%Kiyoshi Shiraishi:CandG lec %
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# �F���_�Əd��

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### Cosmology and Gravitation

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%�ԈႢ�����w�E�������B%LaTeX2.09
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\title{{\small ���Ȗځ@���R�̗���}\\
�F���_�Əd��}
\author{
���΁@���i�R����w���w���j
}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
�F���̑傫�ȃX�P�[���ł́C���R�E�̗͂̂����C�d�͂������Ƃ�
�{���I�Ȗ����������Ă���B�{�u��ł́C�F�����f���C�ʎq�F���_�C
�u���b�N�z�[���Ȃǂ��Ƃ肠���C�Ȃ�ׂ������I�ȗ͊w�̒m����p���āC
�F���ɂ�����d�͂̂��܂��܂ȑ��ʂɂ��čl���Ă݂����B
��ʑ��ΐ����_�̏����Ƃ��̉��p�ɂ��Ă��Љ�C
�j���[�g���̏d�͗��_�Ƃ̔�r�����݂����B
\end{abstract}

\newpage
\tableofcontents

\newpage

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\section{Introduction}
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�d�͍͂ł��Â�����l�ނɔF�����ꂽ�u�́v�ł��낤�B
\footnote{�����Ƃ��C�u�́v�̊T�O�̓j���[�g���ȍ~���H}
�j���[�g��(1643-1727)�ȗ��̐����Ȋw�Ƃ��Ă̘g�g�݂̒��ŁC�d�͂̊T�O��
�A�C���V���^�C��(1879-1955)�̈�ʑ��Θ_�Ƃ��Č������݂��B

�d�͉͂F���̗l�X�ȃX�P�[���Ŗ{���I�Ȗ����������Ă���B
����͏d�͓͂����I�Ȓ����̂Ȃ��������͂ł���Ƃ��������ɗR������B
\footnote{�ŋ߁C�p�C�I�j�A�P�O�y�тP�P�������z�n���ӕ��ŗ\�����Ȃ�����
���󂯂Ă���Ƃ����񍐂�����iPhys. Rev. Lett. {\bf 81} (1998) 2858 ���j�B
�{�����Ƃ���ƁC�d�͂������̃X�P�[���Ɉˑ����Ă���\��������B}

�{�u��ł͗͊w����ъ�b���w�̒m����O��Ƃ���B
\footnote{�W���u��Ȃ̂ŁC�܂Ƃ܂������̂ɂ��������C
����Ŏ��Ԃ̐��񂪂��邽�߁B���₪�L��΂��̓s�x�������悤�B}
�u��^�C�g���́C�u�F���_�Əd�́v�����C�m�[�g
\footnote{���̕��������Ă��鎞�_�ł́C�u�m�[�g�v�Ƃ͌�����C
�u�����v�ł���I}
�������Ă���i�K�Łu�d�́v�̂ق��ɗ͂������Ă��܂����B
\footnote{����ł����c�������Ƃ����X����B�X�E�B���O�o�C\cite{�o�[�W���[1}�C
���O�����W���_�̂��ƂȂǁB}
\footnote{���Θ_�I����ł��c�������Ƃ́C�d�͂ƃX�s���C�d�͎q�̃X�s���C
��̗��_�I�Ɍ����d�͗��_�̗B�ꐫ�C�ȂǁB}
����ɁC�F���_�̕��������ɊY���ƂȂ��Ă��܂����B
\footnote{�F���_�I����ł��c�������Ƃ́C�����ς��L�肷����B}
�����ł́u�F���_�v�́u�F�������v�Ɖ��߂��ė~�����B

\newpage

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\section{�j���[�g���d��}
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\subsection{�n��ƓV��}
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����$m$�̕��̂ɊO��$\vF$�������Ă���Ƃ��C���̉^����������
\be
\vF=m\va
\ee
�Ə�����B������$\va$�͕��̂̉����x�B

�n�\�߂��ł́C���������$z$�����Ƃ�Əd�͂�
\be
F_z=-mg
\ee
�ŕ\�����B$g$�͏d�͉����x�łق�$9.8 {\rm m/s}$�Ƃ����l�����B
�j���[�g���́C�V�̂̉^���̉�͂���C����$M$�̕��̂ɂ���Ď���$m$�̕��̂ɓ����͂�
\be
F_r=-\frac{GMm}{r^2}
\ee
�ł��邱�Ƃ𓱂����B$F_r$�͗�$\vF$�̓��a�i�����j�����B������������$G$�͏d�͒萔�B
\footnote{$G$�͖{���Ɂu�萔�v���H�Ƃ����̂͋����[���b��ł��邪�C����͊����B}
����͂����钆�S�͂̓T�^��ł���B

�n���̎���$M_{\oplus}$���n�����S�ɏW�܂������̂Ƃ��Čv�Z�����
\footnote{���̐�������Appendix}
�n��̏d�͉����x��
\footnote{�n��̏d�͖͂{���͖��L���͂Ɓu���S�́v�Ƃ̘a�B
�ȍ~���C���̂悤�ȑ�G�c�Șb�������B}
\be
g=\frac{GM_{\oplus}}{R_{\oplus}^2}
\ee
�����������Œn���͔��a$R_{\oplus}$�̋��Ƌߎ������B

���C���͂��������n���̂܂��́C
���a$r$�̉~�O��������̑���$v$�ŉ^�����Ă���Ƃ����
\footnote{$\omega$�͈�ʂɊp���x�C$T$�͎����ƌĂ΂��B}
\be
\frac{v^2}{r}=r\omega^2=\frac{GM_{\oplus}}{r^2}
\ee

\be
r\left(\frac{2\pi}{T_{moon}}\right)^2=g\frac{R_{\oplus}^2}{r^2}
\ee
�X�̒l�����Ă݂悤�B

�Ƃ���ŁC�j���[�g���̃����S�̎q���͐��E�e�n�ɂ���B

�j���[�g���́C�����S�������n���̏d�͂ɂЂ���Ă��邱�Ƃ�
�C�������B

���͗����Ă���I���Ƃ͎��̂悤�ɂ��Ă킩��B
�Z�����ԊԊu$\Delta t$�̊ԂɁC���͒n����
\bea
\sqrt{r^2+(v\Delta t)^2}-r&\approx& r\left(1+\frac{(v\Delta t)^2}{2r^2}-1\right) \\
&=&\frac{1}{2}\frac{v^2}{r}{\Delta t}^2
\eea
�����߂Â��Ă���C���Ȃ킿�u�����Ă���v�B����́u���R�����v�Ɠ��l�ł���B

�j���[�g���́C�R�̒��ォ�琅���ɕ��̂�ł��o���G�����Ă���B
�������\������������΁C���̂͒n�ʂɗ������ɁC���ɖ߂��Ă���B
�u�l�H�q���v��\�������H�Ƃ����Ă��悢�B

\bigskip

��ɁC�p���̃L���x���f�B�V��(1731-1810)�͎������ŏd�͂̋t���摥�����؂����B

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\subsection{�d�̓|�e���V����}
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$\Phi$�F�d�̓|�e���V����

\be
\vF=-m\vnabla\Phi
\ee

�d�̓|�e���V�����́C�P�ʎ��ʓ�����̏d�͂ɂ��ʒu�G�l���M�[�ł���B

�n�\�ł�
\be
\Phi=gz
\ee

\begin{quote}
ex.
����-�R���ԂɃg���l�����@���āC���̒���
�d�͂ɂ��^���Œʉ߂���B
�ŒZ���Ԃł�����悤�ȃg���l���̌�����߂�B���C�͖�������B
\end{quote}

��ʂɎ��ʖ��x$\rho(\vr)$���^�����Ă���Ƃ��C�d�̓|�e���V�����͎���
���߂���B
\be
\Phi(\vr)=-G\int_V \frac{\rho(\vr')}{|\vr-\vr'|}d^3\vr'
\ee

�d�̓|�e���V�����̖������ׂ���������
\be
\nabla^2\Phi=4\pi G \rho
\label{poisson}
\ee
�ł���B$\nabla^2$�̓��v���V�A���ƌĂ΂��B
�܂��C���̌�̕��������|�A�\���������ƌĂԁB

\bigskip

���S�͂̏ꍇ�i���Ώ́j
\be
F_r=-m\frac{\partial\Phi(r)}{\partial r}
\ee

\be
\nabla^2\Phi(r)=\frac{1}{r^2}\frac{\partial}{\partial r}r^2
\frac{\partial\Phi(r)}{\partial r}
\ee

������(\ref{poisson})�̍��ӂ𔼌a$R$�̋��̓����ő̐ϐϕ������
\bea
\int_{��}\nabla^2\Phi(r) d^3\vr&=&4\pi\int_0^R
\frac{1}{r^2}\frac{\partial}{\partial r}r^2
\frac{\partial\Phi(r)}{\partial r} r^2 dr \nn
&=&4\pi\left[r^2
\frac{\partial\Phi(r)}{\partial r}\right]_0^R
\eea
���_��$\Phi$�������Ɖ��肷���
\be
4\pi\left.R^2
\frac{\partial\Phi(r)}{\partial r}\right|_{r=R}
\ee

����C������(\ref{poisson})�̉E�ӂ̐ϕ���
\be
4\pi G\int_{��}\rho(r) d^3\vr=4\pi GM(R)
\ee
�����ŁC$M(R)$�͋����ɂ��鎿�ʁB

���������āC
\be
\left.\frac{\partial\Phi(r)}{\partial r}\right|_{r=R}=\frac{GM(R)}{R^2}
\ee
���Ώ̂̂Ƃ��̃|�A�\���������́u�������v�́C����Ŏ����ł������ȁH

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\subsection{������}
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���̖��������͂Ȃ������邩�H\cite{�c���]1}

��ʂɁC�d�͂́u���v���i���������ςɂ����΁j�����͂ƂȂ�B

\bigskip

���ɂ��d�͂̋����C�n�\�ƒn�����S�ł̍���
\be
\frac{GM_{moon}}{(r-R_{\oplus})^2}-\frac{GM_{moon}}{r^2}\approx
2\frac{GM_{moon}}{r^3}R_{\oplus}
\ee

�n���ɂ��d�́C���ϊC���ʂƂ�������$a$��������オ�����ꍇ�̍���
\be
\frac{GM_{\oplus}}{R_{\oplus}^2}-\frac{GM_{\oplus}}{(R_{\oplus}+a)^2}\approx
2\frac{GM_{\oplus}}{R_{\oplus}^3}a
\ee

����炪�ނ荇���Ă�Ƃ����
\be
\frac{M_{moon}}{r^3}R_{\oplus}=
\frac{M_{\oplus}}{R_{\oplus}^3}a
\ee

\be
M_{moon}=
\frac{a r^3}{R_{\oplus}^4}M_{\oplus}
\ee

�l�����Ă݂悤�B
$a\approx 50 {\rm cm}$�ł悢���H

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\subsection{�P�v���[�̑�R�@���F������͂���}
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\noindent
�P�v���[�̖@��

�P�@�f���̉^���͑ȉ~�O�������B

�Q�@�ʐϑ��x���

�R�@�e�f���̋O���̕��ϔ��a�̂R��ƌ��]�����̂Q��̔�͈��

\bigskip

�f���̉^���͑��z�ɂ��d�͂ɂ̂ݎx�z����Ă���ł��낤�B
��������Ƌ��ʂ���ʂ�
\be
GM_{\odot}
\ee
�ł��낤�B������$M_{\odot}$�͑��z���ʁB
\be
\frac{GM_{\odot}m}{r^2}
\ee
���͂̎����������Ƃ���C$GM_{\odot}$�̎�����
\be
(����)^3/(����)^2
\ee

���������āC�X�̘f���́u�O�����a�v�̂R�抄��u���]�����v�̂Q���
���ʂł��낤�C�Ɛ��������B
����͌�ɍĂъm���߂�B

\newpage

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\section{�A�C���V���^�C���d��}
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�ЂƂƂ���̂��񑩂��Ƃ́C�w�K���Ă���Ɖ��肷��B
\footnote{���������΁C�A�C���V���^�C���̋K��͂�낵���̂ł��傤���H}

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\subsection{���n��}
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����$m$�̗��q���l����B���q�̍�p(action)�́C���E���̒����ɔ�Ⴗ��B
\be
I=-mc\int ds=-mc\int \sqrt{-g_{\mu\nu}
\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}d\tau
\ee
�v��$g_{\mu\nu}$�́C���W�ŕ\���������́u�͂�����v������킷�B

�ϕ������ɂ��C��p���ɒl�ƂȂ�^�������������B
���̏ꍇ�C���E���̒���($\int ds$)���ɒl�ƂȂ�ƌ����Ă��ǂ��B

Euler-Lagrange eq.
\be
\frac{d}{d\tau}\left(\frac{\partial L}{\partial \dot{x}^{\mu}}\right)-
\frac{\partial L}{\partial x^{\mu}}=0
\ee

�i������
\be
I=\int L d\tau
\ee
�����
\be
\dot{x}^{\mu}=\frac{dx^{\mu}}{d\tau}
\ee
�j�͍��̏ꍇ�ȉ��̂悤�ɂȂ�B
\be
\frac{d}{d\tau}\left(
\frac{-g_{\mu\nu}}{\sqrt{-g_{\lambda\sigma}\dot{x}^{\lambda}\dot{x}^{\sigma}}}
\frac{dx^{\nu}}{d\tau}\right)-
\frac{-\frac{\partial g_{\alpha\beta}}{\partial x^{\mu}}}%
{2\sqrt{-g_{\lambda\sigma}\dot{x}^{\lambda}\dot{x}^{\sigma}}}
\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}=0
\ee

\be
\frac{d}{ds}\left(g_{\mu\nu}\frac{dx^{\nu}}{ds}\right)-
\frac{1}{2}\frac{\partial g_{\alpha\beta}}{\partial x^{\mu}}
\frac{dx^{\alpha}}{ds}\frac{dx^{\beta}}{ds}=0
\ee

\be
g_{\mu\nu}\frac{d^2x^{\nu}}{ds^2}+
\frac{\partial g_{\mu\beta}}{\partial x^{\alpha}}
\frac{dx^{\alpha}}{ds}\frac{dx^{\beta}}{ds}-
\frac{1}{2}\frac{\partial g_{\alpha\beta}}{\partial x^{\mu}}
\frac{dx^{\alpha}}{ds}\frac{dx^{\beta}}{ds}=0
\ee

�v�ʂ́u�t�v
\be
g^{\lambda\mu}g_{\mu\nu}=\delta^{\lambda}_{\nu}
\ee
���g����
\be
\frac{d^2x^{\lambda}}{ds^2}+
\Gamma^{\lambda}_{\alpha\beta}\frac{dx^{\alpha}}{ds}\frac{dx^{\beta}}{ds}=0
\label{geodesic}
\ee
�Ə�����B������������
(metric) connection $\Gamma^{\lambda}_{\alpha\beta}$��
\be
\Gamma^{\lambda}_{\alpha\beta}=\frac{1}{2}g^{\lambda\mu}\left(
\frac{\partial g_{\mu\beta}}{\partial x^{\alpha}}+
\frac{\partial g_{\alpha\mu}}{\partial x^{\beta}}-
\frac{\partial g_{\alpha\beta}}{\partial x^{\mu}}
\right)
\ee
�ŗ^������B

(\ref{geodesic})���u���n���̎��v�ƌĂԁB

���ԋ�Ԃ����킹�āu����v�ƌĂԁB�u����v���u���R�v�ȂƂ�
$g_{\mu\nu}=\eta_{\mu\nu}$�Ə����C
\be
-ds^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=-c^2dt^2+dx^2+dy^2+dz^2
\ee
�ł���B

���đ��n���̎��͂����ɂ��ďd�͂������Ă���Ƃ��̗��q�̉^����\���̂��H
�j���[�g���ߎ����l���Ă݂�B���q�̑��x�͌����ɔ�ׂď\���������Ƃ��C
�d�͂̑傫�����ア�Ƃ���B
\be
ds\approx cdt
\ee
�Ȃ̂�
\be
\frac{d^2x^{i}}{dt^2}+
c^2\Gamma^{i}_{00}\approx 0
\ee
�܂�
\be
\Gamma^{i}_{00}\approx -\frac{1}{2}
\frac{\partial g_{00}}{\partial x^{i}}
\ee
�ł��邩��C
�j���[�g���̉^��������
\be
\frac{d^2x^{i}}{dt^2}=-\frac{\partial\Phi}{\partial x^i}
\ee
�Ƃ���ׂ��
\be
g_{00}\approx -\left(1+\frac{2\Phi}{c^2}\right)
\ee
�ƂȂ��Ă���΂悢�B

\begin{quote}
���ꂪ�������Ӗ��������Ă��邩���m���߂邽�߁C
�ŏ��̍�p�Ƀj���[�g���ߎ����g���Ă݂�B
\bea
I&\approx& -mc\int\sqrt{1+\frac{2\Phi}{c^2}-\frac{\vv^2}{c^2}}cdt \nn
&\approx& -mc\int\left(1-\frac{\vv^2}{2c^2}+\frac{\Phi}{c^2}\right)cdt \nn
&\approx& \int\left(-mc^2+\frac{1}{2}m\vv^2-m\Phi\right)dt \nn
\eea
�m���ɁC�i�萔�������āj�|�e���V����������ꍇ�̌ÓT�͊w�̍�p�ƂȂ��Ă���B
\end{quote}

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\subsection{�A�C���V���^�C��������}
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�ł́C�|�e���V���������߂�i\ref{poisson}�j�ɑΉ��������͉����낤���B
���ꂪ�A�C���V���^�C���������ł���B
\be
R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}
\ee
������$R_{\mu\nu}$�̓��b��ȗ��C$R=g^{\mu\nu}R_{\mu\nu}$��
�X�J���[�ȗ��ƌĂ΂��ʂł���B

�A�C���V���^�C���������͎��̂悤�ɂ�������B
\be
R_{\mu\nu}=\frac{8\pi G}{c^4}\left(T_{\mu\nu}-\frac{1}{2}T g_{\mu\nu}\right)
\ee
������������$T\equiv T_{\mu\nu}g^{\mu\nu}$�B

�G�l���M�[�^���ʃe���\��$T_{\mu\nu}$�̒��g�͌�ɏڂ������ׂ邪�C�j���[�g���ߎ�
�ł́C���͓��͖�������āC
\be
T_{00}\approx \rho c^2
\ee
�܂�
\be
T\approx -\rho c^2
\ee
�ł���B

\be
\Gamma^{i}_{00}\approx -\frac{1}{2}
\frac{\partial g_{00}}{\partial x^{i}}=
\frac{1}{c^2}\frac{\partial \Phi}{\partial x^{i}}
\ee
�Ȃ̂�
\be
R_{00}\approx \frac{\partial}{\partial x^{i}}\Gamma^{i}_{00}
=\frac{1}{c^2}\nabla^2\Phi
\ee
���
\be
\nabla^2\Phi=4\pi G\rho
\ee
���Č�����B

\newpage

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\section{�O���^��}
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\subsection{�~�O��(N)}
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�{���́C�d�S�̂܂��̑��Ή^�����l���Ȃ��Ă͂Ȃ�Ȃ����C
�Е��̎��ʂ����ɑ傫���Ƃ��āC����𒆐S�Ƃ����O���^���i�����~�^���j��
�l����B

\begin{quote}
ex.
���z-�ؐ��̏d�S�͂ǂ��ɂ��邩�H
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�Î~�q��}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�q���̎������n���̎��]�����Ɠ����l�H�q���B

\begin{quote}
ex.
�Î~�q���̒n�\����̍����͂����قǂ��H
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{���F�����x}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�n�\���肬��̐l�H�q���̑����B
\be
\frac{v_{}^2}{R_{\oplus}}=\frac{GM_{\oplus}}{R_{\oplus}^2}
\ee
���
\be
v_{}=\sqrt{\frac{GM_{\oplus}}{R_{\oplus}}}=\sqrt{g R_{\oplus}}
\ee

\begin{quote}
ex.
�ǂ̂��炢�̒l���H
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{���F�����x}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�n���̏d�͂Ɉ�����Ă��ǂ��ė��Ȃ����߂́C�n�\�ł̍Œᑬ�x�B
\footnote{���R�C�ł��グ��͐��i�͎͂g��Ȃ��Ƃ���B}

���F�����x�E�E�E��11.2 km/s�C�����̖�R�O�{

\be
\frac{1}{2}v_{}^2-\frac{GM_{\oplus}}{R_{\oplus}}=0
\ee
�S�͊w�I�G�l���M�[���[���E�E�E�������Ɂi�M���M���j���B�ł���B
\be
v_{}=\sqrt{\frac{2GM_{\oplus}}{R_{\oplus}}}=\sqrt{2gR_{\oplus}}
\ee

\begin{quote}
ex.
��C���̕��q�̑��x����F�����x�Ɣ�ׂĂ݂悤�B
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{��O�F�����x}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

���z�̏d�͂Ɉ�����Ă��ǂ��ė��Ȃ����߂́C�n���O���ł̍Œᑬ�x�B
\be
\frac{1}{2}v_{}^2-\frac{GM_{\odot}}{R_{AU}}=0
\ee
$R_{AU}$�́i���ρj�n�����]�O�����a���P�V���P�ʁB

\be
v_{}=\sqrt{\frac{2GM_{\odot}}{R_{AU}}}=\sqrt{2}\frac{2\pi R_{AU}}{T_{�P�N}}
\ee

�Ƃ����̂́C�ԈႢ�B

���]���x�������Ă܂�����˂��B

���������߂Ă��������B

\begin{quote}
ex.
�ǂ̂��炢�̒l���H
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�E�o���x}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

����$M$�C���a$R$�̓V�̂���̒E�o���x��
\be
v_e=\sqrt{\frac{2GM}{R}}
\ee

\begin{quote}
ex.
���z�\�ʂ���̒E�o���x�́H�@�i��$620 {\rm km/s}$�j
\end{quote}

\begin{quote}
ex.
��ʂ̐��\�ʂŁC�E�o���x���u�����v�ƂȂ�Ƃ��̏����́H

�������C�����ɋ߂��^�����j���[�g���͊w�ł͎�舵���Ȃ��̂����E�E�E
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�P�v���[�̑�R�@��}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�f���̋O���𔼌a$a$�̉~�O���Ƃ���B
\be
\frac{GM_{\odot}}{a^2}=\frac{v^2}{a}=a\omega^2=a\left(\frac{2\pi}{T}\right)
\ee
$T$�͌��]�����B

����������
\be
4\pi^2\frac{a^3}{T^2}=GM_{\odot}
\ee

\bigskip

�q���̉^������f���̎��ʂ��킩��E�E�E�Ȃǂ̉��p�Ɏg����B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�X�J�C�t�b�N}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{quote}
ex.
�X�J�C�t�b�N�i�n�C�����C���ɂ��j�����낤�B\cite{princeton}
���x��l�̃��[�v�̒[���ԓ���̗L��n�_�ɂ���C�����ɉ��тĂ���B
���[�v�ƒn�ʂ͐ڒ��͂��Ă��Ȃ��B���[�v�̒����͂ǂꂭ�炢�H
�i$1.5\times 10^5 {\rm km}$�j
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�ȉ~�O��(N)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�ɍ��W�ŉ���}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�O����$x$-$y$���ʏ�ɂ���Ƃ���B

�ɍ��W$r,\phi$�ł���킷�B
\be
\left\{\begin{array}{l}
x=r\cos\phi \\
y=r\sin\phi
\end{array}
\right.
\ee

�G�l���M�[�ۑ����
\be
E=T+V=\frac{1}{2}m(\dot{r}^2+r^2\dot{\phi}^2)+V(r)
\ee
�͈��B�����ŁC$\dot{~}$�͎��Ԃɂ�������\���B

�p�^���ʕۑ��ɂ��
\be
m r^2 \dot{\phi}=\ell
\ee
�͈��B

����������
\be
\dot{r}^2=\frac{2}{m}\left(E-V(r)-\frac{\ell^2}{2mr^2}\right)
\ee
������
\be
V(r)=-\frac{k}{r}=-\frac{GMm}{r}
\ee

���̎��̍��ӂ͐��Ȃ̂ŁC�E�ӂ����ƂȂ�悤��$r$�̒l�����Ƃ肦�Ȃ��B

\bigskip

�O����\������
\be
\frac{dr}{d\phi}=
\frac{\dot{r}}{\dot{\phi}}=
\frac{mr^2}{\ell}\sqrt{\frac{2}{m}\left(E-V(r)-\frac{\ell^2}{2mr^2}\right)}
\ee

�ȒP�̂��߁C�ϐ���ς���B
\be
u=\frac{1}{r}
\ee
�����
\be
\frac{du}{d\phi}=
-\sqrt{\frac{2mE}{\ell^2}+\frac{2mku}{\ell^2}-u^2}
\ee

�ϕ��i�����W�j
\be
\int\frac{dx}{\sqrt{a+bx+cx^2}}=\frac{1}{\sqrt{-c}}\cos^{-1}\frac{-b+2cx}{b^2-4ac}
\ee
��p�����
\be
\phi=\phi_0-\cos^{-1}\left[
\frac{\frac{\ell^2 u}{mk}-1}{\sqrt{1+\frac{2E\ell^2}{mk^2}}}\right]
\ee

\begin{quote}
�܂��́C�ȉ��̂悤�ɂ��ĉ����Ă��ǂ��B
\be
\left(\frac{du}{d\phi}\right)^2=
\frac{2mE}{\ell^2}+\frac{2mku}{\ell^2}-u^2
\ee
�̗��ӂ�$\phi$�Ŕ������������
\be
\frac{d^2u}{d\phi^2}+u=\frac{mk}{\ell^2}
\ee
�i����͒P�U�����I($u-mk/\ell^2$)�j
\end{quote}

������ɂ������
\be
\frac{1}{r}=C\left(1+\epsilon\cos(\phi-\phi_0)\right)
\ee

\be
C=\frac{mk}{\ell^2}=\frac{GMm^2}{\ell^2}
\ee

\be
\epsilon=\sqrt{1+\frac{2E\ell^2}{mk^2}}=\sqrt{1+\frac{2E\ell^2}{G^2M^2m^3}}
\ee

\bea
& &E>0 \rightarrow \epsilon>1~~~~~~~~~�o�Ȑ�\\
& &E=0 \rightarrow \epsilon=1~~~~~~~~~������\\
& &E<0 \rightarrow \epsilon<1~~~~~~~~~�ȉ~�@�i\epsilon=0 �~�j
\eea
�ƂȂ��Ă���B

$\epsilon$�́i���ɑȉ~�̏ꍇ�j�C���S���ƌĂ΂��B

\bigskip

�O�ɂ��q�ׂ����C
�Q�̂̉^���͐��m�ɂ͏d�S�n�Ŏ�舵���ׂ��ł���B
���ɓ�d�����̉^���ɂ��ẮC���炩�ł���B
�f���n�ɂ��āC���̘f���̏d�͂ɂ��ۓ����d�v�Ȃ̂�
�����܂ł��Ȃ��B

�Q�̂̉^����
���ڂ������͂�\cite{�]��1}\cite{�o�[�W���[1}�Ȃǂ�����B
�O���ʂ��s�ςł��邱�Ƃ�������Əؖ����Ă���B
\footnote{�ڂ����v�킸�x�N�g�����g�������Ȃ����B}

�j���[�g���̓v�����L�s�A�ɂ����āC�􉽊w�I���o�����Ă���B
�t�@�C���}��\cite{�t�@�C���}��1}��ǂ߁B�܂��C
������h���Z�J�[���ɂ�������Q�ƁB

\bigskip

�E���͂��t����łȂ��ꍇ�̌�����\cite{�x1}������B
\begin{quote}
ex. ���S�́i���́j�ɂ��^���ŁC���_�̋O����$r=2a\cos\phi$�ŕ\�����Ƃ��C
�͂̑傫����$1/r^5$�ɔ�Ⴗ�邱�Ƃ������B
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�ȉ~}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$E<0$�̂Ƃ�

$r$�̍ő�l
\be
r_{max}=C^{-1}\left(1-\epsilon\right)^{-1}
\ee

$r$�̍ŏ��l
\be
r_{min}=C^{-1}\left(1+\epsilon\right)^{-1}
\ee

\be
a=\frac{r_{max}+r_{min}}{2}=C^{-1}\left(1-\epsilon^2\right)^{-1}
\ee
$a$�͋O�������a�C���邢�͕��ϋO�����a�Ƃ�΂��B

$\phi_0=0$�Ƃ����
\be
\frac{(r\cos\phi+a\epsilon)^2}{a^2}+\frac{(r\sin\phi)^2}{b^2}=
\frac{(x+a\epsilon)^2}{a^2}+\frac{y^2}{b^2}=1
\ee
������
\be
b=\sqrt{1-\epsilon^2} a
\ee
����͑ȉ~�̕\���́C�����ЂƂ̌�B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�P�v���[�̑�O�@��(N)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�p�^���ʕۑ�������
\be
\frac{1}{2}r^2\dot{\phi}=\frac{\ell}{2m}
\ee

�����ϕ����邱�Ƃɂ��C�O���i�Ɉ͂܂ꂽ�j�ʐς����߂���B
\be
\int_0^T\frac{1}{2}r^2\dot{\phi}dt=\pi ab=\frac{\ell}{2m}T
\ee
$T$�́i���]�j�����B

$b$�����܂����������
\be
\frac{\ell}{2m}T=\pi ab=\pi a^{3/2}\sqrt{\frac{\ell^2}{mk}}
\ee
�ƂȂ�C���Ȃ킿
\be
T=2\pi a^{3/2}\sqrt{\frac{m}{k}}=\frac{2\pi}{\sqrt{GM_{\odot}}} a^{3/2}
\ee

\be
GM_{\odot}=4\pi^2\frac{a^3}{T^2}
\ee
�P�v���[�̑�R�@���𓾂��B$a$�݂̂�p���C���S�����܂܂Ȃ��\���ɂȂ��Ă���B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{��l���x}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

��l���ʖ��x$\rho_0$��������̐����l���悤�B������u�n���v�Ƃ���B
\be
F_r=-\frac{GMm}{r^2}
\ee

���S����$r$�̋������Ɋ܂܂�鎿�ʂ�
\be
M=\frac{4\pi}{3}r^3 \rho_0
\ee
�Ȃ̂�
\be
F_r=-\frac{4\pi Gm}{3}\rho_0 r
\ee

\be
V(r)=m\Phi(r)=\frac{1}{2}\frac{4\pi Gm}{3}\rho_0 r^2
\ee

�P�U�����������|�e���V�����ł���B
�p�U�����͒n���ł�
\be
\omega=\sqrt{\frac{g}{R_{\oplus}}}\approx 1.2\times 10^{-3} {\rm s^{-1}}
\ee

��ʂɑȉ~�O���ł��邪�C���S�͗��œ_�̒��_�i�u���S�v�j�B�n�������ɂ������
�ǂ�ȋO���ł��C�������������B���̎����́C�n�\���ꂷ��̐l�H�q���Ƃ������B
�i��W�T���ň���B�j

\begin{quote}
ex.
(future ex.) ���Θ_�ł́H�i��l���x�̐��C���n���̎��C�������B�j
\end{quote}

\begin{quote}
ex.
�����̃g���l�����@�����Ƃ��i�K���������S��ʂ�Ȃ��j�C���̒���
�d�͂ɂ���ĉ^�����镨�͕̂Г��ǂ̂��炢�̎��ԂŒʉ߂��邩�H���C�͖�������B
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�f���̋O��(R)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�O���̎�(R)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�A�C���V���^�C���̏d�͗��_����ʑ��Θ_�ōl�@����B�j���[�g���̏d�͂�
�ǂ̂悤�ɂǂ̂��炢����Ă���̂��B

�j���[�g���ߎ��ł͈�v�������Ȃ��Ƃ͊��Ɍ����B

���������āC�����Ȏ���̌v�ʂ̉�
\footnote{�A�C���V���^�C���������̉�}
���g���B

����$M$�̋��Ώ̂Ȑ��̂܂��i�O���j�̎���͎��̐^�󋅑Ώ̉���
�\�����B
\be
-ds^2=-\left(1-\frac{r_g}{r}\right)c^2 dt^2+\frac{1}{1-\frac{r_g}{r}}dr^2+
r^2\left(d\theta^2+\sin^2\theta d\phi^2\right)
\ee
������
\be
r_g\equiv\frac{2GM}{c^2}
\ee
�ł���B���̉����V�����@���c�V���g���Ƃ����C$r_g$���V�����@���c�V���g���a�C
�܂��͂Ƃ��ɏd�͔��a�ƌĂԁB

\bigskip

�O���͐ԓ��ʏ�i$\theta=\pi/2$�j�ɂ���Ɖ��肷��B

���n���̎�����C�ȉ��̂R�̎���������B

���̂P�F
\be
\left(1-\frac{r_g}{r}\right)c\dot{t}=\frac{E_R}{mc^2}
\ee
����̓G�l���M�[�ۑ��ɑΉ��B�����ł�$\dot{~}$��$s$�ɂ�������\���B

���̂Q�F
\be
r^2\dot{\phi}=\frac{\ell}{mc}
\ee
����͊p�^���ʕۑ��ɑΉ��B

���̂R�F�i���̂P�C���̂Q�������āj
\be
-\frac{E_R^2/(m^2c^4)}{1-\frac{r_g}{r}}+\frac{\dot{r}^2}{1-\frac{r_g}{r}}+
\frac{\ell^2}{m^2c^2r^2}=-1
\ee

���Ȃ킿
\be
\dot{r}^2=\frac{E_R^2}{m^2c^4}-\left(1-\frac{r_g}{r}\right)-
\frac{\ell^2}{m^2c^2r^2}\left(1-\frac{r_g}{r}\right)
\ee
���̍��ӂ͐�������C�E�ӂ����ɂȂ�悤��$r$�͈̔͂��������Ȃ��B
�i(N)�̂Ƃ��Ɣ�ׂ悤�B�j

\bigskip

�O���̌��\����
\be
-\frac{r^4E_R^2/(\ell^2c^2)}{1-\frac{r_g}{r}}+
\frac{1}{1-\frac{r_g}{r}}\frac{\dot{r}^2}{\dot{\phi}^2}+r^2
=-\frac{r^4m^2c^2}{\ell^2}
\ee

\be
\frac{1}{r^4}\left(\frac{dr}{d\phi}\right)^2+\frac{1}{r^2}=
\frac{E_R^2-m^2c^4}{\ell^2c^2}+\frac{m^2c^2r_g}{\ell^2 r}+\frac{r_g}{r^3}
\ee

\be
\left(\frac{du}{d\phi}\right)^2+u^2=
\frac{2mE}{\ell^2}+\frac{2mku}{\ell^2}+r_g u^3
\ee

�Ō�̍����u��ʑ��Θ_�I���ʁv�B
\footnote{�u��Ԃ̋Ȃ����Ă���v���ʂł��邱�Ƃɒ���}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�����ߓ��_�̈ړ�}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�����ߓ��_�̈ړ��E�E�E���̘f���̐ۓ��Ȃǂ�
�j���[�g���d�͂Ő����ł��Ȃ����́C
�P�O�O�N������S�R�b�ł��邱�Ƃ��m���Ă����B

�����ł̋ߎ��@�͎��\cite{DInv}�ɂ��B

�O��������������
\be
\frac{d^2u}{d\phi^2}+u-\frac{3}{2}r_g u^2=
\frac{mk}{\ell^2}
\ee

����̉���W�J�ŋ��߂�B
\be
u=u_0+u_1+\cdots
\ee
������
\be
u_0=C(1+\epsilon\cos\phi)
\ee
�͖��ۓ����ł���C
\be
|u_1|\approx r_g/a^2 << |u_0|
\ee
�͐ۓ��ɂ�鏬���Ȃ���B

�������ɑ�����ēW�J�̂P����
\bea
\frac{d^2u_1}{d\phi^2}+u_1&=&\frac{3}{2}r_g C^2(1+\epsilon\cos\phi)^2 \nn
&=&\frac{3}{2}r_g C^2(1+\epsilon^2/2)
+3r_g C^2 \epsilon\cos\phi
+\frac{3}{4}r_g C^2 \epsilon^2\cos 2\phi
\eea
���̕������̉���
\be
u_1=\frac{3}{2}r_g C^2(1+\epsilon^2/2)
+\frac{3}{2}r_g C^2 \epsilon\phi\sin\phi
-\frac{1}{4}r_g C^2 \epsilon^2\cos 2\phi
\ee

�����I�ȕ����́C�O���̌�ɂ͊֌W���邪�C�ߓ��_�ړ��ɂ͊�^���Ȃ��B
���������Ď����I�łȂ��������������
\bea
u&\approx&C(1+\epsilon\cos\phi)+\frac{3}{2}r_g C^2 \epsilon\phi\sin\phi \nn
&\approx&C\left(1+\epsilon\cos\phi+\frac{3}{2}r_g C \epsilon\phi\sin\phi\right) \nn
&\approx&C\left(1+\epsilon\cos\left[\phi\left(1-\frac{3}{2}r_g C\right)\right]\right)
\eea
�Ƌߎ��ł���B

�ߓ��_�ɖ߂�܂ł�
\be
\frac{2\pi}{1-\frac{3}{2}r_g C}\approx
2\pi\left(1+\frac{3}{2}r_g C\right)
\ee
�̊p�x�i�܂Ȃ���΂Ȃ�Ȃ��B���������ċߓ��_�̈ʒu�͊p�x
\be
\Delta\phi=\frac{6\pi GM_{\odot}}{(1-\epsilon^2) a c^2}
\ee
������������ɐi��ł������ƂɂȂ�B

\begin{quote}
ex.
�l�����Ă݂�B
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{���̋O��(R)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�����̋O��

\be
\left(1-\frac{r_g}{r}\right)c\dot{t}=E
\ee

\be
r^2\dot{\phi}=L
\ee

$ds^2=0$���
\be
-\frac{E^2}{1-\frac{r_g}{r}}+\frac{\dot{r}^2}{1-\frac{r_g}{r}}+
\frac{L^2}{r^2}=0
\ee

\be
\dot{r}^2=E^2-\frac{L^2}{r^2}\left(1-\frac{r_g}{r}\right)
\ee

\begin{quote}
ex.
�������~�O���������Ƃ͉\���H
\end{quote}

\bigskip

\begin{quote}
\underline{�z���C�Y���߂��̉�}
\bea
\frac{d\phi}{dt}=\left(1-\frac{r_g}{r}\right)\frac{\lambda}{r^2} \\
\frac{dr}{dt}=-\left(1-\frac{r_g}{r}\right)
\sqrt{1-\left(1-\frac{r_g}{r}\right)\frac{\lambda^2}{r^2}}
\eea
������$\lambda=L/E$�B

$r=r_g+\Delta$�C$\Delta << r_g$�Ƃ����
\bea
\frac{d\phi}{dt}=\frac{\lambda}{r_g^3}\Delta \\
\frac{d\Delta}{dt}=-\frac{\Delta}{r_g}
\sqrt{1-\frac{\lambda^2}{r_g^3}\Delta}
\eea
���̉���
\bea
\phi=\frac{2 r_g}{\lambda}\tanh\frac{t}{2 r_g} \\
\Delta=\frac{r_g^3 / \lambda^2}{\cosh^2\frac{t}{2 r_g}}
\eea
\end{quote}

\bigskip

�O����\������
\be
\left(\frac{du}{d\phi}\right)^2+u^2=
\frac{1}{b^2}+r_g u^3
\ee
�ƂȂ�B�O�Ɠ��l��$u=1/r$�Ƃ��C$b=|L/E|$�ł���B

�����������
\be
\frac{d^2u}{d\phi^2}+u=\frac{3}{2}r_g u^2
\ee

�O�Ɠ������W�J�ŋ��߂�B
\be
u=u_0+u_1+\cdots
\ee
������
\be
u_0=\frac{\sin\phi}{b}
\ee
�͖��ۓ����i���_����$b$���ꂽ�����I�j�ł���C
\be
|u_1|\approx r_g/b^2 << |u_0|
\ee
�͐ۓ��ɂ�鏬���Ȃ���B

�������ɑ�����ēW�J�̂P����
\be
\frac{d^2u_1}{d\phi^2}+u_1=\frac{3}{2}r_g \frac{\sin^2\phi}{b^2}
\ee
���̕������̉���
\be
u_1=\frac{1}{2}\frac{r_g}{b^2}(1+C_0\cos\phi+\cos^2\phi)
\ee
������$C_0$�͔C�Ӓ萔�B

$u_0$��$\phi$��$0$�����$\pi$��$0$�C�����������ɒB���Ă���B
�ۓ����ł́C$-\theta_1$�����$\pi+\theta_2$��$0$�ɂȂ�Ƃ���B
������$\theta_1,\theta_2$�͂Ƃ��ɏ��������̊p�x�ł��邱�Ƃ����҂����B

\bea
& &-\frac{\theta_1}{b}+\frac{1}{2}\frac{r_g}{b^2}(2+C_0)=0 \\
& &-\frac{\theta_2}{b}+\frac{1}{2}\frac{r_g}{b^2}(2-C_0)=0
\eea

���ǁC��������$\Delta\theta$�����Ȃ���B�����
\be
\Delta\theta=\theta_1+\theta_2=2\frac{r_g}{b}
\ee
�Ƃ�����B

���z�ɂ����̋Ȃ���ŁC��ԋȂ���̂͑��z�\�ʂ��ꂷ���ʂ���Ȃ̂�
\be
\Delta\theta=\frac{4GM_{\odot}}{R_{\odot}c^2}\approx 1.75''
\ee
���H���ɔw��̍P���̈ʒu������Č�����B�i�ʂ̎��̎ʐ^�Ɣ�r�j

�d�g���ł��ϑ��ł���B

\bigskip

��͂̎��ʂɂ��w��̋�͂��c��ŁC�܂����ɖ��邭�C�܂������Ɍ�����悤�Ȍ��ۂ�
�d�̓����Y���ہi���ʁj�Ƃ����C�ŋߑ����̗Ⴊ�������Ă���B

\newpage

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\section{��������d��}
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\subsection{���ƃj���[�g���d��}
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\subsubsection{�Ð����t}
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���́C�i�ߎ��I�Ɂj���Ώ̎��ȏd�͌n�ł���B
�����̑Η��C������̕��ˁC�������ϓ�����щ�]�͂����ł͍l���Ȃ��B
�܂��C�j���[�g���I�ɍl�@����B

�ÓI�Ȑ��̓����ł́C���͌��z�Əd�͂̂荇�������藧���Ă���i�Ð��i���j���t�j�B
�i�Ⴆ�΁C\cite{�]����1}�Q�ƁB�j

���͌��z�Əd�͂̂荇���́C�ȉ��̎��ɂ܂Ƃ߂���B
\be
\frac{dP}{dr}=-\frac{GM(r)}{r^2}\rho
\ee
\be
\frac{dM}{dr}=4\pi r^2\rho(r)
\ee
�����̎��ƁC��ԕ������i����$P$�Ǝ��ʖ��x$\rho$�̊Ԃ̊֌W���j������΁C
���̍\�������߂邱�Ƃ��ł���B

\bigskip

��G�c�ɕ]������ƁC
\be
\frac{P_c}{R}\approx\frac{GM\rho_c}{R^2}
\ee
�����ŁC$R$�͐��̔��a�C
\be
M\approx\frac{4\pi}{3}R^3\rho_c
\ee
�͐��̎��ʁB���͂Ǝ��ʖ��x�͒��S�ł̒l�i�܂��͕��ϒl�j�ő�\����B

���̂Ƃ����̎��ʂ�
\be
M\approx\sqrt{\frac{P_c^3}{G^3 \rho_c^4}}
\ee
�܂����̔��a��
\be
R\approx\sqrt{\frac{P_c}{G \rho_c^2}}
\ee
�ƌ��ς��邱�Ƃ��ł���B

\bigskip

\noindent
\underline{��}

�E���̈��͂������Ƃ��i���x���������j
\be
P_c=���
\ee
\be
M\propto\frac{1}{\rho_c^2}
\ee

�E�C�̂̈��͂������Ƃ�
\be
P_c\approx \rho_c
\ee
\be
M\propto\frac{1}{\sqrt{\rho_c}}
\ee

\begin{quote}
ex.
���z�̏ꍇ�C
\be
P\approx\frac{\rho}{m_H}kT
\ee
\be
M_{\odot}=\frac{R_{\odot}kT}{Gm_H}
\ee
�l�����Ă݂悤�B���x�́H
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�|���g���[�v}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

��ԕ����������̌�ŗ^������Ƃ����|���g���[�v�Ƃ����B
\be
P=K \rho^{1+1/N}
\ee
�i$K$�͒萔�B�j

�Ð����t�̎�����
\be
\frac{1}{r^2}\frac{d}{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right)=
-4\pi G \rho(r)
\ee
�𓾂�̂ŁC����Ƀ|���g���[�v�^�̏�ԕ�������������B
���̍ہC���̂悤�ȕϐ���I�Ԃ��Ƃɂ���B
\be
\rho(r)=\rho_c \phi^N(r),~~~\rho_c=\rho(0),~~~\phi(0)=1
\ee

�����͎��̎��𓾂�B
\be
(N+1) K \rho_c^{1/N}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\phi}{dr}\right)=
-4\pi G \rho_c \phi^N(r)
\ee

�����
\be
a\equiv\sqrt{\frac{(N+1) K \rho_c^{1/N-1}}{4\pi G}}
\ee
\be
\xi\equiv r/a
\ee
�Ƃ�����

\underline{Lane-Emden equation}
\be
\frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2\frac{d\phi}{d\xi}\right)=
-\phi^N~~~~~~~~(\phi(0)=1)
\ee
�𓾂�B

$N=0,1,5$�̂Ƃ��ɂ́C��͓I�ȉ����m���Ă���B

�E$N=0$ ���x���̐�
\be
\phi(\xi)=1-\frac{1}{6}\xi^2
\ee

�E$N=1$
\be
\phi(\xi)=\frac{\sin\xi}{\xi}
\ee

�E$N=5$
\be
\phi(\xi)=(1+\xi^2/3)^{-1/2}
\ee

�܂��C�������ƌĂ΂����̂ł�
$P\propto \rho$�̊֌W������CLane-Emden��������$N=\infty$
�ɑΉ�����B
���̏ꍇ�́C���E�����𖳎������
����ȉ��Ƃ���$\rho\propto r^{-2}$��������B
\footnote{�Q�ߓI�ɂ͓������͓����U�镑��������B}

\bigskip

���̕\�ʂ�
\be
\phi(\xi_0)=0
\ee
�ƂȂ�$\xi_0$�ɑΉ�����̂Ő��̔��a��
\be
R_{\star}=a \xi_0
\ee
�ŗ^������B

\bigskip

���a$\xi$�܂łɊ܂܂�鎿�ʂ�
\bea
M(\xi)&=&\int_0^{\xi}4\pi a^3 \rho_c \phi^N(\xi') {\xi'}^2 d\xi' \nn
&=&-4\pi a^3 \rho_c\int_0^{\xi} \frac{d}{d\xi'}{\xi'}^2\frac{d\phi(\xi')}{d\xi'}  d\xi' \nn
&=&-4\pi a^3 \rho_c{\xi}^2\frac{d\phi(\xi)}{d\xi}
\eea
�Ȃ̂ŁC
���̎��ʂ�
\be
M_{\star}=M(\xi_0)=-4\pi a^3 \rho_c{\xi_0}^2\left.
\frac{d\phi(\xi)}{d\xi}\right|_{\xi=\xi_0}
\ee
�ƂȂ�B

\begin{quote}
ex.
$N=3$�̂Ƃ��C���̎��ʂ͒��S���x�ɂ��Ȃ����Ƃ������B
\end{quote}

\begin{quote}
ex.
$N=0,1,5$�̂Ƃ��C���̔��a�Ǝ��ʂ����߂�B

���̂��̂̏ꍇ�ɁC���a-���ʂ̊֌W���O���t�ɂ���B
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{�k�ނ�����(N)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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�d�q�̂悤�ȃt�F���~���q�i�t�F���~�I���j�́C�ቷ�ŏk�ނƂ������ۂ��N�����B
�t�F���~���q�͓���̏�Ԃ͂P�̗��q������L�ł��Ȃ����߁C���x���[���̏ꍇ�ł�
�t�F���~���q�̋C�͈̂��́i�k�ވ��j�����B

���F�킢���́C�d�q�̏k�ވ��ɂ���Ďx�����Ă���B

�����ł͊ȒP�̂��ߗz�q�Ɠd�q�̐����قړ������Ƃ���B

�d�q�̖��x�i$\approx$�z�q�̖��x�j��
\footnote{�ʑ���Ԃ�$\hbar^3$�̑̐ϒ��ɂP����Ɛ�����B
�i�X�s���̎��R�x�𖳎������B�j}
\be
n\approx p_F^3/\hbar^3
\ee
������$p_F$�̓t�F���~�^���ʁi�d�q�̏W�܂�̒��ł̍ő�̉^���ʁj�ł���B

���ʖ��x�́C�z�q�̊�^�����|�I�Ȃ̂�
\be
\rho\approx m_p n
\ee
�ƂȂ�B$m_p$�͗z�q�̎��ʁB

���͂͂قƂ�Ǔd�q�̏k�ވ��ł���B���͂͂����悻
���q�̉^���G�l���M�[�ɔ�Ⴗ��Ǝv���ėǂ��B

�E�񑊑Θ_�I�ȏꍇ�i��r�I�ᖧ�x�j�ł�
\be
P\approx n p_F^2/m_e\approx \frac{\hbar^2}{m_e m_p^{5/3}}\rho^{5/3}
\ee
�i$m_e$�͓d�q�̎��ʁj

�E���Θ_�I�ȏꍇ�i��r�I�����x�j�ł�
\be
P\approx n p_F c\approx \frac{\hbar c}{m_p^{4/3}}\rho^{4/3}
\ee

\begin{quote}
ex.
���ꂼ��̏ꍇ�C�|���g���[�v��$N$�͂����ɑΉ����邩�B
\end{quote}

\begin{quote}
ex.
���x�ƈ��͂̌����ȕ\���𒲂ׂĂ݂悤�B
\end{quote}

\bigskip

���S���x����r�I�Ⴏ��΁C
\be
P_c\propto \rho_c^{5/3}
\ee
�Ȃ̂ŁC���F�킢���̎��ʂ�
\be
M\propto \sqrt{\rho_c}
\ee

�����x�̏ꍇ��
\be
P_c\propto \rho_c^{4/3}
\ee
���
\be
M\approx \frac{1}{m_p^2}\left(\frac{\hbar c}{G}\right)^{3/2}
=\frac{M_{pl}^3}{m_p^2}=���
\ee
�ƂȂ�B������$M_{pl}$�̓v�����N���ʂƌĂ΂��B

���S���x���������Ă��C���̎��ʂɂ͍ő�l������Ƃ������Ƃł���B
�܂�C����ȏ�傫�Ȏ��ʂ̔��F�킢���́i����ɂ́j���݂ł��Ȃ��B

�����Ȍv�Z�ł͍ő县�ʂ�
\be
M_{Ch}=1.4 M_{\odot}
\ee
�ƂȂ�C�����������h���Z�J�[�����ʂƌĂԁB

\begin{quote}
ex. �ȉ��̂悤�ɂ��Ă��]���ł��邱�Ƃ��m���߂�B

���̗͊w�I�G�l���M�[��
\be
E\approx n R^3 E_k-\frac{GM^2}{R}
\ee
������$R$�͐��̔��a�C

���q������̉^���G�l���M�[��

\be
E_k\approx \frac{p_F^2}{m_e}
\ee
�i�񑊑Θ_�I�j

\be
E_k\approx p_F c
\ee
�i���Θ_�I�j

�G�l���M�[���ł��Ⴍ�Ȃ�悤��$R$�����肷��B
\end{quote}

\bigskip

���l�ɁC�����q���k�ނ��Ă���ꍇ�C�����q����������邱�Ƃ��ł���B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{���̐U��(N)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

���̔����ȐU�����l���Ă݂�B\cite{�]����1}

$r$�̈ʒu�Ɏ��_��u���B�^����������
\be
\frac{d^2r}{dt^2}=-\frac{GM}{r^2}
\ee

���͈�l���x�ŋߎ�����B
\be
M=\frac{4\pi}{3}r^3 \rho
\ee
�����
\be
\frac{d^2r}{dt^2}=-\frac{4\pi G\rho r}{3}
\ee
�P�U���ƂȂ邱�Ƃ��킩��B

�U���̎�����
\be
T=2\pi\sqrt{\frac{3}{4\pi G\rho}}\propto\frac{1}{\sqrt{G\rho}}
\ee

\bigskip

\begin{quote}
ex.
�ȉ��̏ꍇ�C�U���̎����͂ǂꂭ�炢���B

���F�킢���i$1 M_{\odot}$, $10000 {\rm km}$�j

�����q���i$1 M_{\odot}$, $10 {\rm km}$�j
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{���̎��](N)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�������]���Ă���ꍇ�C���S�͂��\�ʏd�͂��z���邱�Ƃ͂Ȃ��̂�
\cite{������1}
\be
R\omega^2<\frac{GM}{R^2}
\ee
������l���x�Ƃ����
\be
M=\frac{4\pi}{3}R^3\rho
\ee
���x�ɉ��������B
\be
\rho>\frac{3}{4\pi}\frac{\omega^2}{G}
\ee

����$1.56\times 10^{-3} {\rm s}$�̃p���T�[�ł�
\be
\omega=\frac{2\pi}{1.56\times 10^{-3} {\rm s}}=4.0\times 10^3 {\rm s^{-1}}
\ee
�Ȃ̂�
\be
\rho>5.7\times 10^{10} {\rm kg/cm^3}
\ee

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{��ʑ��Θ_�I��}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

���Ώ̂Ȍv��
\be
-ds^2=-e^{-2\delta}\Delta dt^2+\frac{dr^2}{\Delta}+
r^2 \left(d\theta^2+\sin^2\theta d\phi^2\right)
\ee
�����肷��B
$\Delta, \delta$��$r$�݂̂̊֐��Ɖ��肷��B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Einstein tensor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Einstein tensor�͎��̂悤�ɒ������B
\be
G^{\mu}_{\nu}\equiv R^{\mu}_{\nu}-\frac{1}{2}R\delta^{\mu}_{\nu}
\ee

\bigskip

��̌v�ʂ���v�Z����ƁiAppendix�j
\bea
G^0_0&=&
2\left\{\sqrt{\Delta}\left[\frac{\sqrt{\Delta}}{r}\right]'+
\left[\frac{\sqrt{\Delta}}{r}\right]^2\right\}-
\left\{\frac{1}{r^2}-\left[
\frac{\sqrt{\Delta}}{r}\right]^2\right\} \nn
&=&\frac{1}{r}\Delta'+\frac{\Delta-1}{r^2}
\eea
\bea
G^0_0-G^1_1&=&R^0_0-R^1_1 \nn
&=&
-2\left[
\sqrt{\Delta}\left(-\delta'+
\frac{1}{2}\frac{\Delta'}{\Delta}\right)\right]
\left[\frac{\sqrt{\Delta}}{r}\right] \nn
&+&2\left\{\sqrt{\Delta}\left[
\frac{\sqrt{\Delta}}{r}\right]'+
\left[\frac{\sqrt{\Delta}}{r}\right]^2\right\} \nn
&=&\Delta\frac{2}{r}\delta'
\eea
�����ŁC${}'$��$r$�ɂ�������\���B

\bigskip

Einstein equation�͎��̂悤�ɂȂ��Ă���B
\be
G^{\mu}_{\nu}=R^{\mu}_{\nu}-\frac{1}{2}R\delta^{\mu}_{\nu}=
\frac{8\pi G}{c^4} T^{\mu}_{\nu}
\ee

\be
R^{\mu}_{\nu}=
\frac{8\pi G}{c^4}\left(T^{\mu}_{\nu}-\frac{1}{2}T\delta^{\mu}_{\nu}\right)
\ee

\bigskip

�����ŁC���S���� (perfect fluid) �����肷���
\be
T^{\mu}_{\nu}=diag.\left(-\rho c^2,P,P,P\right)
\ee
�܂�
\be
T=T^{\mu}_{\mu}=-\rho c^2+3 P
\ee
�ł���B

\bigskip

���S���̂̏ꍇ�C�A�C���V���^�C������������
\bea
G^0_0&=&-\frac{8\pi G}{c^2}\rho \\
G^0_0-G^1_1&=&-\frac{8\pi G}{c^2}\left(\rho+\frac{P}{c^2}\right)
\eea
�𓾁C���̏ꍇ
\bea
\frac{1}{r}\Delta'+
\frac{\Delta-1}{r^2}&=&-\frac{8\pi G}{c^2}\rho \\
\Delta \left[\frac{2}{r}\delta'\right]
&=&-\frac{8\pi G}{c^2}\left(\rho+\frac{P}{c^2}\right)
\eea
�ƂȂ�B
\footnote{�^��i$\rho c^2=P=0$�j�̂Ƃ��C�V�����@���c�V���g����������B}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�ۑ��̎�}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�G�l���M�[�^���ʂ̕ۑ��iconservation�j�̎��i���邢�͗͊w�I���t�̎��j
\be
\nabla_{\mu}T^{\mu\nu}=0
\ee
��K�p���Ă݂�B
\footnote{
\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
}
�i���̎��́C�A�C���V���^�C��������������������B�j

�����������
\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

������
\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
���g�����B

\bigskip

\bea
T^{tt}&=&\frac{1}{e^{-2\delta}\Delta}\rho c^2 \\
T^{rr}&=&\Delta P \\
T^{ij}&=&\frac{1}{r^2}\tilde{g}^{ij}P \\
\eea
�����
\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
�Ȃǂ��g���ƁC
\footnote{$\tilde{g}_{ij}dx^idx^j=d\theta^2+\sin^2\theta d\phi^2$}
�ۑ��̎�
\be
\nabla_{\mu}T^{\mu r}=0
\ee
�͎��̂悤�ɂȂ�B

\be
P'+\left(-\delta'+\frac{1}{2}
\frac{\Delta'}{\Delta}\right)\left(\rho c^2+P\right)=0
\ee

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{TOV������}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�A�C���V���^�C���������Ƃ��킹��ƁC
\be
-P'=\frac{4\pi G}{\Delta}r\left(\frac{1}{8\pi G}
\frac{1-\Delta}{r^2}+\frac{P}{c^4}\right)\left(\rho c^2+P\right)
\ee
�𓾂�B
\be
\Delta=1-\frac{2 G M_r}{c^2 r}
\ee
�Ƃ������Ƃɂ��

\underline{�g�[���}��-�I�b�y���n�C�}�[-���H���R�t(TOV)������}
\be
-P'=\frac{4\pi G~r}{1-\frac{2G M_r}{c^2 r}}
\left(\frac{M_r}{4\pi r^{3}}+\frac{P}{c^2}\right)\left(\rho+\frac{P}{c^2}\right)
\ee
��������B

�j���[�g���ߎ��ł�
\be
-P'=\frac{G M_r}{r^{2}}\rho
\ee
�𓾂�B
������
\be
M_r(r)=4\pi\int_0^r \rho(r') {r'}^{2}dr'
\ee
�ł���B

TOV�������́C��ʑ��Θ_�I�ȐÐ����t�̎��ł���B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{��l���x�̐�(R)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

���x�����l$\rho_0$�̂Ƃ��ɁCTOV�������������Ă݂悤�B
\be
M_r(r)=\frac{4\pi}{3}\rho_0 {r}^{3}
\ee
�ƂȂ邱�Ƃɒ��ӂ���ƁC
\be
\frac{P(r)+\rho_0 c^2}{3P(r)+\rho_0 c^2}=
\frac{\sqrt{1-\frac{r_g}{R}}}{\sqrt{1-\frac{r_g r^2}{R^3}}}
\label{inner}
\ee
�Ɖ�����B
������������$P(R)=0$�ƂȂ�$R$�����̔��a�C
$r_g=\frac{2G}{c^2}\frac{4\pi}{3}\rho_0 {R}^{3}=\frac{2GM_{\star}}{c^2}$�Ƃ����B

\begin{quote}
ex.
�񑊑Θ_�I�|���g���[�v$N=0$�̏ꍇ�Ɣ�r����B
\end{quote}

(\ref{inner})�̍��ӂ�$1/3$�ȏ�ł���̂ŁC���̔��a�ɂ͉���������
\be
R>\frac{9}{8}r_g
\ee
�ł���B

\bigskip

�v�ʂ͎��̂悤�ɂȂ�B�i�������C$r0$�Ȃ̂�$r_*<\pi$�̏ꍇ�j

\bigskip

�d�͎���
\bea
M&=&\frac{N_B}{4\pi}\sqrt{\tilde{g}}\frac{M_{pl}^2}{m_B} \nn
&=&\frac{N_B}{\sqrt{2}(4\pi)^{3/2}}\sqrt{g}\frac{M_{pl}^3}{m_B^2}
\eea

��ʑ��Θ_�I�ȍl�@�i�ő县�ʁE�E�E�j�ɂ��ẮC\cite{Jetzer1}
�Ȃǂ�����B

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\subsection{�d�͕���}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsubsection{�d�͕���(N)}
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���a$r$�̐��̕\�ʂɗ��q������C�����Î~������Ԃ��狅�Ώ̂̂܂�
���k���Ă����Ƃ����\cite{������2}
\be
\frac{1}{2}\dot{r}^2-\frac{GM}{r}=-\frac{GM}{r_0}
\ee
������$\dot{~}$�͎��Ԕ����B���͂Ȃǂ̗͓͂����Ȃ��Ƃ��Ă���B
���������Ă��̏ꍇ�C���̓_�X�g��̕�������Ȃ�Ƃ���B
���̉������߂邽�߁C
\be
r=r_0 \cos^2\frac{\theta}{2}
\ee
�Ƃ�����
\be
\frac{1}{2}r_0^2 \dot{\theta}^2=
\frac{GM}{r_0}\frac{1}{\cos^4\frac{\theta}{2}}
\ee
�Ə����������B���������ĉ���
\be
t=\sqrt{\frac{r_0^3}{8GM}}(\theta+\sin\theta)
\ee
�ƕ\�����B�������C$t=0$�̂Ƃ�$r=r_0$�Ƃ����B

$\theta=\pi$��$r=0$�ɑΉ�����̂ŁC������_�܂Ŏ��k����̂ɂ����鎞�Ԃ�
\be
t_k\approx\sqrt{\frac{1}{G\rho}}
\ee
�̒��x�Ƃ�����B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�d�͕���(R)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

���͖����C�_�X�g����Ȃ鐯�̋��Ώ̏d�͕�����l����Ƃ��C
�v�ʂ͎��̂悤�ɂƂ��($c=1$)�B\cite{Weinberg1}
\be
-ds^2=-dt^2+a^2(t)\left(\frac{dr^2}{1-r^2}+r^2(d\theta^2+
\sin^2\theta d\phi^2)\right)
\ee
�A�C���V���^�C������������
\be
\left(\frac{\dot{a}}{a}\right)^2+\frac{1}{a^2}=\frac{8\pi G}{3}\rho
\ee
�Ƃ����������𓾂�B

\begin{quote}
ex.
���̕������̉��́H�������C$\rho\propto a^{-3}$�Ƃ���B
\end{quote}

\bigskip

�d�͕���(R)�͍ŏ��ɃI�b�y���n�C�}�[�E�V���i�C�_�[�ɂ����
�������ꂽ�B
�I�b�y���n�C�}�[�E�V���i�C�_�[�_��(Phys. Rev. 56 (1939))��
���{���i�ώR�������ɂ��j���u���̎蒟�vVol. 23 �P�X�W�S�N�~��
�ɍڂ��Ă���B�i���肵�ɂ������ȁH�j

\bigskip

�ŋ߂ł��C�d�͕���ɂ͋����[�����ۂ����낢�댩�����Ă��āC
����������ɍs���Ă���B

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\subsection{�u���b�N�z�[��(R)}
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���̓������͂��L���łȂ��Ȃ�ƁC���͏d�͕�����N����
�ی��Ȃ��ׂ�Ă����B

�O�Ɍ����悤�ɁC�j���[�g���I��舵���ł�
�E�o���x�������ƂȂ锼�a�����݂����B

��ʑ��Θ_�I�Ɉ����Ă��C���̏o�Ă����Ȃ��̈悪���݂��邱�Ƃ��킩��B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{����̌v��}
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�A�C���V���^�C���������̐^�󋅑Ώ̉���
\be
ds^2=-\left(1-\frac{r_g}{r}\right)c^2 dt^2+\frac{1}{1-\frac{r_g}{r}}dr^2+
r^2\left(d\theta^2+\sin^2\theta d\phi^2\right)
\ee
������
\be
r_g=\frac{2GM}{c^2}
\ee

\bigskip

�ʂ̌�̏������Ƃ���
$r=R\left(1+\frac{GM}{2Rc^2}\right)^2$�Ƃ����Όv�ʂ�
\bea
ds^2&=&-\left(\frac{1-GM/(2Rc^2)}{1+GM/(2Rc^2)}\right)^2c^2 dt^2 \nn
&+&
\left(1+\frac{GM}{2Rc^2}\right)^4 \left[
dR^2+R^2\left(d\theta^2+\sin^2\theta d\phi^2\right)\right]
\eea
�ƂȂ�B

\bigskip

���a�����̌��̋O����
$ds^2=0$���
\be
\frac{dr}{dt}=\pm\left(1-\frac{r_g}{r}\right)
\ee
�u���̑��x�v�́C$r=r_g$�ɋ߂Â��ƂO�ɂȂ�I

$r_g$������������͌��������o�Ă����Ȃ��B
$r=r_g$�̖ʂ��u���ۂ̒n����(horizon)�v�ƌĂԁB

�܂��C$r_g$���V�����@���c�V���g���a�ƌĂԁB

\bigskip

���z����$M_{\odot}$�̃u���b�N�z�[����
Schwarzshild���a�͖�$3 {\rm km}$�ł���B

\bigskip

$10^6 M_{\odot}$�̎��ʂ̃u���b�N�z�[����
Schwarzshild���a��$\approx 5 R_{\odot}$�ł���B

���̒��x�̃u���b�N�z�[������͒��S�ɑ��݂���ƍl�����Ă���B

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\subsubsection{�u���b�N�z�[���̏���}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�����ł́C$G=\hbar=c=k_B=1$�Ƃ���B

\bigskip

black hole�̔w�i�ł́C���x
\be
T_H=\frac{1}{8\pi M}
\ee
�̌n�ɂ���̂Ɠ����ł���B
$T_H$��Hawking���x�Ƃ����B

���Ȃ݂ɁC
\be
T_H=\frac{\kappa}{2\pi}
\ee
������$\kappa$�̓u���b�N�z�[���\�ʂ̏d�́i�����x�j�B
\footnote{���z���ʂ�black hole�ł́C$T_H\sim 10^{-7} {\rm K}$�B}

\bigskip

�M�͊w�֌W��
\be
dU=TdS-PdV
\ee

�ɂ����āC�����G�l���M�[$U$��black hole�̎���$M$�Ɠ��ꎋ����B
�i������$dV=0$�j

����ɉ��x��Hawking���x�Ƃ�����
\be
dM=T_H dS
\ee

���Ȃ킿
\be
dS=8\pi M dM
\ee

�ϕ������
\be
S=4\pi M^2=\frac{4\pi (2M)^2}{4}=\frac{A}{4}
\ee

������$A$��horizon�̕\�ʐρB

�u���b�N�z�[���̓G���g���s�[�������Ă���I
\footnote{���z���ʂ�black hole�ł́C$S\sim 10^{77}$�B}

\bigskip

��ʂ̃u���b�N�z�[���ł��C
�G���g���s�[$S$��$A/4$�ƕ\����B
\footnote{$G$�𕜊�������ƁC$S=\frac{A}{4G}$�B}

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\subsubsection{���̑��̃u���b�N�z�[��}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

��]���Ă���C�ѓd���Ă���C�Ɍ���(extreme)�C�������́C�᎟���́C
�����_�́C�u���[����́C�E�E�E�C�u���b�N�z�[���B

\bigskip

�܂��C�u���b�N�z�[���̂܂��̍~���~�Ղɂ��ĂȂǁC
�����[���b�肪��������B

\newpage

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\section{��́C��͒c�Əd��}
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\subsection{��͂̉�]�Ȑ�}
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�W���I�ȋ�͂́C$10^{11}$�̐��C$10 {\rm kpc}$�̂Ђ낪��������Ă��āC
�������Ɖ�]���Ă���B

\bigskip

���S����̋�����$r$�̐������̑���$v$�ŉ~�O�������Ă���Ƃ����
\be
\frac{v^2}{r}=\frac{GM(r)}{r^2}
\ee
������$M(r)$�͔��a$r$�̋��̓����Ɋ܂܂�鎿�ʁB

�\��$r$�̑傫���Ƃ���Ȃ�$M(r)$�͈��ƂȂ�͂��B
���������ď\���傫��$r$�ɂ��Ă�
\be
v\propto\frac{1}{\sqrt{r}}
\ee
�ƂȂ�͂��ł���B
�������ϑ�����́C�قƂ�ǂ̋�͂ɂ����āC$r$�����Ȃ�傫���Ă�
\be
v\approx ���
\ee

������$r$�C�c����$v$���v���b�g�����O���t��
��]�Ȑ��ƌĂԁB�قƂ�ǂ̋�͂ŁC��]�Ȑ��͕��R�ł���B

\bigskip

���̌����Ȃ��悤�ȋ�͒��S���痣�ꂽ�Ƃ���ɂ��C���ʂ�����̂��낤���H

����Ȃ��C���̂悤�ȉ����I�����i���́j���C
�u�_�[�N�}�^�[�v�ƌĂ�ł���B

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\subsection{��͒c�Əd��}
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\subsubsection{�����q�n�̃G�l���M�[}
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�����̋�͂͏W�܂��ċ�͒c��������Ă���B

\bigskip

��͂𗱎q�Ƃ݂Ȃ��B�݂��̏d�݂͂̂Ɏx�z����Ă���n�ł�\cite{Duric1}
\be
m_i\ddot{\vr}_i=-G\sum_{j�ii�ȊO�j}\frac{m_im_j(\vr_i-\vr_j)}{|\vr_i-\vr_j|^3}
\ee

$\dot{\vr_i}$����ς��āC$i$�Řa���Ƃ�B���ӂł�
\be
\sum_i m_i\ddot{\vr}_i\cdot\dot{\vr}_i=
\frac{d}{dt}\sum_i\frac{1}{2}m_i\dot{\vr}_i^2=\frac{dK}{dt}
\ee
�Ə������Ƃ��ł��C����E�ӂł�
\bea
-G\sum_{i,j}\frac{m_im_j(\vr_i-\vr_j)\cdot\dot{\vr}_i}{|\vr_i-\vr_j|^3}
&=&-G\sum_{i,j}\frac{m_im_j(\vr_j-\vr_i)\cdot\dot{\vr}_j}{|\vr_j-\vr_i|^3} \nn
&=&\frac{1}{2}G\sum_{i,j}m_im_j\frac{d}{dt}\frac{1}{|\vr_i-\vr_j|} \nn
&=&-\frac{dV}{dt}
\eea
�Ƃ�����̂ŁC
\be
\frac{d}{dt}(K+V)=\frac{dE_{tot}}{dt}=0
\ee
��������B

����͑S�͊w�I�G�l���M�[�̕ۑ���\���Ă���B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�r���A���藝}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

����������\cite{Duric1}
\be
m_i\ddot{\vr}_i=-G\sum_{j�ii�ȊO�j}\frac{m_im_j(\vr_i-\vr_j)}{|\vr_i-\vr_j|^3}
\ee
�ɍ��x��$\vr_i$����ς��C$i$�Řa���Ƃ�B
\bea
\sum_i m_i\ddot{\vr}_i\cdot\vr_i&=&
\frac{d^2}{dt^2}\sum_i\frac{1}{2}m_i\vr_i^2-
\sum_i m_i\dot{\vr}_i^2 \nn
&=&\frac{d^2}{dt^2}I-2K
\eea
�����
\bea
-G\sum_{i,j}\frac{m_im_j(\vr_i-\vr_j)\cdot\vr_i}{|\vr_i-\vr_j|^3}
&=&-G\sum_{i,j}\frac{m_im_j(\vr_j-\vr_i)\cdot\vr_j}{|\vr_j-\vr_i|^3} \nn
&=&-\frac{1}{2}G\sum_{i,j}\frac{m_im_j}{|\vr_i-\vr_j|} \nn
&=&V
\eea
�ƂȂ�B

$I$�̂Q�K�����̍��́C�n���͊w�I�ɕ��t�ł���Ƃ���΁C���ςƂ��ĂO�ɂȂ�̂�
\be
2K+V=0
\ee
�𓾂�B�i�r���A���藝�j

����������
\be
E_{tot}=K+V=-K=\frac{1}{2}V<0
\ee

$K$�͋�͂̉^���̊ϑ�����킩��̂ŁC���ǂ��̊֌W������|�e���V�����̑傫���C
�܂�n�Ɋ܂܂�鎿�ʂ����߂���B

\bigskip

�ϑ�����C�����Ă��鐯�╨������̊�^�ł͉^����������邾���̎��ʂ�
�B���ł��Ȃ����Ƃ��킩��B�����ɂ��u�_�[�N�}�^�[�v�����݂���̂��낤���H

\begin{quote}
�Q�̂ł̃r���A���藝�͊ȒP�ɏؖ��ł���B
�����ł͂���ɊȒP�̂��߁C
$M>>m$�C�܂�����$m$�͓����~�^�������Ă���Ƃ��悤�B

\be
K=\frac{1}{2}mv^2
\ee
�ł���C�܂�
\be
V=-\frac{GMm}{R}
\ee
�ł���B

�^��������
\be
\frac{mv^2}{R}=\frac{GMm}{R^2}
\ee
���
\be
K=-\frac{1}{2}V
\ee
��������B
\end{quote}

\newpage

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\section{�c���F���Əd��}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{�j���[�g���I�F��}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{�[�[���K�[�̃p���h�N�X�ɂ��ĉ������������������ǁC�����B}
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��l�Ȗ��x���z�������C���͖����̕����i�_�X�g�j�����肷��B\cite{Liddle1}
���_����$r$�̈ʒu�ɂ��闱�q�����a�����ɕ����ƂƂ��ɉ^�����Ă���Ƃ��C
�͊w�I�G�l���M�[�ۑ�����
\be
\frac{1}{2}\dot{r}^2-\frac{GM(r)}{r}=���l
\ee
�ł���B������$\dot{~}$�͎��Ԕ����C�܂�
\be
M(r)=\frac{4\pi}{3}r^3\rho
\ee
�ł���B

�����ƂƂ��ɓ������W��
\be
\vr=a(t)\vx
\ee
�i$x$�͈��B�j�ƕ\����$a(t)$�ɑ΂���������́C
���l��$kc^2x^2/2$�Ƃ���ƁC
\be
\left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=\frac{8\pi G}{3} \rho
\label{ex1}
\ee

\bigskip

�n�b�u���̖@���C��͂̋����ƌ�ޑ��x����Ⴗ�邱�ƁC��
\be
\vv=\dot{a}(t)\vx=\frac{\dot{a}}{a}\vr\equiv H\vr
\ee
�̂悤�ɐ��������B

���݂̃n�b�u���萔$H_0$�́C�ϑ�����
\be
H_0=100h {\rm km/s/Mpc}
\ee
\be
h\approx 0.5�0.85
\ee
�ł���B

���Ƃߍ���͒c�̏ꍇ�C
\be
v=1200 {\rm km/s}
\ee
�Ȃ̂�
\be
����=\frac{v}{H_0}=\frac{1200}{100 h} {\rm Mpc}=12 h^{-1} {\rm Mpc}
\ee
�ƂȂ�B
\footnote{��͂̋����͕��ʐԕ��Έ�$z$�ŕ\���B�߂��̋�͂ɂ��ẮC
$z=v/c$�ƂȂ�B���Ƃߍ���͒c�̏ꍇ�C$z\approx 4\times 10^{-3}$�B}

\bigskip

�M�͊w���@��
\be
dE=TdS-PdV
\label{netsu1}
\ee
�ɂ����āC�����G�l���M�[����
\be
E=\frac{4\pi}{3}a^3\rho c^2
\ee
�Ƃ����
\be
\frac{dE}{dt}=4\pi a^2\rho c^2\frac{da}{dt}+
\frac{4\pi}{3}a^3\frac{d\rho}{dt} c^2
\ee
�𓾂�B����C
\be
\frac{dV}{dt}=4\pi a^2\frac{da}{dt}
\ee
�ł��邩��C(\ref{netsu1})��
\be
\dot{\rho}+3\frac{\dot{a}}{a}\left(\rho+\frac{P}{c^2}\right)=0
\label{ex2}
\ee
�ƕ\�����B

\begin{quote}
ex. (\ref{ex1})��(\ref{ex2})����
\be
\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right)
\label{Ein2d}
\ee
�𓱂��B
\end{quote}

\bigskip

(\ref{Ein2d})�́C�ȉ��̂悤�ɂ��Ă��������Ƃ��ł���B\cite{Zee1}

�d�̓|�e���V�����ɂ��Ẵ|�A�\����������
\be
\nabla^2\Phi=4\pi G\rho
\ee

����C���q�̗���ɂ��Ă�Euler��������
\footnote{���͂̌��ʁi�E�ӂ�$-\vnabla P/\rho$�̊�^�j�𖳎������B}
\be
\frac{\partial\vv}{\partial t}+(\vv\cdot\vnabla)\vv=-\vnabla\Phi
\ee

����ɑ΂���A���̎���
\be
\frac{\partial\rho}{\partial t}+\vnabla\cdot(\rho\vv)=0
\ee

�����ŕ����͋�ԓI�Ɉ�l�ȃ_�X�g���Ƃ����
\be
\rho=\frac{C}{a^3(t)}
\ee
�܂��C
\be
\vv=\vx\frac{\dot{a}}{a}
\ee
�Ƃ���B

�|�A�\���������̉���
\be
\Phi=\frac{1}{2}x^2\frac{4\pi G\rho}{3}
\ee
�Ȃ̂ŁC�����p�����
\be
\frac{\ddot{a}}{a}=-\frac{4\pi G\rho}{3}
\ee
���������B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{���Θ_�I�F���_}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

��ʑ��Θ_�ł͎���S�̂��������Ƃ��ł���̂ŁC
�F�����̂̕������c�_���邱�Ƃ��ł���B

\bigskip

��Ԃɂ��āC��l�����������肷��ƁC�������̎�ނ́u��Ԃ̌�v
�����肳���B

\bigskip

�܂��C���R�ȂS������ԂŁC���a�P�́u���ʁv���l����B
�����\����������
\be
x^2+y^2+z^2+w^2=1
\ee
�ł���B

���́u���ʁv��ł͎��̂悤�Ƀp�����[�^���g�����Ƃ��ł���B
\bea
x&=&\sin\chi \sin\theta \cos\phi \\
y&=&\sin\chi \sin\theta \sin\phi \\
z&=&\sin\chi \cos\theta \\
w&=&\cos\chi
\eea

���������āC�u���ʁv��ł̒Z�������̎����
\bea
d\ell^2_{1}&=&dx^2+dy^2+dz^2+dw^2 \nn
&=&d\chi^2+\sin^2\chi(d\theta^2+\sin^2\theta d\phi^2)
\eea
���ꂪ��l�����ȋ�ԕ����̌v�ʂ̈�̌��ł���B

$r=\sin\chi$�Ƃ�����
\be
d\ell^2_{1}=\frac{dr^2}{1-r^2}+r^2(d\theta^2+\sin^2\theta d\phi^2)
\ee

\bigskip

�܂��C
\be
x^2+y^2+z^2-w^2=-1
\ee
�ŕ\�����u���Ȗʁv���l����B

���̏ꍇ
\bea
x&=&\sinh\chi \sin\theta \cos\phi \\
y&=&\sinh\chi \sin\theta \sin\phi \\
z&=&\sinh\chi \cos\theta \\
w&=&\cosh\chi
\eea
�Ƃ������Ƃ��ł���B

�Z�������̎���͍��x�͎��̂悤�ɂƂ�B
\bea
d\ell^2_{-1}&=&dx^2+dy^2+dz^2-dw^2 \nn
&=&d\chi^2+\sinh^2\chi(d\theta^2+\sin^2\theta d\phi^2)
\eea

�������l�����ȋ�ԕ����̌v�ʂ̈�̌��ł���B

$r=\sinh\chi$�Ƃ�����
\be
d\ell_{-1}^2=\frac{dr^2}{1+r^2}+r^2(d\theta^2+\sin^2\theta d\phi^2)
\ee

\bigskip

��l�����ȋ�Ԃ͂��̂悤�ɂ܂Ƃ߂���B
\be
d\ell_{k}^2=\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta d\phi^2)
\ee
������$k$�́C$1,0,-1$�̂����ꂩ�ł���B$k=0$�̏ꍇ�͕��ʂ̃��[�N���b�h�I
��Ԃł���B

\bigskip

\be
-ds^2=-c^2dt^2+a^2(t)d\ell_{k}^2
\ee
�Ƃ����v�ʁi���o�[�g�\��-�E�H�[�J�[�v�ʁj�ŃA�C���V���^�C��������������������
���̈ꕔ�Ƃ���
\be
\left(\frac{\dot{a}}{a}\right)^2+\frac{k}{a^2}=\frac{8\pi G}{3} \rho
\ee
�Ƃ����n�b�u�����������������B������$c=1$�Ƃ����B

\begin{quote}
ex.
(\ref{ex2})�́C���Θ_�I�ɐ��藧�B$\nabla_{\mu}T^{\mu}_{\nu}=0$
�Ɠ����ł��邱�Ƃ������B
\end{quote}

\begin{quote}
ex.
�ȉ��̂悤�ȏꍇ�Ƀn�b�u���������������B

�E���������͂Ȃ��̃_�X�g�̂Ƃ��i$k=1,0,-1$�ɂ��āj(Friedmann universe)
\be
\rho\propto a^{-3}
\ee

�E���������̂悤�ɂقƂ�ǎ��ʂ������Ȃ����q����Ȃ�Ƃ�
�i$k=0$�ɂ��āj
\be
\rho\propto a^{-4}
\ee
\end{quote}

\bigskip

�ŋ߂̊ϑ��ł́C�����̂قƂ�ǂ́i�V�O���j�u�F�����v�ɂ����̂��Ƃ���Ă���B
�F�����̖��x�ƈ��͂ւ̊�^��
\be
P=-\rho c^2
\ee
�Ƃ����ƂĂ���Ȃ��̂ł���B

\begin{quote}
ex.
$k=0$�̂Ƃ��C
�F�����ƃ_�X�g�݂̂��܂ޏꍇ�̃n�b�u���������������B
\end{quote}

�܂��C�̂���̂R�O���̂قƂ�ǂ́C����Ȃ������C�_�[�N�}�^�[�ł���炵���B
\footnote{�F�������̌��f�����̗��_�Ɗϑ�����C
�ʏ�̕����̗ʂ͔��ɏ��Ȃ��炵�����Ƃ��킩���Ă���B}

\newpage

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\section{�����F���Əd��}
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\subsection{�F���̑�K�͍\��}
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���݂̉F���̋�͕��z�̊ϑ�����C
��͂̕��z�ɂ͑傫�ȍ\�������݂��邱�Ƃ��킩��B
�����͂ǂ̂悤�ɂ��Ăł����̂��낤���H

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\subsection{��炬����̍\�����}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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�������z�̏����Ȃ�炬����V�̂��ł���B\cite{�c���]1}

\bigskip

���ʁF$M$

���x�F$\rho$

�傫���F$R$

\bigskip

���x$V$�Ŏ��k���Ă����Ƃ���΁C�\������̎��Ԃ�
\be
t\approx\frac{R}{V}
\ee
�����x�͏d�͂ŗ^������B
\be
\frac{V}{t}\approx\frac{GM}{R^2}
\ee
����������
\footnote{�d�͕���̃^�C���X�P�[���Ɠ����B}
\be
t\approx\frac{1}{\sqrt{G\frac{M}{R^3}}}\approx\frac{1}{\sqrt{G\rho}}
\ee

���������X���k����Ă����Ƃ����
\be
t<\frac{R}{v_s}
\ee
�łȂ���΂Ȃ�Ȃ��B������$v_s$�͉����B
����������
\be
R>v_s t
\ee

���̏ꍇ
$M_J\approx\rho\left(\frac{4\pi}{3}R^3\right)$���W�[���Y���ʂƌĂԁB

\bigskip

$t\approx 1���N$�C
$v_s\approx 100 {\rm km/s}$
�Ƃ����

$R\approx 3.2\times 10^{17} {\rm km}\approx 10^4 {\rm pc}$�C
$\rho\approx 10^{-27} {\rm g/cm^3}$�C
$M_J\approx 1.4\times 10^{36} {\rm kg}\approx 7\times 10^5 M_{\odot}$

\begin{quote}
ex.
���͂Əd�͂̋ύt�������狁�߂Ă��ǂ��B\cite{Zee1}

���ȏd�̓G�l���M�[�̕ϕ�
\be
\Delta\left(\frac{GM^2}{R}\right)\approx
\Delta\left(G\rho^2R^5\right)\approx
G\rho^2R^4\Delta R
\ee

���͂ɂ��d���i�̕ϕ��j
\be
P R^2\Delta R
\ee

����炪�ނ荇���Ƃ����W�[���Y���a
\be
R_J=\frac{1}{\sqrt{G\rho}}\sqrt{\frac{P}{\rho}}
\ee
\end{quote}

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\subsection{��炬�̐���}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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�W�[���Y���ʒ��x�̂�炬���C���͂������Ȃ��Ȃ�Ɛ������͂��߂�B\cite{������2}
\footnote{����͂�����Ƒ�G�c�����邩�ȁH�i���Ƃɂ͓{����H�j}

�O�ɂ��o�Ă�����{�I�ȕ�����
\be
\nabla^2\Phi=4\pi G\rho
\ee
\be
\frac{\partial\vv}{\partial t}+(\vv\cdot\vnabla)\vv=-\vnabla\Phi
\ee
\be
\frac{\partial\rho}{\partial t}+\vnabla\cdot(\rho\vv)=0
\ee
�ɂ����āC
\be
\rho=\frac{C}{a^3(t)}(1+\delta)=\rho_0(1+\delta)
\ee
\be
\vv=\frac{\dot{a}}{a}\vr+\vv_1=\vv_0+\vv_1
\ee
\be
\Phi=\frac{1}{2}x^2\frac{4\pi G\rho}{3}+\Phi_1
\ee
�Ƃ����$\delta, \vv_1, \Phi_1$�̐���ߎ���
\be
\nabla^2\Phi_1=4\pi G\rho_0\delta
\ee
\be
\frac{\partial\vv_1}{\partial t}+(\vv_1\cdot\vnabla)\vv_0+
(\vv_0\cdot\vnabla)\vv_1=-\vnabla\Phi_1
\ee
\be
\frac{\partial\delta}{\partial t}+\vnabla\cdot\vv_1+\vv_0\cdot\vnabla\delta=0
\ee
�𓾂�B

$\frac{d}{dt}\equiv\frac{\partial}{\partial t}+ (\vv_0\cdot\vnabla)$�Ƃ���B�܂��ȒP�̂��ߒ��g���̂�炬�̂�
�l����$\delta(t)$�Ƃ���B
\be
\nabla^2\Phi_1=4\pi G\rho_0\delta
\ee
\be
\frac{d\vv_1}{dt}+\frac{\dot{a}}{a}\vv_1=-\vnabla\Phi_1
\ee
\be
\frac{d\delta}{dt}+\vnabla\cdot\vv_1=0
\ee
����炩��
\bea
\vnabla\cdot\frac{d\vv_1}{dt}+\frac{\dot{a}}{a}\vnabla\cdot\vv_1&=&
-\nabla^2\Phi_1 \nn
&=&-4\pi G\rho_0\delta
\eea
\bea
\frac{d^2\delta}{dt^2}&=&-\frac{d}{dt}\vnabla\cdot\vv_1 \nn
&=&-\vnabla\cdot\frac{d\vv_1}{dt}+\frac{\dot{a}}{a}\vnabla\cdot\vv_1
\eea
������$\vnabla=\frac{1}{a}\vnabla_{\vx}$�ł��邱�Ƃɒ��ӂ���B

�ȏ�̎�����
\be
\ddot{\delta}+2\frac{\dot{a}}{a}\dot{\delta}=4\pi G\rho_0\delta
\ee
�𓾂�B

\bigskip

$a(t)\propto t^{2/3}$�����g����
\be
\delta\propto t^{2/3}\propto a(t)
\ee
���킩��B

\bigskip

�Ⴆ��$\delta\approx 10^{-4}$�̂�炬�́C�F���̑傫����10000�{�ɂȂ鍠�ɂ�
�������ʂ��������炢�ɐ������Ă���B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{very early universe}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�C���t���[�V����}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\rho+3P/c^2<0$�ƂȂ�悤�ȕ����̏ꍇ�C
�F���̑傫���͉����I�ɑ傫���Ȃ�B

\bigskip

����$\rho+P/c^2=0$�̏ꍇ�C
\be
\rho=\rho_v=���
\ee
�ƂȂ邱�Ƃ��킩��̂ŁC�n�b�u���������̉���($k=0$)
\be
a(t)\propto e^{Ht}
\ee
������
\be
H=\sqrt{\frac{8\pi G}{3}\rho_v}
\ee

\bigskip

�C���t���[�V�������̗ʎq�h�炬����͌���̎�ƂȂ�h�炬�ɂȂ�
�ߒ���񋟂���B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{�ʑ�����}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�E�F���Ђ��@Cosmic string

�܂������ȁu���Θ_�I�v�g�Ђ��h�̂���G�l���M�[�^���ʃe���\��
($c=1$)
\be
T^{\mu}_{\nu}=\mu\delta(x)\delta(y)\left(
\begin{array}{cccc}
-1 & & & \\
& 0 & & \\
& & 0 & \\
& & & -1
\end{array}\right)
\ee
������$\mu$�͐����x�B$\mu=v^2$

�����Einstein eq�ɑ������Ɖ���
\be
-ds^2=-dt^2+dz^2+dr^2+(1-8G\mu)r^2d\theta^2
\ee

������
\be
\sqrt{1-8G\mu}\theta=\theta'
\ee
�Ə�����
\be
ds^2=-dt^2+dz^2+dr^2+r^2{d\theta'}^2
\ee
�i���R�ȋ�ԁj

�F���Ђ����܂������ŐÎ~���Ă���΁C�܂��ł͏d�͂������Ȃ��B

������
\be
0<\theta'<2\pi\sqrt{1-8G\mu}
\ee
�łȂ���΂Ȃ�Ȃ��E�E�E�u�~���v�̂悤�ȋ�ԁB

\bigskip

�@$v=�i�哝�ꗝ�_�̃X�P�[���j\sim 10^{16} {\rm GeV}$

\be
G\mu\sim Gv^2\sim 10^{-6}\left(\frac{v}{10^{16}{\rm GeV}}\right)^2
\ee

\bigskip

�@���ڂ��ׂ����Ƃ�

�@�@�Ђ��̋߂��ɂƂǂ܂��Ă��Ă��d�͂������Ȃ�
\footnote{��Ԃ��C�Ђ��̐L�тĂ�������ƂQ������Ԃ̐ςƂƂ炦��΁C
���̊�Ȑ����͂R��������ɂ�����A�C���V���^�C���d�͂̐����ɗR�����邱�Ƃ��킩��B}

�@�@���������݂����Ƃ�����߂��ȁI

�������݂���Ό����ɋ߂��̂ŁC��͂�댯�ł���B
\be
10^{-6}\times 3\times 10^{8} {\rm m/s}=300 {\rm m/s}
\ee
�Ђ��̒ʉߌ�C�����̕����͂��̂��炢�̑��x�łԂ��荇���B

\bigskip

�F���_�ɂ�����g�Ђ��h�@cosmic string�@�̋����[���Ƃ���

�@�@�@���ꎩ�g�̏d�͂͂قƂ�ǂȂ�

�@�A�@�������Əd�͍�p��������͂̎�

�@�B�@�Ђ��̍q�H��ɕ������W�߂遨��K�͍\��

\bigskip

�@�Ђ����������񂠂�ꍇ�C

�@����̂���G�l���M�[���x

�@�P�ʒ���������̎��ʖ��x$\sim \mu$ ($G\mu\sim 10^{-6}$)

\bigskip

\be
\rho_{cs}\propto \frac{\mu}{a^2}
\ee

�n�b�u��������
\be
3\left(\frac{\dot{a}}{a}\right)^2+3\frac{k}{a^2}=8\pi G\rho
\ee

\be
\rho=\rho_m+\rho_{cs}=\frac{\rho_0}{a^3}+\frac{\rho_{cs0}}{a^2}
\ee

�@cosmic strings�̊�^�͋ȗ��Ɠ������x�̂�����

�@�@Friedmann�F���̂ӂ�܂��ɉe���͏��Ȃ�

\newpage

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\section{�ʎq�d��}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$G$�͕����I�Ȏ������������萔�B

�ʎq���͍���B

\bigskip

�v�����N�����@$\sqrt{\frac{G\hbar}{c^4}}\approx 10^{-33} {\rm cm}$

�v�����N���ԁ@$10^{-43} {\rm s}$

�v�����N�G�l���M�[�@$\sqrt{\frac{\hbar c}{G}}c^2\approx 10^{19} {\rm GeV}$

\bigskip

�ȏ�̂悤�ȃX�P�[�������n�ɂ��ẮC
�ʎq�d�͂̌��ʂ������ł��Ȃ��Ȃ�ł��낤�B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{�ʎq�F���_}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�v�ʂ�($c=1$)
\be
-ds^2=\sigma^2\left[-N^2 dt^2+a^2(t)d\ell^2_{1}\right]
\ee
�Ƃ���B������$\sigma^2=\frac{2G}{3\pi}$�ł���B

�d�͂̍�p�i�A�C���V���^�C��-�q���x���g �A�N�V�����j�v���X�F�����̍�p
\be
S=\frac{1}{16\pi G}\int d^4x \sqrt{-g}\left(R-2\Lambda\right)
\ee
�ɂ����̌v�ʂ��������
\be
S=\int L dt=\int N\left[\frac{1}{2}a\left(1-
\frac{\dot{a}^2}{N^2}\right)-a^3\frac{\Lambda}{6}\right]
\ee

\begin{quote}
ex.
$N$�ŕϕ�����ƁC�n�b�u�����������o�邱�Ƃ��m���߂�B
\end{quote}

���������$\Pi$��
\be
\Pi=\frac{\partial L}{\partial\dot{a}}=-\frac{a\dot{a}}{N}
\ee
�Ƃ����
�n�~���g�j�A��$H$��
\bea
H&=&\Pi\dot{a}-L \nn
&=&N\left\{\frac{1}{2}\left[-\frac{1}{a}\Pi^2-a\right]+
\frac{\Lambda}{6}a^3\right\}\equiv N{\cal H}
\eea

$N$�ŕϕ������${\cal H}=0$�I�ƂȂ��Ă���B
�����F���ł́C�����Əd�͂̃G�l���M�[�̑��a�͂O�ł��邱�ƂɂȂ��Ă���B

\bigskip

�����ʎq���̎葱��
\be
\Pi\rightarrow -i\frac{d}{da}~~~~(\hbar=1)
\ee
�ɂ��������C�g���֐�$\Psi$��p�ӂ���
\be
{\cal H}\Psi=0
\ee
��g���������Ƃ��悤�B

\be
\left[-\frac{d^2}{da^2}+a^2-\frac{\Lambda}{3}a^4\right]\Psi=0
\ee

�L���|�e���V����
\be
V_{eff}=a^2-\frac{\Lambda}{3}a^4
\ee
�́u�R�v�����B

\bigskip

�z�[�L���O�@�o�H�ϕ��@�����E���E����

���B�����L���@������(from nothing)�̉F���̑n��

�@�@�@�@�@�@�@�ʎq�͊w�I�g���l�����O

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�����_�Ƃl���_}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�����_�i�X�g�����O���_�j�ł́C�Ђ��̐U�����l�X�ȗ��q��\���B
�����Ђ��i�u������v�j�̗��_�ɂ́C�K���d�͑��ݍ�p���܂܂��i�ĒJ�j�B

�ŋ߁Cbrane�Ƃ����������́u���v�̂悤�ȍ\�������݂����邱�Ƃ��킩�����B
����Ɂu�o�ΐ��v�Ƃ��������Ȃǂ��g���āC
�����_�𓝈ꂵ���u�l���_�v�ɂ��Ă̌���������ł���B

�����͈�̃u���[���̏�ɏZ��ł���\��������B
���̏ꍇ�ɂ��d�͂́C��Ԃ������������B
\footnote{�K�������u���R�Ɂv����Ȃ����B}

���̂悤�ȍl�����ɂ��ƁC�Ȃ��d�͂��������̗͂����ɒ[�Ɂu�ア�v
�������R�ɐ����ł���B

�������̑��݂̂��߁Cmm ���炢�ŋt���摥����̂��ꂪ���҂����B
\footnote{�ׂ����̓��f���ɂ�邪�C�Ⴆ�ΒP����
$r$���ꂽ�Q���̂̊Ԃ̏d�͂̑傫���́C��Ԃ̎�����$D$�̂Ƃ��C
$\propto 1/r^{D-1}$�ƂȂ�B���̂悤�ȉe�����Z�������Ŋ��҂����B}

\newpage

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\section{�d�͔g}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�A������̏d�͔g���o�̊m�F�̌��тɂ��C
�n���X�C�e�C���[��1993�N�m�[�x���܂���܂����B

\bigskip

�����ł́C���R�Ȏ��󂩂�̏����Ȃ���ɂ��čl����B
\be
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}
\ee
$h_{\mu\nu}$�̓Y�����̏グ������$\eta^{\mu\nu}$���ł����Ȃ��B

\bigskip

���[�}���ȗ���
\be
R_{\lambda\sigma\beta\alpha}\approx -\frac{1}{2}\left(
\partial_{\lambda}\partial_{\beta}h_{\alpha\sigma}-
\partial_{\lambda}\partial_{\alpha}h_{\beta\sigma}-
\partial_{\sigma}\partial_{\beta}h_{\lambda\alpha}+
\partial_{\sigma}\partial_{\alpha}h_{\lambda\beta}
\right)
\ee
���b��ȗ���
\be
R_{\sigma\alpha}\approx -\frac{1}{2}\left(
\Box h_{\alpha\sigma}-
\partial_{\alpha}\partial_{\lambda}h^{\lambda}_{\sigma}-
\partial_{\sigma}\partial_{\lambda}h^{\lambda}_{\alpha}+
\partial_{\sigma}\partial_{\alpha}h
\right)
\ee
������
\be
h\equiv h^{\mu}_{\mu}=\eta^{\mu\nu}h_{\mu\nu}
\ee

�X�J���[�ȗ���
\be
R\approx \eta^{\mu\nu}R_{\mu\nu}=
-\Box h+
\partial_{\sigma}\partial_{\lambda}h^{\lambda\sigma}
\ee

�A�C���V���^�C���e���\����
\bea
& &R_{\sigma\alpha}-\frac{1}{2}g_{\sigma\alpha}R\approx
R_{\sigma\alpha}-\frac{1}{2}\eta_{\sigma\alpha}R \nn
&\approx& -\frac{1}{2}\left(
\Box h_{\alpha\sigma}-
\partial_{\alpha}\partial_{\lambda}h^{\lambda}_{\sigma}-
\partial_{\sigma}\partial_{\lambda}h^{\lambda}_{\alpha}+
\partial_{\sigma}\partial_{\alpha}h
-\Box h \eta_{\sigma\alpha}+
\partial_{\beta}\partial_{\lambda}h^{\lambda\beta}
\eta_{\sigma\alpha}
\right)
\eea
�ƂȂ�B

\bigskip

������
\be
\tilde{h}_{\mu\nu}\equiv h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h
\ee
�Ƃ��C
�Q�[�W����
\footnote{$g^{\mu\nu}\Gamma_{\mu\nu}=0$�Ɠ����B}
\be
\partial_{\mu}\tilde{h}^{\mu}_{\nu}\equiv 0
\ee
���ۂ��ƁC�A�C���V���^�C�����������
\be
\Box \tilde{h}_{\mu\nu}=-\frac{16\pi G}{c^4}T_{\mu\nu}
\ee
�𓾂�B

\bigskip

���̕������́C�����̕ۑ��̎��Ɩ������Ȃ��B

\bigskip

�܂��C�����̂Ȃ��^��ł́C�g���������ł���B

\bigskip

�A�����J�ł�LIGO�C
���{�ł�TAMA300�Ȃǂ̏d�͔g�ϑ����u�������I�ϑ����n�߂�H

\newpage

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�萔}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
������\cite{PDG1}����̂����B

$\begin{array}{rcl} ���� & c & 299792458 {\rm m/s} \\ �d�͒萔 & G & 6.67259(85)\times 10^{-11} {\rm m^3 kg^{-1} s^{-2}} \\ �V���P�� & AU & 1.4959787066(2)\times 10^{8} {\rm km} \end{array}$

$\begin{array}{rcl} �v�����N���� & \sqrt{\hbar c/G} & 1.221047(79)\times 10^{19} {\rm GeV/c^2} \\ & &=2.17671(14)\times 10^{-8} {\rm kg} \\ �p�[�Z�N & {\rm pc} & 3.0856775807(4)\times 10^{16} {\rm m} \\ & & =3.262\ldots {\rm ly} \\ ���N & {\rm ly} & 0.3066\ldots {\rm pc} \\ & &=0.9461\ldots\times 10^{16} {\rm m} \\ ���z�̃V���o���c�V���g���a & 2GM_{\odot}/c^2 & 2.95325008 {\rm km} \\ ���z���� & M_{\odot} & 1.98892(25)\times 10^{30} {\rm kg} \\ ���z�i�ԓ��j���a & R_{\odot} & 6.96\times 10^{8} {\rm m} \\ �n������ & M_{\oplus} & 5.97370(76)\times 10^{24} {\rm kg} \\ �n���i�ԓ��j���a & R_{\oplus} & 6.378140\times 10^{6} {\rm m} \end{array}$

$\begin{array}{rcl} ��͌n���S�܂��̑��z�̉�]���x & & 220(20) {\rm km/s} \\ ��͌n���S�Ƒ��z�̋��� & & 8.0(5) {\rm kpc} \end{array}$

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\subsection{���k�̂���d�̓|�e���V����}
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\be
\Phi(r)=-2\pi a^2 G\lambda
\int_0^{\pi}\frac{\sin\theta d\theta}{\sqrt{r^2+a^2-2ar\cos\theta}}
\ee

\be
\int_{-1}^{1}\frac{dx}{\sqrt{r^2+a^2-2arx}}=
\left[\frac{\sqrt{r^2+a^2-2arx}}{-ar}\right]_{-1}^{1}
\ee

\bea
& &r>a �̂Ƃ�~~~\frac{2}{r} \\
& &ra �̂Ƃ�~~~\Phi=\frac{G 4\pi a^2\lambda}{r} \\
& &r

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