac {\partial f_{2}}{\partial x_{1}}}\left(1-x_{2}^{2}\right),\quad \Gamma _{44}^{1}=-{\dfrac {1}{2}}{\dfrac {1}{f_{1}}}{\dfrac {\partial f_{4}}{\partial x_{1}}},)]
[math( \Gamma _{21}^{2}=-{\dfrac {1}{2}}{\dfrac {1}{f_{2}}}{\dfrac {\partial f_{2}}{\partial x_{1}}},\quad \Gamma _{22}^{2}=-{\dfrac {x_{2}}{1-x_{2}^{2}}},\quad \Gamma _{33}^{2}=-x_{2}\left(1-x_{2}^{2}\right),)]
[math( \Gamma _{31}^{3}=-{\dfrac {1}{2}}{\dfrac {1}{f_{2}}}{\dfrac {\partial f_{2}}{\partial x_{1}}},\quad \Gamma _{32}^{3}=+{\dfrac {x_{2}}{1-x_{2}^{2}}},)]
[math( \Gamma _{41}^{4}=-{\dfrac {1}{2}}{\dfrac {1}{f_{4}}}{\dfrac {\partial f_{4}}{\partial x_{1}}})]
4. 옙센 정리
노르웨이의 외르그 토프테 옙센(노르웨이어: Jørg Tofte Jebsen)이 1921년에 발표한 이러한 옙센 정리의 아이디어는 본인이 그의 논문에서 기술한 바와 같이 힐베르트 액션으로 이해될수있는 1917년의 힐베르트가 사용한 일반적(비정적) 구형 대칭사례(general—hence not static—spherically symmetric case)의 접근방법으로부터 얻을수있는 여러 가능한 미분방정식의 새로운 해를 예상함으로써 이러한 아이디어를 제안하고 있다.[14][15]4.1. 리치텐서
옙센(Jebsen)이 1921년에 발표한 옙센 정리(Jebsen theorem)의 리치 텐서(리치곡률텐서)[math( R_{mn} = \dfrac{\partial}{\partial x^n}\displaystyle\sum_{k} \begin{bmatrix}mk\\ k \end{bmatrix}- \displaystyle\sum_{k}\dfrac{\partial}{\partial x^k} \begin{bmatrix}mn\\ k \end{bmatrix}+\displaystyle\sum_{k,l} \begin{pmatrix} \begin{bmatrix}mk\\ l \end{bmatrix} \begin{bmatrix}nl\\ k \end{bmatrix} - \begin{bmatrix}mn\\ k \end{bmatrix}\begin{bmatrix}kl\\ l \end{bmatrix} \end{pmatrix} )]
아서 스탠리 에딩턴(A. S. Eddington)이 1921년과 1923년에 발표한 에딩턴 방법(Eddington method)의 리치곡률텐서
[math( G_{\mu\nu} = -\dfrac{\partial}{\partial x_{\alpha}} \{\mu\nu,\alpha\} +\{\mu\alpha,\beta \}\{\nu\beta,\alpha \} + \dfrac{\partial^2}{\partial x_{\mu}\partial x_{\nu}} log\sqrt{-g} -\{\mu\nu,\alpha\} \dfrac{\partial}{\partial x_{\alpha}} log\sqrt{-g} )] [마]37.2
4.2. 비앙키 항등식
1902년 루이지 비앙키(Luigi Bianchi)가 비앙키 항등식을 제안할때 리만기호(Riemann symbol)인 리만-크리스토펠 곡률 텐서를 사용한 리만-리치 항등식(Riemann-Ricci identities)을 도입하였다.[가][math( (rk,ih) = \dfrac{\partial }{\partial x_h} \begin{bmatrix} ri \\ k \end{bmatrix} - \dfrac{\partial }{\partial x_i} \begin{bmatrix} rh \\ k \end{bmatrix} + \displaystyle\sum_{\lambda,\mu}^{1...n} A_{\lambda \mu} \begin{Bmatrix} \begin{bmatrix} rh \\ \lambda \end{bmatrix}\cdot \begin{bmatrix} ik \\ \mu \end{bmatrix} - \begin{bmatrix} ri \\ \lambda \end{bmatrix}\cdot \begin{bmatrix} hk \\ \mu \end{bmatrix} \end{Bmatrix} )]
비앙키 항등식을 보여주는 리만-리치 항등식(Riemann-Ricci identities)은 리치곡률텐서를 표현하는 옙센 정리의 기초형태를 제공할수있다.
4.3. 옙센 정리
옙센 정리((Jebsen theorem)의 기본형[math( R_{mn} = \dfrac{\partial}{\partial x}\displaystyle\sum_{k} \begin{bmatrix}mk\\ k \end{bmatrix}- \displaystyle\sum_{k}\dfrac{\partial}{\partial x} \begin{bmatrix}mn\\ k \end{bmatrix}+\displaystyle\sum_{k,l} \begin{pmatrix} \begin{bmatrix}mk\\ l \end{bmatrix} \begin{bmatrix}nl\\ k \end{bmatrix} - \begin{bmatrix}mn\\ k \end{bmatrix}\begin{bmatrix}kl\\ l \end{bmatrix} \end{pmatrix} )]
3색인 크리스토펠 심볼(three-index symbols)의 계산
[math(ds^2 = g_{11}dr^2 + g_{22}d\theta^2 + g_{33}d\phi^2 + g_{44}dt^2 )]
[math(ds^2 = -\lambda dr^2 -r^2 d\theta^2 -r^2 \sin^2\theta d\phi^2 + v dt^2 )]로부터
[math( g_{\mu\nu} = \begin{pmatrix} -\lambda & 0 & 0 & 0 \\ 0 & -r^2 & 0 & 0 \\ 0 & 0 & -r^2 \sin^2\theta & 0 \\ 0 & 0 & 0 & v \end{pmatrix} )]
[math(\{ \mu\nu,\alpha\} = \dfrac{1}{2}g^{\sigma \alpha} \left( \dfrac{\partial g_{\mu\sigma}}{\partial x_{ \nu}} + \dfrac{\partial g_{\sigma\nu}}{\partial x_{\mu}} - \dfrac{\partial g_{\nu\mu}}{\partial x_{\sigma}} \right) )]이다. 그리고 [math( g^{\square^1 \square^2} , \square^1 \neq \square^2 = 0 )] 이므로
[math(\{ \mu\mu,\mu \} = \dfrac{1}{2}g^{\mu\mu} \dfrac{\partial g_{\mu\mu}}{\partial x_{\mu}} , \{ \mu\mu,\nu \} = -\dfrac{1}{2}g^{\nu\nu} \dfrac{\partial g_{\mu\mu}}{\partial x_{\nu}} ,\{ \mu\nu,\nu \} = \dfrac{1}{2}g^{\nu\nu} \dfrac{\partial g_{\nu\nu}}{\partial x_{\mu}} , \{ \mu\nu,\sigma \} = 0)]
따라서
[math(\{ 11,1 \} = \dfrac{1}{2}\dfrac{1}{\lambda} \lambda' = \Gamma^{1}_{11}= \begin{bmatrix} 11\\ 1 \end{bmatrix} = \dfrac{\Lambda'}{2\Lambda} )]
[math(\{ 22,1 \} = -\dfrac{1}{2} 2r\lambda^{-1} = \Gamma^{1}_{22} = \begin{bmatrix} 22\\ 1 \end{bmatrix} = - \dfrac{r}{\Lambda} )]
[math(\{ 33,1 \} = -\dfrac{1}{2} 2r\sin^2\theta\lambda^{-1} = \Gamma^{1}_{33} = \begin{bmatrix} 33\\ 1 \end{bmatrix} = - \dfrac{r}{\Lambda}\sin^2\theta )]
[math(\{ 44,1 \} = \dfrac{1}{2} v'\lambda^{-1} = \Gamma^{1}_{44} = \begin{bmatrix} 44\\ 1 \end{bmatrix} = \dfrac{V'}{2\Lambda} )]
[math(\{ 12,2 \} = \dfrac{1}{2} \dfrac{1}{r^2}2r = \dfrac{1}{r}=\Gamma^{2}_{12}= \Gamma^{2}_{21} = \begin{bmatrix} 12\\ 2 \end{bmatrix}= \begin{bmatrix} 21\\ 2 \end{bmatrix} = \dfrac{1}{r} )]
[math(\{ 33,2 \} = - \dfrac{1}{2} \dfrac{1}{r^2} r^2 2\sin\theta\cos\theta = -\sin\theta\cos\theta = \Gamma^{2}_{33} = \begin{bmatrix} 33\\ 2 \end{bmatrix} = -\sin\theta\cos\theta )]
[math(\{ 13,3 \} = \dfrac{1}{2} \dfrac{1}{r^2 \sin^2 \theta}2r\sin^2 \theta = \dfrac{1}{r}=\Gamma^{3}_{13}= \Gamma^{3}_{31} = \begin{bmatrix} 13\\ 3 \end{bmatrix}= \begin{bmatrix} 31\\ 3 \end{bmatrix} =\dfrac{1}{r} )]
[math(\{ 23,3 \} = \dfrac{1}{2} \dfrac{1}{r^2 \sin^2 \theta} r^2 2\sin\theta\cos\theta = \dfrac{\cos\theta}{\sin\theta} = \cot\theta = \Gamma^{3}_{23} = \begin{bmatrix} 23\\ 3 \end{bmatrix} = \cot\theta )]
[math(\{ 14,4 \} = \dfrac{1}{2} \dfrac{1}{v}v' = \Gamma^{4}_{14}= \Gamma^{4}_{41} = \begin{bmatrix} 14\\ 4 \end{bmatrix} = \begin{bmatrix} 41\\ 4 \end{bmatrix} = \dfrac{V'}{2 V} )]
[math(\{ 41,1 \} = \{ 14,1 \} = \begin{bmatrix} 14\\ 1 \end{bmatrix} =\begin{bmatrix} 41\\ 1 \end{bmatrix} = 0)],[math(\{ 11,4 \} = \begin{bmatrix} 11\\ 4 \end{bmatrix} = 0)],[math(\{ 44,4 \} = \begin{bmatrix} 44\\ 4 \end{bmatrix} = 0)]등 나머지는 [math( 0)]
텐서 행렬로 정리헤 보면
[math(\Gamma^{1}_{mn} = \left( \begin{array}{rrrr} \dfrac{\Lambda'}{2\Lambda} \;\; && 0 \;\; && 0 \;\; && 0 \\ 0 \;\; && - \dfrac{r}{\Lambda} \;\; && 0 \;\; && 0 \\ 0 \;\; && 0 \;\; && - \dfrac{r}{\Lambda}\sin^2\theta \;\; && 0 \\ 0 \;\; && 0 \;\; && 0 \;\; && \dfrac{V'}{2\Lambda} \end{array} \right) )]
[math(\Gamma^{2}_{mn} = \left( \begin{array}{rrrr} 0 \;\; && \dfrac{1}{r} \;\; && 0 \;\; && 0 \\ \dfrac{1}{r} \;\; && 0 \;\; && 0 \;\; && 0 \\ 0 \;\; && 0 \;\; && -\sin \theta \cos \theta \;\; && 0 \\ 0 \;\; && 0 \;\; && 0 \;\; && 0 \end{array} \right) )]
[math(\Gamma^{3}_{mn} = \left( \begin{array}{rrrr} 0 \;\; && 0 \;\; && \dfrac{1}{r} \;\; && 0 \\ 0 \;\; && 0 \;\; && \cot \theta \;\; && 0 \\ \dfrac{1}{r} \;\; && \cot \theta \;\; && 0 \;\; && 0 \\ 0 \;\; && 0 \;\; && 0 \;\; && 0 \end{array} \right) )]
[math(\Gamma^{4}_{mn} = \left( \begin{array}{rrrr} 0 \;\; && 0 \;\; && 0 \;\; && \dfrac{V'}{2 V} \\ 0 \;\; && 0 \;\; && 0 \;\; && 0 \\ 0 \;\; && 0 \;\; && 0 \;\; && 0 \\ \dfrac{V'}{2 V} \;\; && 0 \;\; && 0 \;\; && 0 \end{array} \right) )]
와 같은 행렬텐서를 얻을수있다.
4.4. 계산 예
옙센 정리((Jebsen theorem) 기본형으로부터 행렬텐서를 사용한 리치텐서 계산 예시[math( R_{mn}=R_{11} = \dfrac{\partial}{\partial x}\displaystyle\sum_{k} \begin{bmatrix}mk\\ k \end{bmatrix}- \displaystyle\sum_{k}\dfrac{\partial}{\partial x} \begin{bmatrix}mn\\ k \end{bmatrix}+\displaystyle\sum_{k,l} \begin{pmatrix} \begin{bmatrix}mk\\ l \end{bmatrix} \begin{bmatrix}nl\\ k \end{bmatrix} - \begin{bmatrix}mn\\ k \end{bmatrix}\begin{bmatrix}kl\\ l \end{bmatrix} \end{pmatrix} )]
계산해보면
[math( R_{11} = \dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 1 \end{bmatrix} +\begin{bmatrix}12\\ 2 \end{bmatrix} + \begin{bmatrix}13\\ 3 \end{bmatrix} + \begin{bmatrix}14\\ 4 \end{bmatrix} \right) - \left( \dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 1 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 2 \end{bmatrix} \right)+ \dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 3 \end{bmatrix} \right) +\dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 4 \end{bmatrix} \right) \right)
)]
[math( +\left( \begin{bmatrix} 11 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix}11 \\ 2 \end{bmatrix} \begin{bmatrix} 12 \\ 1 \end{bmatrix} + \begin{bmatrix}11 \\ 3 \end{bmatrix} \begin{bmatrix} 13 \\ 1 \end{bmatrix} + \begin{bmatrix}11 \\ 4 \end{bmatrix} \begin{bmatrix} 14 \\ 1 \end{bmatrix} \right) +\left( \begin{bmatrix} 12 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 2 \end{bmatrix}+ \begin{bmatrix}12 \\ 2 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix}12 \\ 3 \end{bmatrix} \begin{bmatrix} 13 \\ 2 \end{bmatrix}+ \begin{bmatrix}12 \\ 4 \end{bmatrix} \begin{bmatrix} 14 \\ 2 \end{bmatrix} \right) +\left( \begin{bmatrix} 13 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 3 \end{bmatrix}+ \begin{bmatrix}13 \\ 2 \end{bmatrix} \begin{bmatrix} 12 \\ 3 \end{bmatrix} + \begin{bmatrix}13 \\ 3 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix}+ \begin{bmatrix}13 \\ 4 \end{bmatrix} \begin{bmatrix} 14 \\ 3 \end{bmatrix} \right) +\left( \begin{bmatrix} 14 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 4 \end{bmatrix}+ \begin{bmatrix}14 \\ 2 \end{bmatrix} \begin{bmatrix} 12 \\ 4 \end{bmatrix} + \begin{bmatrix}14 \\ 3 \end{bmatrix} \begin{bmatrix} 13 \\ 4 \end{bmatrix}+ \begin{bmatrix}14 \\ 4 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) )]
[math( -\left( \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix}11 \\ 2 \end{bmatrix} \begin{bmatrix} 21 \\ 1 \end{bmatrix}+ \begin{bmatrix}11 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 2 \end{bmatrix} + \begin{bmatrix}11 \\ 2 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix} + \begin{bmatrix}11 \\ 2 \end{bmatrix} \begin{bmatrix} 24 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix}11 \\ 3 \end{bmatrix} \begin{bmatrix} 31 \\ 1 \end{bmatrix}+ \begin{bmatrix}11 \\ 3 \end{bmatrix} \begin{bmatrix} 32 \\ 2 \end{bmatrix} + \begin{bmatrix}11 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 3 \end{bmatrix} + \begin{bmatrix}11 \\ 3 \end{bmatrix} \begin{bmatrix} 34 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix}11 \\ 4 \end{bmatrix} \begin{bmatrix} 41 \\ 1 \end{bmatrix}+ \begin{bmatrix}11 \\ 4 \end{bmatrix} \begin{bmatrix} 42 \\ 2 \end{bmatrix} + \begin{bmatrix}11 \\ 2 \end{bmatrix} \begin{bmatrix} 43 \\ 3 \end{bmatrix} + \begin{bmatrix}11 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 4 \end{bmatrix} \right) )]
정리하면[18]
[math( R_{11} = \dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 1 \end{bmatrix} +\begin{bmatrix}12\\ 2 \end{bmatrix} + \begin{bmatrix}13\\ 3 \end{bmatrix} + \begin{bmatrix}14\\ 4 \end{bmatrix} \right) - \dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 1 \end{bmatrix} \right) + \begin{bmatrix}11\\ 1 \end{bmatrix}\begin{bmatrix}11\\ 1 \end{bmatrix} + \begin{bmatrix}12\\ 2 \end{bmatrix}\begin{bmatrix}12\\ 2 \end{bmatrix} + \begin{bmatrix}13\\ 3 \end{bmatrix}\begin{bmatrix}13\\ 3 \end{bmatrix} + \begin{bmatrix}14\\ 4 \end{bmatrix}\begin{bmatrix}14\\ 4 \end{bmatrix} -\left( \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) )]
정리하면
[math( R_{11} =\dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 1 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left(\begin{bmatrix}12\\ 2 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left(\begin{bmatrix}13\\ 3 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left(\begin{bmatrix}14\\ 4 \end{bmatrix} \right) - \dfrac{\partial}{\partial x}\left( \begin{bmatrix}11\\ 1 \end{bmatrix} \right) + \begin{bmatrix}11\\ 1 \end{bmatrix}^2 + \begin{bmatrix}12\\ 2 \end{bmatrix}^2 + \begin{bmatrix}13\\ 3 \end{bmatrix}^2 + \begin{bmatrix}14\\ 4 \end{bmatrix}^2 -\begin{bmatrix} 11 \\ 1 \end{bmatrix}^2 - \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} - \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} - \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} )]
[math( = \dfrac{\partial}{\partial x}\left(\begin{bmatrix}12\\ 2 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left(\begin{bmatrix}13\\ 3 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left(\begin{bmatrix}14\\ 4 \end{bmatrix} \right) + \begin{bmatrix}12\\ 2 \end{bmatrix}^2 + \begin{bmatrix}13\\ 3 \end{bmatrix}^2 + \begin{bmatrix}14\\ 4 \end{bmatrix}^2 - \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} - \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} - \begin{bmatrix}11 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} )]
[math( = \dfrac{\partial}{\partial x}\left(\dfrac{1}{r} \right) + \dfrac{\partial}{\partial x}\left(\dfrac{1}{r} \right) + \dfrac{\partial}{\partial x}\left( \dfrac{V'}{2V} \right) + \left(\dfrac{1}{r}\right)^2 + \left(\dfrac{1}{r}\right)^2 + \left( \dfrac{V'}{2V} \right)^2 - \dfrac{\Lambda'}{2\Lambda} \dfrac{1}{r} - \dfrac{\Lambda'}{2\Lambda}\dfrac{1}{r} - \dfrac{\Lambda'}{2\Lambda} \dfrac{V'}{2V} )]
[math( = -\left(\dfrac{1}{r} \right)^{2} - \left(\dfrac{1}{r} \right)^{2} + \dfrac{V'}{2V} + \left(\dfrac{1}{r}\right)^2 + \left(\dfrac{1}{r}\right)^2 + \left( \dfrac{V'}{2V} \right)^2 - \dfrac{\Lambda'}{2\Lambda} \dfrac{1}{r} - \dfrac{\Lambda'}{2\Lambda}\dfrac{1}{r} - \dfrac{\Lambda'}{2\Lambda} \dfrac{V'}{2V} )]
[math( = \dfrac{V'}{2V} + \left( \dfrac{V'}{2V} \right)^2 - \dfrac{\Lambda'}{2\Lambda} \dfrac{1}{r} - \dfrac{\Lambda'}{2\Lambda}\dfrac{1}{r} - \dfrac{\Lambda'}{2\Lambda} \dfrac{V'}{2V} )]
[math( = \dfrac{V'}{2V} + \dfrac{1}{4}\left(\dfrac{V'}{V} \right)^2 - 2\dfrac{\Lambda'}{2\Lambda} \dfrac{1}{r} - \dfrac{1}{4}\dfrac{\Lambda'V'}{\Lambda V} )]
[math( = \dfrac{V'}{2V} + \dfrac{1}{4}\left(\dfrac{V'}{V} \right)^2 - \dfrac{\Lambda'}{\Lambda r} - \dfrac{1}{4}\dfrac{\Lambda'V'}{\Lambda V} )] 을 얻고 [math( \lambda' = \dfrac{\Lambda'}{\Lambda} , v' = \dfrac{V'}{V} )] 를 취해서
[math( R_{11} = \dfrac{1}{2} v' -\dfrac{1}{4} v'' - \dfrac{\lambda '}{r} -\dfrac{1}{4}\lambda ' v' )] [사]38.61 을 조사할 수 있다.
[math( R_{22} = \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 21\\ 1 \end{bmatrix} +\begin{bmatrix} 22\\ 2 \end{bmatrix} + \begin{bmatrix} 23\\ 3 \end{bmatrix} + \begin{bmatrix} 24\\ 4 \end{bmatrix} \right) - \left( \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 22\\ 1 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 22\\ 2 \end{bmatrix} \right)+ \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 22\\ 3 \end{bmatrix} \right) +\dfrac{\partial}{\partial x}\left( \begin{bmatrix} 22\\ 4 \end{bmatrix} \right) \right)
)]
[math( +\left( \begin{bmatrix} 21 \\ 1 \end{bmatrix} \begin{bmatrix} 21 \\ 1 \end{bmatrix}+ \begin{bmatrix} 21 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 1 \end{bmatrix} + \begin{bmatrix} 21 \\ 3 \end{bmatrix} \begin{bmatrix} 23 \\ 1 \end{bmatrix} + \begin{bmatrix} 21 \\ 4 \end{bmatrix} \begin{bmatrix} 24 \\ 1 \end{bmatrix} \right) +\left( \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 21 \\ 2 \end{bmatrix}+ \begin{bmatrix} 22 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 2 \end{bmatrix} + \begin{bmatrix} 22 \\ 3 \end{bmatrix} \begin{bmatrix} 23 \\ 2 \end{bmatrix}+ \begin{bmatrix} 22 \\ 4 \end{bmatrix} \begin{bmatrix} 24 \\ 2 \end{bmatrix} \right) +\left( \begin{bmatrix} 23 \\ 1 \end{bmatrix} \begin{bmatrix} 21 \\ 3 \end{bmatrix}+ \begin{bmatrix} 23 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 3 \end{bmatrix} + \begin{bmatrix} 23 \\ 3 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix}+ \begin{bmatrix} 23 \\ 4 \end{bmatrix} \begin{bmatrix} 24 \\ 3 \end{bmatrix} \right) +\left( \begin{bmatrix} 24 \\ 1 \end{bmatrix} \begin{bmatrix} 21 \\ 4 \end{bmatrix}+ \begin{bmatrix} 24 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 4 \end{bmatrix} + \begin{bmatrix} 24 \\ 3 \end{bmatrix} \begin{bmatrix} 23 \\ 4 \end{bmatrix}+ \begin{bmatrix} 24 \\ 4 \end{bmatrix} \begin{bmatrix} 24 \\ 4 \end{bmatrix} \right) )]
[math( -\left( \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 22 \\ 2 \end{bmatrix} \begin{bmatrix} 21 \\ 1 \end{bmatrix}+ \begin{bmatrix} 22 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 2 \end{bmatrix} + \begin{bmatrix} 22 \\ 2 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix} + \begin{bmatrix} 22 \\ 2 \end{bmatrix} \begin{bmatrix} 24 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 22 \\ 3 \end{bmatrix} \begin{bmatrix} 31 \\ 1 \end{bmatrix}+ \begin{bmatrix} 22 \\ 3 \end{bmatrix} \begin{bmatrix} 32 \\ 2 \end{bmatrix} + \begin{bmatrix} 22 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 3 \end{bmatrix} + \begin{bmatrix} 22 \\ 3 \end{bmatrix} \begin{bmatrix} 34 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 22 \\ 4 \end{bmatrix} \begin{bmatrix} 41 \\ 1 \end{bmatrix}+ \begin{bmatrix} 22 \\ 4 \end{bmatrix} \begin{bmatrix} 42 \\ 2 \end{bmatrix} + \begin{bmatrix} 22 \\4 \end{bmatrix} \begin{bmatrix} 43 \\ 3 \end{bmatrix} + \begin{bmatrix} 22 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 4 \end{bmatrix} \right) )]
정리하면
[math( R_{22} = \dfrac{\partial}{\partial x}\left( 0+0 + \begin{bmatrix} 23\\ 3 \end{bmatrix} + 0 \right) - \left( \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 22\\ 1 \end{bmatrix} \right) + 0+ 0 +0 \right) +\left( 0+ \begin{bmatrix} 21 \\ 2 \end{bmatrix}\begin{bmatrix} 22 \\ 1 \end{bmatrix} + 0+ 0 \right) +\left( \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 21 \\ 2 \end{bmatrix}+ 0 + 0+0 \right) + \left( 0+0 + \begin{bmatrix} 23 \\ 3 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix}+ 0 \right) +\left( 0+ 0 + 0+ 0 \right) -\left( \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) -\left( 0+ 0+0+0 \right) -\left( 0+ 0+0+0 \right) -\left( 0+ 0+0+0\right) )]
[math( = \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 23\\ 3 \end{bmatrix} \right) - \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 22\\ 1 \end{bmatrix} \right) +\left( \begin{bmatrix} 21 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 1 \end{bmatrix}\right) +\left( \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 21 \\ 2 \end{bmatrix}\right) + \left( \begin{bmatrix} 23 \\ 3 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix} \right) -\left( \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 22 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) )]
[math( = \dfrac{\partial}{\partial x}\left( cot\theta \right) - \dfrac{\partial}{\partial x}\left(-\dfrac{r}{\Lambda} \right) +\left( -\dfrac{r}{\Lambda}\dfrac{1}{r} \right) +\left( -\dfrac{r}{\Lambda}\dfrac{1}{r} \right) + \left( cot\theta cot\theta \right) -\left( \left(-\dfrac{r}{\Lambda}\dfrac{\Lambda'}{2\Lambda} \right)+\left( -\dfrac{r}{\Lambda} \dfrac{1}{r} \right) + \left(-\dfrac{r}{\Lambda}\dfrac{1}{r}\right) +\left( -\dfrac{r}{\Lambda}\dfrac{V'}{2V} \right) \right) )]
[math( = \dfrac{\partial}{\partial x}\left( cot\theta \right) + \dfrac{\partial}{\partial x}\left(\dfrac{r}{\Lambda} \right) -\left(\dfrac{r}{\Lambda}\dfrac{1}{r} \right) -\left(\dfrac{r}{\Lambda}\dfrac{1}{r} \right) + \left( cot^2\theta \right) +\left(\dfrac{r}{\Lambda}\dfrac{\Lambda'}{2\Lambda} \right) +\left(\dfrac{r}{\Lambda} \dfrac{1}{r} \right) + \left(\dfrac{r}{\Lambda}\dfrac{1}{r}\right) +\left(\dfrac{r}{\Lambda}\dfrac{V'}{2V} \right) )]
[math( = -1 + \dfrac{1}{\Lambda} +\left(\dfrac{r\Lambda'}{2\Lambda^2} \right) +\left(\dfrac{r V'}{\Lambda 2V} \right) )]
[math( R_{22} = e^{-\lambda} \left( 1 + \dfrac{1}{2}r(v' - \lambda') \right) -1 )][사]38.62 을 조사할 수 있다.
[math( R_{33} = \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 31\\ 1 \end{bmatrix} +\begin{bmatrix} 32\\ 2 \end{bmatrix} + \begin{bmatrix} 33\\ 3 \end{bmatrix} + \begin{bmatrix} 34\\ 4 \end{bmatrix} \right) - \left( \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 1 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 2 \end{bmatrix} \right)+ \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 3 \end{bmatrix} \right) +\dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 4 \end{bmatrix} \right) \right)
)]
[math( +\left( \begin{bmatrix} 31 \\ 1 \end{bmatrix} \begin{bmatrix} 31 \\ 1 \end{bmatrix}+ \begin{bmatrix} 31 \\ 2 \end{bmatrix} \begin{bmatrix} 32 \\ 1 \end{bmatrix} + \begin{bmatrix} 31 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 1 \end{bmatrix} + \begin{bmatrix} 31 \\ 4 \end{bmatrix} \begin{bmatrix} 34 \\ 1 \end{bmatrix} \right) +\left( \begin{bmatrix} 32 \\ 1 \end{bmatrix} \begin{bmatrix} 31 \\ 2 \end{bmatrix}+ \begin{bmatrix} 32 \\ 2 \end{bmatrix} \begin{bmatrix} 32 \\ 2 \end{bmatrix} + \begin{bmatrix} 32 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 2 \end{bmatrix}+ \begin{bmatrix} 32 \\ 4 \end{bmatrix} \begin{bmatrix} 34 \\ 2 \end{bmatrix} \right) +\left( \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 31 \\ 3 \end{bmatrix}+ \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 32 \\ 3 \end{bmatrix} + \begin{bmatrix} 33 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 3 \end{bmatrix}+ \begin{bmatrix} 33 \\ 4 \end{bmatrix} \begin{bmatrix} 34 \\ 3 \end{bmatrix} \right) +\left( \begin{bmatrix} 34 \\ 1 \end{bmatrix} \begin{bmatrix} 31 \\ 4 \end{bmatrix}+ \begin{bmatrix} 34 \\ 2 \end{bmatrix} \begin{bmatrix} 32 \\ 4 \end{bmatrix} + \begin{bmatrix} 34 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 4 \end{bmatrix}+ \begin{bmatrix} 34 \\ 4 \end{bmatrix} \begin{bmatrix} 34 \\ 4 \end{bmatrix} \right) )]
[math( -\left( \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 21 \\ 1 \end{bmatrix}+ \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 2 \end{bmatrix} + \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix} + \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 24 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 33 \\ 3 \end{bmatrix} \begin{bmatrix} 31 \\ 1 \end{bmatrix}+ \begin{bmatrix} 33 \\ 3 \end{bmatrix} \begin{bmatrix} 32 \\ 2 \end{bmatrix} + \begin{bmatrix} 33 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 3 \end{bmatrix} + \begin{bmatrix} 33 \\ 3 \end{bmatrix} \begin{bmatrix} 34 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 33 \\ 4 \end{bmatrix} \begin{bmatrix} 41 \\ 1 \end{bmatrix}+ \begin{bmatrix} 33 \\ 4 \end{bmatrix} \begin{bmatrix} 42 \\ 2 \end{bmatrix} + \begin{bmatrix} 33 \\4 \end{bmatrix} \begin{bmatrix} 43 \\ 3 \end{bmatrix} + \begin{bmatrix} 33 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 4 \end{bmatrix} \right) )]
정리하면
[math( R_{33} = \dfrac{\partial}{\partial x}\left( 0 +0+ 0 + 0 \right) - \left( \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 1 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 2 \end{bmatrix} \right)+ 0+ 0\right) +\left( 0+ 0+ \begin{bmatrix} 31 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 1 \end{bmatrix} + 0 \right) +\left( 0+ 0 + \begin{bmatrix} 32 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 2 \end{bmatrix}+ 0 \right) +\left( \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 31 \\ 3 \end{bmatrix}+ \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 32 \\ 3 \end{bmatrix} + 0+ 0\right) +\left( 0+ 0 + 0+ 0 \right) -\left( \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) -\left( 0+ 0 + \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix} + 0 \right) -\left( 0+0+0+0\right) -\left( 0+0+0+0 \right) )]
[math( = - \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 1 \end{bmatrix} \right) - \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 2 \end{bmatrix} \right) +\left(\begin{bmatrix} 31 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 1 \end{bmatrix} \right) +\left( \begin{bmatrix} 32 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 2 \end{bmatrix} \right) +\left( \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 31 \\ 3 \end{bmatrix}+ \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 32 \\ 3 \end{bmatrix} \right) -\left( \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 33 \\ 2 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix} \right) )]
[math( = - \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 1 \end{bmatrix} \right) - \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 33\\ 2 \end{bmatrix} \right) +\left(\begin{bmatrix} 31 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 1 \end{bmatrix} \right) +\left( \begin{bmatrix} 32 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 2 \end{bmatrix} \right)- \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix} - \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} - \begin{bmatrix} 33 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} )]
[math( = - \dfrac{\partial}{\partial x}\left( -\dfrac{r}{\Lambda}sin^2\theta \right) - \dfrac{\partial}{\partial x}\left( -sin\theta cos\theta \right) +\left(\left( \dfrac{1}{r} \right) \left(-\dfrac{r}{\Lambda}sin^2\theta \right) \right) +\left( (cot\theta) (-sin\theta cos\theta) \right)- \left( -\dfrac{r}{\Lambda}sin^2\theta \dfrac{\Lambda'}{2\Lambda}\right) - \left(-\dfrac{r}{\Lambda}sin^2\theta \dfrac{1}{r} \right) - \left(- \dfrac{r}{\Lambda}sin^2\theta \dfrac{V'}{2V} \right) )]
[math( = \dfrac{\partial}{\partial x}\left(\dfrac{r}{\Lambda}sin^2\theta \right) + \dfrac{\partial}{\partial x}\left( sin\theta cos\theta \right) - \dfrac{1}{r}\dfrac{r}{\Lambda}sin^2\theta - cos^2\theta +\left( \dfrac{r}{\Lambda}sin^2\theta \dfrac{\Lambda'}{2\Lambda}\right) + \left(\dfrac{r}{\Lambda}sin^2\theta \dfrac{1}{r} \right) +\left( \dfrac{r}{\Lambda}sin^2\theta \dfrac{V'}{2V} \right) )]
[math( = \dfrac{\partial}{\partial x}\left(\dfrac{r}{\Lambda}sin^2\theta \right) + \dfrac{\partial}{\partial x}\left( sin\theta cos\theta \right) - cos^2\theta +\left( \dfrac{r}{\Lambda}sin^2\theta \dfrac{\Lambda'}{2\Lambda}\right) +\left( \dfrac{r}{\Lambda}sin^2\theta \dfrac{V'}{2V} \right) )]
[math( =\left( \dfrac{1}{\Lambda}sin^2\theta\right) -sin^{2}\theta +\left( \dfrac{r}{\Lambda}sin^2\theta \dfrac{\Lambda'}{2\Lambda}\right) +\left( \dfrac{r}{\Lambda}sin^2\theta \dfrac{V'}{2V} \right) )]
[math( R_{33} = sin^{2}\theta e^{-\lambda} \left( 1 + \dfrac{1}{2}r(v' - \lambda') \right) -sin^{2}\theta )][나]38.63 을 조사할 수 있다.
[math( R_{44} = \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 41\\ 1 \end{bmatrix} +\begin{bmatrix} 42\\ 2 \end{bmatrix} + \begin{bmatrix} 43\\ 3 \end{bmatrix} + \begin{bmatrix} 44\\ 4 \end{bmatrix} \right) - \left( \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 44\\ 1 \end{bmatrix} \right) + \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 44\\ 2 \end{bmatrix} \right)+ \dfrac{\partial}{\partial x}\left( \begin{bmatrix} 44\\ 3 \end{bmatrix} \right) +\dfrac{\partial}{\partial x}\left( \begin{bmatrix} 44\\ 4 \end{bmatrix} \right) \right)
)]
[math( +\left( \begin{bmatrix} 41 \\ 1 \end{bmatrix} \begin{bmatrix} 41 \\ 1 \end{bmatrix}+ \begin{bmatrix} 41 \\ 2 \end{bmatrix} \begin{bmatrix} 42 \\ 1 \end{bmatrix} + \begin{bmatrix} 41 \\ 3 \end{bmatrix} \begin{bmatrix} 43 \\ 1 \end{bmatrix} + \begin{bmatrix} 41 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 1 \end{bmatrix} \right) +\left( \begin{bmatrix} 42 \\ 1 \end{bmatrix} \begin{bmatrix} 41 \\ 2 \end{bmatrix}+ \begin{bmatrix} 42 \\ 2 \end{bmatrix} \begin{bmatrix} 42 \\ 2 \end{bmatrix} + \begin{bmatrix} 42 \\ 3 \end{bmatrix} \begin{bmatrix} 43 \\ 2 \end{bmatrix}+ \begin{bmatrix} 42 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 2 \end{bmatrix} \right) +\left( \begin{bmatrix} 43 \\ 1 \end{bmatrix} \begin{bmatrix} 41 \\ 3 \end{bmatrix}+ \begin{bmatrix} 43 \\ 2 \end{bmatrix} \begin{bmatrix} 42 \\ 3 \end{bmatrix} + \begin{bmatrix} 43 \\ 3 \end{bmatrix} \begin{bmatrix} 43 \\ 3 \end{bmatrix}+ \begin{bmatrix} 43 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 3 \end{bmatrix} \right) +\left( \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 41 \\ 4 \end{bmatrix}+ \begin{bmatrix} 44 \\ 2 \end{bmatrix} \begin{bmatrix} 42 \\ 4 \end{bmatrix} + \begin{bmatrix} 44 \\ 3 \end{bmatrix} \begin{bmatrix} 43 \\ 4 \end{bmatrix}+ \begin{bmatrix} 44 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 4 \end{bmatrix} \right) )]
[math( -\left( \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 44 \\ 2 \end{bmatrix} \begin{bmatrix} 21 \\ 1 \end{bmatrix}+ \begin{bmatrix} 44 \\ 2 \end{bmatrix} \begin{bmatrix} 22 \\ 2 \end{bmatrix} + \begin{bmatrix} 44 \\ 2 \end{bmatrix} \begin{bmatrix} 23 \\ 3 \end{bmatrix} + \begin{bmatrix} 44 \\ 2 \end{bmatrix} \begin{bmatrix} 24 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 44 \\ 3 \end{bmatrix} \begin{bmatrix} 31 \\ 1 \end{bmatrix}+ \begin{bmatrix} 44 \\ 3 \end{bmatrix} \begin{bmatrix} 32 \\ 2 \end{bmatrix} + \begin{bmatrix} 44 \\ 3 \end{bmatrix} \begin{bmatrix} 33 \\ 3 \end{bmatrix} + \begin{bmatrix} 44 \\ 3 \end{bmatrix} \begin{bmatrix} 34 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 44 \\ 4 \end{bmatrix} \begin{bmatrix} 41 \\ 1 \end{bmatrix}+ \begin{bmatrix} 44 \\ 4 \end{bmatrix} \begin{bmatrix} 42 \\ 2 \end{bmatrix} + \begin{bmatrix} 44 \\4 \end{bmatrix} \begin{bmatrix} 43 \\ 3 \end{bmatrix} + \begin{bmatrix} 44 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 4 \end{bmatrix} \right) )]
정리하면
[math( R_{44} = \dfrac{\partial}{\partial x}\left(0 +0 + 0 + 0 \right) - \dfrac{\partial}{\partial x}\left( \begin{bmatrix}44\\ 1 \end{bmatrix} +0+0+0\right) +\left( 0+ 0 + 0 + \begin{bmatrix} 41 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 1 \end{bmatrix} \right) +\left( 0+ 0 + 0+ 0 \right) +\left( 0+ 0+ 0+ 0 \right) +\left( \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 41 \\ 4 \end{bmatrix}+ 0 + 0+ 0\right) -\left( \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) -\left( 0+ 0 + 0 + 0\right) -\left( 0+ 0 + 0 + 0 \right) -\left( 0+ 0 + 0 + 0 \right) )]
[math( = -\dfrac{\partial}{\partial x}\left( \begin{bmatrix}44\\ 1 \end{bmatrix}\right) +\left( \begin{bmatrix} 41 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 1 \end{bmatrix} \right) +\left( \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 41 \\ 4 \end{bmatrix} \right) -\left( \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix}+ \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} + \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} + \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 14 \\ 4 \end{bmatrix} \right) )]
[math( = -\dfrac{\partial}{\partial x}\left( \begin{bmatrix}44\\ 1 \end{bmatrix}\right) + \begin{bmatrix} 41 \\ 4 \end{bmatrix} \begin{bmatrix} 44 \\ 1 \end{bmatrix} - \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 11 \\ 1 \end{bmatrix} - \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 12 \\ 2 \end{bmatrix} - \begin{bmatrix} 44 \\ 1 \end{bmatrix} \begin{bmatrix} 13 \\ 3 \end{bmatrix} )]
[math( = -\dfrac{\partial}{\partial x}\left( \dfrac{V'}{2\Lambda} \right) + \dfrac{V'}{2V} \dfrac{V'}{2\Lambda} - \dfrac{V'}{2\Lambda} \dfrac{\Lambda'}{2\Lambda} - \dfrac{V'}{2\Lambda}\dfrac{1}{r} - \dfrac{V'}{2\Lambda}\dfrac{1}{r} )]
[math( = -\dfrac{V'}{2\Lambda} + \dfrac{1}{4}\dfrac{V''}{V\Lambda} - \dfrac{1}{4}\dfrac{V'\Lambda'}{\Lambda\Lambda} - \dfrac{V'}{\Lambda r} )]
[math( = \dfrac{1}{\Lambda}\left( -\dfrac{1}{2}V' + \dfrac{1}{4}\dfrac{V''}{V} - \dfrac{1}{4}\dfrac{V'\Lambda'}{\Lambda} - \dfrac{V'}{ r} \right) )]
[math( R_{44} = e^{v -\lambda }\left( -\dfrac{1}{2} v' +\dfrac{1}{4} v'' +\dfrac{1}{4}\lambda ' v' - \dfrac{v'}{r} \right) )][사]38.64
5. 관련 문서
[가] Jebsen, J. T. ,Über die allgemeinen kugelsymmetrischen Lösungen der Einsteinschen Gravitationsgleichungen im Vakuum. (German) JFM 48.1037.02 , Ark. för Mat., Astron. och Fys. 15, No. 18, 9 p. (1921). #[나] Birkhoff, G. D. Relativity and modern physics. With the cooperation of R. E. Langer. (English) JFM 49.0619.01 Cambridge: Harvard University Press, XI u. 283 S. 8∘(1923) #[다] Relativity Thermodynamics And Cosmology 1943 Richard Chace Tolman, P250,P251, §100 P254~257https://archive.org/details/in.ernet.dli.2015.177229[바] (arXiv:gr-qc/0103103v1 28 Mar 2001)General Birkhoff’s Theorem ,Amir H. Abbassi ,Department of Physics, School of Sciences, Tarbiat Modarres University,#[5] J. T. Jebsen,(English)On the general spherically symmetric solutions of Einstein’s gravitational equations in vacuo(Über die allgemeinen kugelsymmetrischen Lösungen der Einsteinschen Gravitationsgleichungen im Vakuum 1921), Published online: 22 November 2005 Springer-Verlag ,Gen. Relativ. Gravit. (2005) 37(12): 2253–2259 DOI 10.1007/s10714-005-0168-y[라] Proc Natl Acad Sci U S A. 1933 May; 19(5): 559–563.doi: 10.1073/pnas.19.5.559 PMCID: PMC1086067 PMID: 16587786 ' Values of Tμν and Christoffel Symbols for a Line Element of Considerable Generality,Herbert Dingle https://www.pnas.org/doi/epdf/10.1073/pnas.19.5.559[마] THE MATHEMATICAL THEORY OF RELATIVITY BY A. S. EDDINGTON, M.A., M.Sc., F.R.S. ,PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE 1923 #[다] [마] [가] [바] [바] 아인슈타인의 이론에 따른 질량 점의 중력장에 대해서 (Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie) Royal Prussian Academy of Science (Reimer, Berlin 1916, pp. 189-196) 저자: 카를 슈바르츠실트(Karl Schwarzschild) https://ko.wikisource.org/wiki/%EC%95%84%EC%9D%B8%EC%8A%88%ED%83%80%EC%9D%B8%EC%9D%98_%EC%9D%B4%EB%A1%A0%EC%97%90_%EB%94%B0%EB%A5%B8_%EC%A7%88%EB%9F%89_%EC%A0%90%EC%9D%98_%EC%A4%91%EB%A0%A5%EC%9E%A5%EC%97%90_%EB%8C%80%ED%95%B4%EC%84%9C[바] [14] Hilbert: Die Grundlagen der Physik II. Nachr. d. K. Ges. d. Wiss. zu G¨ottingen (1917)[15] Schwarzschild: ¨Uber das Gravitationsfeld eines Massenpunktes. Sitz. der. Preuss. Akad. d. Wiss. 189 (1916)[마] [가] Rendiconti by Accademia nazionale dei Lincei. Classe di scienze fisiche, matematiche e naturali Language Italian Volume ser.5:v.11:sem.1 (1902) Matematica - Sui simboli a quattro indici e sulla curvatura di Riemann. Nota del Socio Luigi Bianchi P3-7https://archive.org/details/rendiconti51111902acca/page/n9/mode/2up[18] 구텐베르크 프로젝트 - Calculus Made Easy , Silvanus P. Thompson 1914 2nd edition ,THE MACMILLAN CO. P17 CHAPTER IV. SIMPLEST CASES https://www.gutenberg.org/files/33283/33283-pdf.pdf [사] THE MATHEMATICAL THEORY OF RELATIVITY BY A. S. EDDINGTON, M.A., M.Sc., F.R.S. ,PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE 1923 #[사] [나] [사]