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04/06/2000
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曲率の計算:球対称な場合


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\title{曲率の計算:
球対称な場合}
\author{白石 清(山口大院理工)}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
Einstein方程式の球対称な解を求めるために。
\end{abstract}

\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{時空}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$(N+1)$次元時空のmetric}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

球対称
\be
ds^2=-e^{-2\delta}\Delta dt^2+\frac{dr^2}{\Delta}+
r^2 d\Omega_{N-1}^2
\ee

$\Delta, \delta$は$r$のみの関数と仮定する。

\subsection{vielbein}

\be
e^0=e^{-\delta}\sqrt{\Delta}dt
\ee

\be
e^1=\frac{1}{\sqrt{\Delta}}dr
\ee

\be
e^A=r\tilde{e}^A
\ee

ここで
\be
d\tilde{e}^A+\tilde{\omega}^A{}_B\wedge\tilde{e}^B=0
\ee

$A, B=2,\dots,N$とする。


\bigskip

\be
de^0=\left(-\delta'+\frac{1}{2}
\frac{\Delta'}{\Delta}\right)dr\wedge e^0=
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)e^1\wedge e^0
\ee

\be
de^1=0
\ee

\be
de^A=\frac{1}{r}dr\wedge e^A+r d\tilde{e}^A=
\frac{\sqrt{\Delta}}{r} e^1\wedge e^A-
\tilde{\omega}^A{}_B\wedge e^B
\ee

\newpage

\subsection{spin connection}

\bea
\omega^0{}_1&=&
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right) e^0 \\
\omega^A{}_1&=&\frac{\sqrt{\Delta}}{r} e^A \\
\omega^A{}_B&=&\tilde{\omega}^A{}_B 
\eea


\subsection{curvature 2-form}

\bea
& &\Theta^0{}_1=d\omega^0{}_1 \nn
&=&-\left\{\sqrt{\Delta}\left[
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]'+
\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]^2\right\}e^0\wedge e^1
\eea

\bea
\Theta^A{}_1&=&d\omega^A{}_1+\omega^A{}_B\wedge\omega^B{}_1 \nn
&=&-\left\{\sqrt{\Delta}\left[
\frac{\sqrt{\Delta}}{r}\right]'+\left[
\frac{\sqrt{\Delta}}{r}\right]^2\right\}e^A\wedge e^1
\eea

\bea
\Theta^A{}_B&=&d\omega^A{}_B+\omega^A{}_C\wedge\omega^C{}_B+
\omega^A{}_1\wedge\omega^1{}_B \nn
&=&\left\{\frac{1}{r^2}-\left[
\frac{\sqrt{\Delta}}{r}\right]^2\right\}e^A\wedge e^B
\eea


\bea
\Theta^0{}_A&=&\omega^0{}_1\wedge\omega^1{}_A \nn
&=&-\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]
\left[\frac{\sqrt{\Delta}}{r}\right] e^0\wedge e^A
\eea

\newpage

\subsection{Riemann tensor}

\bea
R^{01}{}_{01}&=&
-\left\{\sqrt{\Delta}\left[
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]'+
\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]^2\right\} \\
R^{A1}{}_{B1}&=&-\left\{\sqrt{\Delta}\left[
\frac{\sqrt{\Delta}}{r}\right]'+
\left[\frac{\sqrt{\Delta}}{r}\right]^2\right\}\delta^A_B \\
R^{AB}{}_{CD}&=&\left\{\frac{1}{r^2}-\left[
\frac{\sqrt{\Delta}}{r}\right]^2\right\}\left(
\delta^A_C\delta^B_D-\delta^A_D\delta^B_C\right) \\
R^{0A}{}_{0B}&=&
-\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]
\left[\frac{\sqrt{\Delta}}{r}\right]\delta^A_B 
\eea

\newpage

\subsection{Ricci tensor}

\bea
R^0_0&=&R^{01}{}_{01}+R^{0A}{}_{0A} \nn
&=&
-\left\{\sqrt{\Delta}\left[
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]'+
\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]^2\right\} \nn
&-&(N-1)\left[
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]
\left[\frac{\sqrt{\Delta}}{r}\right] 
\eea

\bea
R^1_1&=&R^{01}{}_{01}+R^{A1}{}_{A1} \nn
&=&
-\left\{\sqrt{\Delta}\left[
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]'+
\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]^2\right\} \nn
&-&(N-1)\left\{\sqrt{\Delta}\left[
\frac{\sqrt{\Delta}}{r}\right]'+
\left[\frac{\sqrt{\Delta}}{r}\right]^2\right\} 
\eea

\bea
R^A_B&=&R^{0A}{}_{0B}+R^{A1}{}_{B1}+R^{AC}{}_{BC} \nn
&=&\left(
-\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]
\left[\frac{\sqrt{\Delta}}{r}\right]\right. \nn
&-&\left\{\sqrt{\Delta}\left[
\frac{\sqrt{\Delta}}{r}\right]'+
\left[\frac{\sqrt{\Delta}}{r}\right]^2\right\} \nn
&+&\left.(N-2)\left\{\frac{1}{r^2}-\left[
\frac{\sqrt{\Delta}}{r}\right]^2\right\}\right)
\delta^A_B
\eea

\newpage

\subsection{scalar curvature}

\bea
R&=&
-2\left\{\sqrt{\Delta}\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]'+
\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]^2\right\} \nn
&-&2(N-1)\left[\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]
\left[\frac{\sqrt{\Delta}}{r}\right] \nn
&-&2(N-1)\left\{\sqrt{\Delta}\left[
\frac{\sqrt{\Delta}}{r}\right]'+
\left[\frac{\sqrt{\Delta}}{r}\right]^2\right\} \nn
&+&(N-1)(N-2)\left\{\frac{1}{r^2}-\left[
\frac{\sqrt{\Delta}}{r}\right]^2\right\} 
\eea


\subsection{Einstein tensor}

\be
G^{\mu}_{\nu}\equiv R^{\mu}_{\nu}-\frac{1}{2}R\delta^{\mu}_{\nu}
\ee

\bigskip

\bea
G^0_0&=&
(N-1)\left\{\sqrt{\Delta}\left[\frac{\sqrt{\Delta}}{r}\right]'+
\left[\frac{\sqrt{\Delta}}{r}\right]^2\right\} \nn
&-&\frac{(N-1)(N-2)}{2}\left\{\frac{1}{r^2}-\left[
\frac{\sqrt{\Delta}}{r}\right]^2\right\} 
\eea

\bea
G^0_0-G^1_1&=&R^0_0-R^1_1 \nn
&=&
-(N-1)\left[
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]
\left[\frac{\sqrt{\Delta}}{r}\right] \nn
&+&(N-1)\left\{\sqrt{\Delta}\left[
\frac{\sqrt{\Delta}}{r}\right]'+
\left[\frac{\sqrt{\Delta}}{r}\right]^2\right\} 
\eea

\newpage

\subsection{使う式}

\be
G^0_0=\frac{N-1}{2r}\Delta'+\frac{(N-1)(N-2)}{2}\frac{\Delta-1}{r^2}
\ee

\be
G^0_0-G^1_1=\Delta\frac{N-1}{r}\delta'
\ee



\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Einstein equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\be
G^{\mu}_{\nu}=R^{\mu}_{\nu}-\frac{1}{2}R\delta^{\mu}_{\nu}=
8\pi G T^{\mu}_{\nu}
\ee

\bigskip

$(N+1)$ 次元では
\be
R^{\mu}_{\nu}=
8\pi G\left(T^{\mu}_{\nu}-\frac{1}{N-1}T\delta^{\mu}_{\nu}\right)
\ee

\bigskip

完全流体 (perfect fluid) を仮定すると
\be
T^{\mu}_{\nu}=diag.\left(-\rho,P,P,\cdots,P\right)
\ee

とすると
\be
T=T^{\mu}_{\mu}=-\rho+N P
\ee

\bigskip

\bea
G^0_0&=&-8\pi G\rho \\
G^0_0-G^1_1&=&-8\pi G\left(\rho+P\right) \\
\eea

\bea
\frac{N-1}{2r}\Delta'+\frac{(N-1)(N-2)}{2}
\frac{\Delta-1}{r^2}&=&-8\pi G\rho \\
\Delta \left[\frac{N-1}{r}\delta'\right]
&=&-8\pi G\left(\rho+P\right)
\eea

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{conservation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\be
\nabla_{\lambda}T^{\mu\nu}=\partial_{\lambda}T^{\mu\nu}+
\Gamma^{\mu}_{\lambda\sigma}T^{\sigma\nu}+
\Gamma^{\nu}_{\lambda\sigma}T^{\mu\sigma}
\ee

これより
\bea
\nabla_{\mu}T^{\mu\nu}&=&\partial_{\mu}T^{\mu\nu}+
\Gamma^{\mu}_{\mu\sigma}T^{\sigma\nu}+
\Gamma^{\nu}_{\mu\sigma}T^{\mu\sigma} \nn
&=&\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}T^{\mu\nu}\right)+
\Gamma^{\nu}_{\mu\sigma}T^{\mu\sigma}
\eea

ここで
\be
\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}=
\frac{1}{2}g^{\lambda\sigma}\partial_{\mu}g_{\lambda\sigma}=
\Gamma^{\lambda}_{\mu\lambda}
\ee

を使った。

\bigskip

\bea
T^{tt}&=&\frac{1}{e^{-2\delta}\Delta}\rho \\
T^{rr}&=&\Delta P \\
T^{ij}&=&\frac{1}{r^2}\tilde{g}^{ij}P \\
\eea

および
\bea
\Gamma^r_{tt}&=&e^{-2\delta}\Delta^2\left(-
\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right) \\
\Gamma^r_{rr}&=&-
\frac{1}{2}\frac{\Delta'}{\Delta} \\
\Gamma^r_{ij}&=&-\Delta r \tilde{g}_{ij} \\
\eea

などを使うと,保存の式,あるいは力学的平衡の式
\be
\nabla_{\mu}T^{\mu r}=0
\ee

は次のようになる。
\be
P'+\left(-\delta'+\frac{1}{2}
\frac{\Delta'}{\Delta}\right)\left(\rho+P\right)=0
\ee




\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{重力平衡の式}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

アインシュタイン方程式とあわせると,

\be
-P'=\frac{8\pi G}{(N-1)\Delta}r\left(\frac{(N-1)(N-2)}{16\pi G}
\frac{1-\Delta}{r^2}+P\right)\left(\rho+P\right)
\ee

\be
\Delta=1-\frac{16\pi G M_r}{(N-1)A_{N-1}r^{N-2}}
\ee
とおくと
\be
-P'=\frac{8\pi G}{(N-1)\left(1-\frac{16\pi G M_r}{(N-1)A_{N-1}r^{N-2}}\right)}
r\left(\frac{(N-2)M_r}{A_{N-1}r^{N}}+P\right)\left(\rho+P\right)
\ee

ニュートン近似では
\be
-P'=\frac{8\pi G(N-2)M_r}{(N-1)A_{N-1}r^{N-1}}\rho
\ee
を得る。

\bigskip

なお,
\be
M_r(r)=A_{N-1}\int_0^r \rho(r) {r'}^{N-1}dr'
\ee
である。

\be
A_{N-1}=\frac{2\pi^{N/2}}{\Gamma\left(N/2\right)}
\ee

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Appendix}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\be
ds^2=-e^{-2\delta}\Delta dt^2+\frac{dr^2}{\Delta}+
r^2 d\Omega_{N-1}^2
\ee

$\Delta, \delta$は$r$のみの関数と仮定する。

\be
d\Omega_{N-1}^2=\tilde{g}_{ij}d\tilde{x}^i\tilde{x}^j~~~~~(i,j=2,\dots,N)
\ee

\bigskip

\bea
\Gamma^t_{rt}&=&-\delta'+\frac{1}{2}\frac{\Delta'}{\Delta} \\
\Gamma^r_{tt}&=&e^{-2\delta}\Delta^2\left(-
\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right) \\
\Gamma^r_{rr}&=&-\frac{1}{2}\frac{\Delta'}{\Delta} \\
\Gamma^r_{ij}&=&-\Delta r \tilde{g}_{ij} \\
\Gamma^{i}_{rj}&=&\frac{1}{r}\delta_j^i \\
\Gamma^{i}_{jk}&=&\tilde{\Gamma}^{i}_{jk}
\eea

\bigskip

\bea
\Gamma^{\lambda}_{t\lambda}&=&0 \\
\Gamma^{\lambda}_{r\lambda}&=&\Gamma^{t}_{rt}+\Gamma^{r}_{rr}+\Gamma^{k}_{rk}=
-\delta'+\frac{N-1}{r} \\
\Gamma^{\lambda}_{i\lambda}&=&\tilde{\Gamma}^{k}_{ik}
\eea

\bigskip

\be
R_{\sigma\nu}=
\partial_{\lambda}\Gamma^{\lambda}_{\nu\sigma}-
\partial_{\nu}\Gamma^{\lambda}_{\sigma\lambda}+
\Gamma^{\lambda}_{\rho\lambda}\Gamma^{\rho}_{\nu\sigma}-
\Gamma^{\lambda}_{\nu\rho}\Gamma^{\rho}_{\sigma\lambda}
\ee

\bigskip

\bea
R_{tt}&=&\partial_r\Gamma^r_{tt}-0+
\Gamma^r_{tt}\Gamma^{\lambda}_{r\lambda}-
2\Gamma^r_{tt}\Gamma^{t}_{tr} \nn
&=&\left[e^{-2\delta}\Delta^2\left(-
\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]' \nn
&+&e^{-2\delta}\Delta^2\left(-
\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right)
\left[-\delta'+\frac{N-1}{r}\right] \nn
&-&2e^{-2\delta}\Delta^2\left(-
\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right)^2 \nn
&=&e^{-2\delta}\Delta^2\left(-
\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right)' \nn
&+&e^{-2\delta}\Delta^2\left(-
\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right)
\left[-\delta'+\frac{N-1}{r}+
\frac{\Delta'}{\Delta}\right]
\eea

\bea
R_{rr}&=&\partial_r\Gamma^r_{rr}-
\partial_r\Gamma^{\lambda}_{r\lambda}+
\Gamma^r_{rr}\Gamma^{\lambda}_{r\lambda}-\left(
{\Gamma^t_{rt}}^2+{\Gamma^r_{rr}}^2+\Gamma^i_{rj}\Gamma^j_{ri}\right) \nn
&=&\left(-\frac{1}{2}\frac{\Delta'}{\Delta}\right)'-
\left[-\delta'+\frac{N-1}{r}\right]'
+\left(-\frac{1}{2}\frac{\Delta'}{\Delta}\right)
\left[-\delta'+\frac{N-1}{r}\right] \nn
&-&\left(-
\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right)^2-
\left(-\frac{1}{2}\frac{\Delta'}{\Delta}\right)^2
-\frac{N-1}{r^2} \nn
&=&
-\left[-\delta'+\frac{N-1}{r}+
\frac{1}{2}\frac{\Delta'}{\Delta}\right]'
+\left(-\frac{1}{2}\frac{\Delta'}{\Delta}\right)
\left[-\delta'+\frac{N-1}{r}+
\frac{1}{2}\frac{\Delta'}{\Delta}\right] \nn
&-&
\left(-\delta'+\frac{1}{2}\frac{\Delta'}{\Delta}\right)^2
-\frac{N-1}{r^2}
\eea

\bea
R_{ij}&=&\partial_r\Gamma^r_{ij}+\partial_k\Gamma^k_{ij}-
\partial_{i}\Gamma^{\lambda}_{j\lambda}+
\Gamma^r_{ij}\Gamma^{\lambda}_{r\lambda}+
\Gamma^k_{ij}\Gamma^{\lambda}_{k\lambda}-\left(
2\Gamma^{k}_{ri}\Gamma^r_{jk}+
\Gamma^{k}_{\ell i}\Gamma^{\ell}_{jk}\right) \nn
&=&\left[-\Delta r\right]'\tilde{g}_{ij}+
\partial_k\tilde{\Gamma}^k_{ij}-
\partial_{i}\tilde{\Gamma}^{k}_{jk}+\left[-\Delta r\right]
\left[-\delta'+\frac{N-1}{r}\right]\tilde{g}_{ij} \nn
&+&\tilde{\Gamma}^k_{ij}\tilde{\Gamma}^{\ell}_{k\ell}
-2\left[-\Delta\right]\tilde{g}_{ij}-
\tilde{\Gamma}^{k}_{\ell i}\tilde{\Gamma}^{\ell}_{jk} \nn
&=&\tilde{R}_{ij}-\Delta r
\left[-\delta'+\frac{N-2}{r}+
\frac{\Delta'}{\Delta}\right]\tilde{g}_{ij}
\eea




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\bibitem{???} ???

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\end{document} 

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