1. 개요
옙센-버크호프 정리(Jebsen-Birkhoff theorem)는 일반 상대론에서 진공, 구형 대칭(spherical symmetry)인 해가 특정조건에서뿐만아니라 주어진 조건에서 일반적으로 유일함을 설명할수있는 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. 이는 구형 대칭성의 진공 공간에서 정적일뿐만아니라 동적인 시간의 함수(t)를 대입해도 그 결과가 같다는 의미를 제공한다. 이는 당시 카를 슈바르츠실트(Karl Schwarzschild)의 이해를 이어받아 이를 뛰어넘는 일반적인 확장성과 보편성을 보여줄뿐만아니라 비로서 시공간(spacetime)에서 우주를 이해하는 길을 열어줬다는데 있어서 시사하는 박가 크다.
[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 포함) 구형 대칭사례(general—hence not static—spherically symmetric case)의 접근방법으로부터 얻을수있는 여러 가능한 미분방정식의 새로운 해를 예상함으로써 이러한 아이디어를 제안하고 있다.[14][15]따라서 슈바르츠실트해가 전제하는 조건하에 보여주는 현상은 단순한 선택적 결과물(특히 정적조건 및 제약된 공간적 조건)이 아니라 물리 법칙에 의해 강제되는 자연스러운(시간의 흐름속에서도 동적인) 유일한 형태임을 시공간(spacetime)에서 입증한 중요한 작업일뿐만아니라 시공간의 대칭적 구조를 활용해 이를 이해하는 과정으로 매우 단순화했다는데 점에서 역시 중요한 작업으로 평가받는다.
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_{00}dt^2 + g_{11}dr^2 + g_{22}d\theta^2 + g_{33}d\phi^2 )]
시그니처는 시공간을 변별하는 시간 성분(𝑡)과 공간 성분(r,θ,ϕ)에서 (-,+,+,+)를 도입한다.
따라서[math(ds^2 = - v dt^2 + \lambda dr^2 +r^2 d\theta^2 +r^2 \sin^2\theta d\phi^2 )]를 구성하고 다음 계량텐서행렬을 얻을수있다.
[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} )]
계량텐서 역행렬은
[math( g^{\mu\nu} = \begin{pmatrix} \dfrac{1}{\lambda} & 0 & 0 & 0 \\ 0 & \dfrac{1}{r^2} & 0 & 0 \\ 0 & 0 & \dfrac{1}{r^2 \sin^2\theta} & 0 \\ 0 & 0 & 0 & -\dfrac{1}{v} \end{pmatrix} )]
이다.
[math(\{ \mu\nu,\rho \} = \dfrac{1}{2}g^{\lambda \rho} \left( \dfrac{\partial g_{\mu\lambda}}{\partial x_{ \nu}} + \dfrac{\partial g_{\lambda\nu}}{\partial x_{\mu}} - \dfrac{\partial g_{\nu\mu}}{\partial x_{\lambda}} \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,\lambda \} = 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
1921년의 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} +0+0+0\right) + \left( \begin{bmatrix}11\\ 1 \end{bmatrix}\begin{bmatrix}11\\ 1 \end{bmatrix} +0+0+0 \right) + \left( 0+ \begin{bmatrix}12\\ 2 \end{bmatrix}\begin{bmatrix}12\\ 2 \end{bmatrix}+0+0 \right) + \left(0+0+\begin{bmatrix}13\\ 3 \end{bmatrix}\begin{bmatrix}13\\ 3 \end{bmatrix} +0\right) + \left(0+0+0+\begin{bmatrix}14\\ 4 \end{bmatrix}\begin{bmatrix}14\\ 4 \end{bmatrix} \right) -\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( 0+0+0+0\right) -\left( 0+0+0+0\right) -\left( 0+0+0+0\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} +\left(\dfrac{V'}{2V} \right)^2 - 2\dfrac{\Lambda'}{2\Lambda} \dfrac{1}{r} - \dfrac{1}{4}\dfrac{\Lambda'V'}{\Lambda V} )]
[math( \dfrac{1}{\square^2} )]에서 [math( \sqrt{-g_{\square\square}} )]경우를 계산하고
[math( = \dfrac{V'}{2V} - \dfrac{1}{4}\dfrac{V''}{V^2} - \dfrac{\Lambda'}{\Lambda r} - \dfrac{1}{4}\dfrac{\Lambda'V'}{\Lambda V} )] 을 얻을수있다. [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} -\dfrac{1}{2}\left(\dfrac{r\Lambda'}{\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( =\left( \dfrac{1}{\Lambda}sin^2\theta\right) -sin^{2}\theta -\left(sin^2\theta \dfrac{1}{2}\dfrac{r\Lambda'}{\Lambda^2}\right) +\left( sin^2\theta \dfrac{rV'}{\Lambda2V} \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{V'}{2\Lambda} + \dfrac{1}{4}\dfrac{V''}{V\Lambda} + \dfrac{1}{4}\dfrac{V'\Lambda'}{\Lambda^2} - \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 P254~257
https://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 P3-7
https://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
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[사]
[나]
[사]