1. 개요
리치 스칼라 곡률(Ricci scalar curvature), 간단히 스칼라 곡률은 리만 곡률 텐서의 주대각합(trace)으로 표현되는 리치 곡률 텐서의 대각합으로 표현될 수 있다. 이로써 스칼라 곡률은 리만 다양체의 곡률(curvature)을 표현할 수 있고 따라서 리만다양체의 곡률이 조사될 수 있다.1.1. 예시
[math( R = R^{\mu}_{\mu} = g^{\mu\nu} R_{\mu\nu} = g^{11}R_{11} +g^{22}R_{22}+g^{33}R_{33} + g^{44}R_{44} )]2. 총스칼라곡률
스칼라 곡률은 리만 다양체의 한 점에서의 모든 단면 곡률들의 평균값을 나타냄으로써 스칼라 곡률 함수를 주어진 공간에서 적분한 총스칼라곡률(total scalar curvature)을 얻을 수 있다.[1]3. 리만 다양체의 곡률
리만-크리스토펠 곡률 텐서 [math(R_{321}^{4} = \Gamma^{4}_{32,1} - \Gamma^{4}_{31,2} + \Gamma^{0}_{32} \Gamma^{4}_{01} - \Gamma^{0}_{31} \Gamma^{4}_{02} )]일 때 이것으로부터 4 대신 2를 대입하면[math(R_{321}^{4} = R_{321}^{2} = \Gamma^{2}_{32,1 } - \Gamma^{2}_{31, 2} +\Gamma^{0}_{32} \Gamma^{2}_{01} - \Gamma^{0}_{31} \Gamma^{2}_{02} = R_{31} =R_{13} )]을 조사할 수 있다.
이것은 대각합(trace)의 텐서축약(tensor contraction)을 보여준다.
리치 텐서[math( R_{13} = \Gamma^{2}_{32,1 } - \Gamma^{2}_{31, 2} +\Gamma^{0}_{32} \Gamma^{2}_{01} - \Gamma^{0}_{31} \Gamma^{2}_{02} =S )]일 때 이것으로부터 거리(계량)함수의 메트릭텐서(metric tensor)가 주어지면
[math( g^{13} R_{13} = R_{1}^{1} = R )]
이 R은 대각합(trace)의 텐서축약(tensor contraction)에서 곡률을 보여준다.
[math( g^{13} R_{13} = R_{1}^{1} = R )]
이 R을 크리스토펠 기호로 표현하면
[math( R = g^{13} R_{13} = g^{13} \left( \Gamma^{2}_{32,1 } - \Gamma^{2}_{31, 2} +\Gamma^{0}_{32} \Gamma^{2}_{01} - \Gamma^{0}_{31} \Gamma^{2}_{02} \right) )]
4. 에너지-모멘텀 텐서
스칼라 곡률(R)은 아인슈타인 장 방정식(Einstein field equations)에서 시공간(spacetime)함수를 담당하는 아인슈타인 텐서에서 뿐만아니라 중력장(gravitational force field)을 담당하는 스트레스-에너지 텐서(stress–energy tensor)또는 에너지-모멘텀 텐서(energy-momentum tensor) [math( kT_{\mu\nu} )]에서 이들을 연결해주는 주요한 성분(component)이다. [2]5. 아인슈타인 텐서 행렬
슈바르츠실트(Schwarzschild)가 1916년에 발표한 편미분으로 표현되는 아인슈타인 텐서의 크리스토펠 기호의 대칭행렬 [3][가][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}}})]
5.1. 리치 텐서 행렬
편미분으로 표현되는 크리스토펠 심볼을 사용한 계량텐서함수에 대한 슈바르츠실트(Schwarzschild)의 아인슈타인 장 방정식 대칭행렬로부터 리치 텐서 행렬을 얻을 수 있다.[가]
[math(\Gamma^{1}_{\mu\nu} = \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^{1}_{11} =\dfrac{\lambda'}{2\lambda} , \Gamma^{1}_{22}= \dfrac{-r}{\lambda}, \Gamma^{1}_{33} = \dfrac{-r }{\lambda}sin^2 \theta , \Gamma^{1}_{44}= \dfrac{-v'}{2\lambda} )]
[math(\Gamma^{2}_{\mu\nu} = \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^{2}_{12}= \Gamma^{2}_{21}= \dfrac{1}{r} , \Gamma^{2}_{33} = -\sin \theta \cos \theta)]
[math(\Gamma^{3}_{\mu\nu} = \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^{3}_{13}=\Gamma^{3}_{31}= \dfrac{1}{r} ,\Gamma^{3}_{23}=\Gamma^{3}_{32}= \dfrac{\cos\theta}{\sin\theta}=\cot\theta )]
[math(\Gamma^{4}_{\mu\nu} = \left( \begin{array}{rrrr} 0 \;\; && 0 \;\; && 0 \;\; && \dfrac{v'}{2v} \\ 0 \;\; && 0 \;\; && 0 \;\; && 0 \\ 0 \;\; && 0 \;\; && 0 \;\; && 0 \\ \dfrac{v'}{2v} \;\; && 0 \;\; && 0 \;\; && 0 \end{array} \right) )]
[math( \Gamma^{4}_{14} = \Gamma^{4}_{41} = \dfrac{v'}{2v} )]
5.1.1. 리치-쿠르바스트로 텐서
리치 텐서(리치-쿠르바스트로 텐서)[math(R_{321}^{4} = R_{321}^{2} = \Gamma^{2}_{31,2 } - \Gamma^{2}_{21, 3} + \Gamma^{2}_{20}\Gamma^{0}_{31} - \Gamma^{2}_{30} \Gamma^{0}_{21} = R_{31} =R_{13} )]5.2. 계량텐서행렬
[math(ds^2 = g_{11}dr^2 + g_{22}d\theta^2 + g_{33}d\phi^2 + g_{44}dt^2 )][나](38.33)
[math(g_{\mu\nu} = \left( \begin{array}{rrrr} -\lambda(r) \;\; && 0 \;\; && 0 \;\; && 0 \\ 0 \;\; && -r^2 \;\; && 0 \;\; && 0 \\ 0 \;\; && 0 \;\; && -r^2\sin^2 \theta \;\; && 0 \\ 0 \;\; && 0 \;\; && 0 \;\; && v(r) \end{array} \right) )]
계량텐서(메트릭 텐서,metric tentor)행렬 값
[math(g_{11} = -\lambda(r), g_{22} =-r^2,g_{33} =-r^2 sin^2 \theta,g_{44} = v(r))]
5.3. 리치 스칼라 곡률
[math( R = R^{\mu}_{\mu} = g^{\mu\nu} R_{\mu\nu} = g^{11}R_{11} +g^{22}R_{22}+g^{33}R_{33} + g^{44}R_{44} )]슈바르츠실트 계량텐서[math(g_{11} = \lambda(r), g_{22} =r^2,g_{33} =r^2 sin^2 \theta,g_{44} = v(r))]로부터 계량텐서(metric tentor)의 역원
[math(g^{11} = -\dfrac{1}{\lambda(r)}, g^{22} =-\dfrac{1}{r^2},g^{33} =-\dfrac{1}{r^2 sin^2 \theta},g^{44} = +\dfrac{1}{v(r)} )]를 얻을 수 있다.
6. 아인슈타인 텐서 계산
리치텐서 행렬로부터[math( R_{11} = \dfrac{1}{2} v' -\dfrac{1}{4}\lambda ' v' -\dfrac{1}{4} v'' - \dfrac{\lambda '}{r} )] [나]38.61
[math( R_{22} = e^{-\lambda} \left( 1 + \dfrac{1}{2}r(v' - \lambda') \right) -1 )][나]38.62
[math( R_{33} = sin^{2}\theta e^{-\lambda} \left( 1 + \dfrac{1}{2}r(v' - \lambda') \right) -sin^{2}\theta )][나]38.63
[math( R_{44} = e^{v -\lambda }\left( -\dfrac{1}{2} v' +\dfrac{1}{4}\lambda ' v' +\dfrac{1}{4} v'' - \dfrac{v'}{r} \right) )][나]38.64
정리하면
[math( R_{11} = \dfrac{1}{2} v' -\dfrac{1}{4}\lambda ' v' -\dfrac{1}{4} v'' - \dfrac{\lambda '}{r} )]
[math( \lambda' = \dfrac{\lambda'}{\lambda} ,v' = \dfrac{v'}{v})] 로 놓으면
[math( R_{11} = \dfrac{1}{2} \dfrac{v'}{v} -\dfrac{1}{4}\dfrac{\lambda' v'}{\lambda v} -\dfrac{1}{4} \dfrac{v''}{vv} - \dfrac{1}{r}\dfrac{\lambda '}{\lambda} )]
계속해서
[math( R_{22} = e^{-\lambda} \left( 1 + \dfrac{1}{2}r(v' - \lambda') \right) -1 )]
[math( \lambda' = \dfrac{\lambda'}{\lambda} ,v' = \dfrac{v'}{v} , e^{-\lambda}=\dfrac{1}{\lambda} ,e^{v-\lambda })] 를 [math(\dfrac{v}{\lambda})] 로 놓으면
[math( R_{22} = \dfrac{1}{\lambda} \left( 1 + \dfrac{1}{2}r \left(\dfrac{v'}{v} - \dfrac{\lambda'}{\lambda} \right) \right) -1 )]
[math( R_{33} = sin^{2}\theta e^{-\lambda} \left( 1 + \dfrac{1}{2}r(v' - \lambda') \right) -sin^{2}\theta )]
[math( R_{33} = sin^{2}\theta R_{22} )]
[math( R_{44} = e^{v -\lambda }\left( -\dfrac{1}{2} v' +\dfrac{1}{4}\lambda ' v' +\dfrac{1}{4} v'' - \dfrac{v'}{r} \right) )]
[math( R_{44} = \dfrac{v}{\lambda } \left( -\dfrac{1}{2} \dfrac{v'}{v} +\dfrac{1}{4}\dfrac{\lambda ' v'}{\lambda v} +\dfrac{1}{4} \dfrac{v''}{vv} - \dfrac{1}{r}\dfrac{v'}{v} \right) )]
이를 정리해보면
[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} )]
따라서 리치 스칼라 곡률 R은
[math(R = R^{\mu}_{\mu} = ga^{\mu\nu} R_{\mu\nu} = g^{11}R_{11} +g^{22}R_{22}+g^{33}R_{33} + g^{44}R_{44} )]
[math( R = -\dfrac{1}{\lambda (r)} R_{11} -\dfrac{1}{r^2}R_{22} -\dfrac{1}{r^2 sin^2 \theta}R_{33} +\dfrac{1}{v(r)}R_{44} )]
우선
[math( -\dfrac{1}{r^2}R_{22} - \dfrac{1}{r^2 sin^2 \theta}R_{33} = -\dfrac{1}{r^2}R_{22} - \dfrac{1}{r^2 sin^2 \theta}R_{22}sin^2 \theta = -\dfrac{1}{r^2 }R_{22} -\dfrac{1}{r^2 }R_{22} = -2\dfrac{1}{r^2 }R_{22} )]
또는
[math( -\dfrac{1}{r^2}R_{22} - \dfrac{1}{r^2 sin^2 \theta}R_{33} = - \dfrac{1}{r^2} \left( \dfrac{rv' }{2v \lambda} + \dfrac{1}{\lambda} -\dfrac{r\lambda' }{2\lambda^2} - 1 \right) - \dfrac{1}{r^2 sin^2 \theta} \left( \left( \dfrac{rv' }{2v \lambda} + \dfrac{1}{\lambda} -\dfrac{r\lambda' }{2\lambda^2} - 1 \right) sin^2\theta \right) )]
[math(= -\dfrac{1}{r^2} \left( \dfrac{rv' }{2v \lambda} + \dfrac{1}{\lambda} -\dfrac{r\lambda' }{2\lambda^2} - 1 \right) - \dfrac{1}{r^2 } \left( \dfrac{rv' }{2v \lambda} + \dfrac{1}{\lambda} -\dfrac{r\lambda' }{2\lambda^2} - 1 \right) )]
[math(= - \dfrac{2}{r^2 } \left( \dfrac{rv' }{2v \lambda} + \dfrac{1}{\lambda} -\dfrac{r\lambda' }{2\lambda^2} - 1 \right) )]
[math(= -\dfrac{2}{r^2 } \dfrac{rv' }{2v \lambda} + \dfrac{2}{r^2 } \dfrac{r\lambda' }{2\lambda^2} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
[math(= -\dfrac{1}{r } \dfrac{v' }{v \lambda} + \dfrac{1}{r } \dfrac{\lambda' }{\lambda^2} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
계속해서
[math( g^{11}R_{11}g^{44}R_{44} = -\dfrac{1}{\lambda} \left( \dfrac{v' }{2v} - \dfrac{v }{4v^2} -\dfrac{v'\lambda' }{4v \lambda} - \dfrac{1}{r}\dfrac{\lambda'}{\lambda} \right) + \dfrac{1}{v} \left(-\dfrac{v' }{2\lambda} + \dfrac{v' \lambda' }{4\lambda^2} +\dfrac{v }{4v \lambda} - \dfrac{1}{r}\dfrac{v'}{\lambda} \right) )]
[math( = - \dfrac{v' }{2v\lambda} + \dfrac{v }{4v^2\lambda} + \dfrac{v'\lambda' }{4v \lambda\lambda} + \dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} -\dfrac{v' }{2\lambda v} + \dfrac{v' \lambda' }{4\lambda^2 v} +\dfrac{v }{4v \lambda v} - \dfrac{1}{r}\dfrac{v'}{\lambda v} )]
[math( = - \dfrac{v' }{2v\lambda} -\dfrac{v' }{2\lambda v} +\dfrac{v }{4v \lambda v} + \dfrac{v }{4v^2\lambda}+ \dfrac{v' \lambda' }{4\lambda^2 v} +\dfrac{v'\lambda' }{4v \lambda\lambda} + \dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - \dfrac{1}{r}\dfrac{v'}{\lambda v} )]
[math( = -2\dfrac{v' }{2\lambda v} + 2\dfrac{v' }{4v^2\lambda} +2\dfrac{v'\lambda' }{4v \lambda\lambda} + \dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - \dfrac{1}{r}\dfrac{v'}{\lambda v} )]
[math( = -\dfrac{v' }{\lambda v} + \dfrac{v'' }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + \dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - \dfrac{1}{r}\dfrac{v'}{\lambda v} )]
정리하면
[math(R = -\dfrac{v' }{\lambda v} + \dfrac{v'' }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + \dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - \dfrac{1}{r}\dfrac{v'}{\lambda v} -\dfrac{1}{r } \dfrac{v' }{v \lambda} + \dfrac{1}{r } \dfrac{\lambda' }{\lambda^2} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
[math(R = -\dfrac{v' }{\lambda v} + \dfrac{v'' }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
아인슈타인 장 방정식 [math( G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R )] 으로부터 아인슈타인 텐서를 얻을 수 있다.
[math( G_{11} = R_{11} - \frac{1}{2}g_{11}R )] 아인슈타인 텐서를 얻을 수 있다.
[math( G_{11} = \left(\dfrac{v' }{2v} - \dfrac{v }{4v^2} -\dfrac{v'\lambda' }{4v \lambda} - \dfrac{1}{r}\dfrac{\lambda'}{\lambda} \right) - \dfrac{1}{2}- \lambda \left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{11} = \dfrac{2}{\lambda} \left(\dfrac{v' }{2v} - \dfrac{v }{4v^2} -\dfrac{v'\lambda' }{4v \lambda} - \dfrac{1}{r}\dfrac{\lambda'}{\lambda} \right) + \left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{11} = \dfrac{2v' }{2v\lambda} - \dfrac{2v }{4v^2\lambda} -\dfrac{2v'\lambda' }{4v \lambda\lambda} - \dfrac{2}{r}\dfrac{\lambda'}{\lambda\lambda} + \left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{11} = \dfrac{v' }{v\lambda} - \dfrac{v }{2v^2\lambda} -\dfrac{v'\lambda' }{2v \lambda^2} - \dfrac{2}{r}\dfrac{\lambda'}{\lambda^2} + \left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{11} = \dfrac{v' }{v\lambda} - \dfrac{v }{2v^2\lambda} -\dfrac{v'\lambda' }{2v \lambda^2} - \dfrac{2}{r}\dfrac{\lambda'}{\lambda^2} -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
[math( G_{11} = \dfrac{v' }{v\lambda}-\dfrac{v }{\lambda v} - \dfrac{v }{2v^2\lambda}+ \dfrac{v'' }{2v^2\lambda} -\dfrac{v'\lambda' }{2v \lambda^2}+\dfrac{v'\lambda' }{2v \lambda\lambda} - \dfrac{2}{r}\dfrac{\lambda'}{\lambda^2} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
[math( G_{11} = - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
계속해서
[math( G_{22})] (생략)
[math( G_{33})] (생략)
[math( G_{44} = R_{44} - \frac{1}{2}g_{44}R )]
[math( G_{44} = \left( -\dfrac{v' }{2\lambda} + \dfrac{v' \lambda' }{4\lambda^2} +\dfrac{v }{4v \lambda} - \dfrac{1}{r}\dfrac{v'}{\lambda} \right) - \dfrac{1}{2} v \left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{44} = -\dfrac{2}{v}\left( -\dfrac{v' }{2\lambda} + \dfrac{v' \lambda' }{4\lambda^2} +\dfrac{v }{4v \lambda} - \dfrac{1}{r}\dfrac{v'}{\lambda} \right) +\left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{44} = -\dfrac{2}{v}\left( -\dfrac{v' }{2\lambda} + \dfrac{v' \lambda' }{4\lambda^2} +\dfrac{v }{4v \lambda} - \dfrac{1}{r}\dfrac{v'}{\lambda} \right) +\left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{44} = +\dfrac{2 v' }{2\lambda v} - \dfrac{2 v' \lambda' }{4\lambda^2 v} -\dfrac{2 v }{4v \lambda v} + \dfrac{2}{rv}\dfrac{v'}{\lambda} +\left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{44} = +\dfrac{ v' }{\lambda v} - \dfrac{ v' \lambda' }{2\lambda^2 v} -\dfrac{ v }{2v \lambda v} + \dfrac{2}{rv}\dfrac{v'}{\lambda} +\left( -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) \right) )]
[math( G_{44} = +\dfrac{ v' }{\lambda v} - \dfrac{ v' \lambda' }{2\lambda^2 v} -\dfrac{ v }{2v \lambda v} + \dfrac{2}{rv}\dfrac{v'}{\lambda} -\dfrac{v' }{\lambda v} + \dfrac{v }{2v^2\lambda} +\dfrac{v'\lambda' }{2v \lambda\lambda} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + \dfrac{2}{r^2 } \left( 1-\dfrac{1}{\lambda} \right) )]
[math( G_{44} = +\dfrac{ v' }{\lambda v}-\dfrac{v' }{\lambda v} - \dfrac{ v' \lambda' }{2\lambda^2 v} +\dfrac{v'\lambda' }{2v \lambda\lambda} -\dfrac{ v }{2v \lambda v}+ \dfrac{v }{2v^2\lambda} + \dfrac{2}{rv}\dfrac{v'}{\lambda} - 2\dfrac{1}{r}\dfrac{v'}{\lambda v} + 2\dfrac{1}{r}\dfrac{\lambda'}{\lambda^2} + \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) )]
7. TOV 방정식
[math( G_{11},G_{44} )] 값을 사용해서 TOV 방정식을 조사할 수 있다.8. 관련 문서
[1]
\[Submitted on 3 Aug 2001 (v1), last revised 12 Mar 2002 (this version, v2)\]Constant Scalar Curvature Metrics on Connected Sums Dominic Joyce doi:10.48550/arXiv.math/0108022
https://arxiv.org/abs/math/0108022
[2]
Gravitation, Charles W. Misner , Kip S. Thorne, John Archibald Wheeler (1973)W. H. Freeman
http://fma.if.usp.br/~mlima/teaching/PGF5292_2021/Misner_Gravitation.pdf
[3]
(영역 Lluís Bel) Uber das Gravitationsfeld eines Massenpunktes nach der Einstenschen Theorie ,Karl Schwarzschild
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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)
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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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