<\/span><\/h2>\n\n\n\n\u7d50\u5c40\u3001\u8a08\u91cf$g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$\u3067$h_{\\mu\\nu}$\u306e\u72ec\u7acb\u306a\u6210\u5206\u3068\u3057\u3066\u306f\u3001<\/p>\n\n\n\n
$$ \\left\\{ \\begin{aligned} h_{11}&=-h_{22} \\\\ h_{12}&=h_{21} \\end{aligned} \\right. $$<\/p>\n\n\n\n
\u306e\u4e8c\u3064\u3068\u306a\u308b\u306e\u3067\u3001<\/p>\n\n\n\n
$$\ng_{\\mu\\nu}=\n\\begin{pmatrix}\n -1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1\n\\end{pmatrix}\n+\n\\begin{pmatrix}\n 0 & 0 & 0 & 0 \\\\\n 0 & A & B & 0 \\\\\n 0 & B & -A & 0 \\\\\n 0 & 0 & 0 & 0\n\\end{pmatrix}\n$$<\/p>\n\n\n\n
\u8a08\u91cf\u306e\u3046\u3061\u5909\u5316\u3057\u306a\u3044\u90e8\u5206\u306f\u7701\u7565\u3057\u3066\u3001<\/p>\n\n\n\n
$$\ng_{ij}=\n\\begin{pmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{pmatrix}\n+\n\\begin{pmatrix}\n A & B \\\\\n B & -A\n\\end{pmatrix}\n$$<\/p>\n\n\n\n
\u3068\u8868\u3059\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u3053\u306b\u51fa\u3066\u304f\u308b$A$\u3068$B$\u304c\u6ce2\u306e\u3088\u3046\u306b\u5909\u5316\u3059\u308b\u306e\u3067\u3001<\/p>\n\n\n\n
$$ \\left\\{ \\begin{aligned} A&=a\\sin(\\omega_at+\\delta_a) \\\\ B&=b\\sin(\\omega_bt+\\delta_b) \\end{aligned} \\right. $$<\/p>\n\n\n\n
\u3068\u304a\u304f\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
$ds^2=g_{ij}dx^idx^j$\u306a\u306e\u3067\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n ds^2&=g_{11}dx^2+g_{12}dxdy+g_{21}dydx+g_{22}dy^2 \\\\\n &=g_{11}dx^2+2g_{12}dxdy+g_{22}dy^2 \\\\\n &=(1+A)dx^2+2Bdxdy+(1-A)dy^2\n\\end{align*}\n$$<\/p>\n\n\n\n
\u3053\u308c\u3092\u5e73\u9762\u56f3\u306e\u4e0a\u3067\u8868\u3059\u305f\u3081\u306b\u306f\u3001\u4f55\u3089\u304b\u306e\u5ea7\u6a19\u5909\u63db\u3092\u884c\u3063\u3066$ds^2=g’_{11}dx’^2+g’_{22}dy’^2$\u3068\u8868\u305b\u308b\u3088\u3046\u306b\u3059\u308c\u3070\u3088\u3044\u3067\u3059\u3002
\u884c\u5217\u3067\u8868\u305b\u3070\u3001<\/p>\n\n\n\n
$$\n\\begin{pmatrix}\n dx & dy\n\\end{pmatrix}\n\\begin{pmatrix}\n 1+A & B \\\\\n B & 1-A\n\\end{pmatrix}\n\\begin{pmatrix}\n dx \\\\\n dy\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n dx’ & dy’\n\\end{pmatrix}\n\\begin{pmatrix}\n X & 0 \\\\\n 0 & Y\n\\end{pmatrix}\n\\begin{pmatrix}\n dx’ \\\\\n dy’\n\\end{pmatrix}\n$$<\/p>\n\n\n\n
\u3068\u306a\u308b\u3088\u3046\u306a$X$\u3068$Y$\u3092\u898b\u3064\u3051\u308c\u3070\u3088\u3044\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001$dx’$\u3068$dy’$\u3092\u4eee\u306b\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
$$\n\\begin{pmatrix}\n dx’ \\\\\n dy’\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n s & t \\\\\n u & v\n\\end{pmatrix}\n\\begin{pmatrix}\n dx \\\\\n dy\n\\end{pmatrix}\n$$<\/p>\n\n\n\n
\u8ee2\u7f6e\u3092\u3068\u308c\u3070\u3001<\/p>\n\n\n\n
$$\n\\begin{pmatrix}\n dx’ & dy’\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n dx & dy\n\\end{pmatrix}\n\\begin{pmatrix}\n s & u \\\\\n t & v\n\\end{pmatrix}\n$$<\/p>\n\n\n\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u308c\u3092\u5148\u306e\u5f0f\u306e\u53f3\u8fba\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001<\/p>\n\n\n\n
$$
\\begin{align*}
&\\begin{pmatrix}
dx’ & dy’
\\end{pmatrix}
\\begin{pmatrix}
X & 0 \\\\
0 & Y
\\end{pmatrix}
\\begin{pmatrix}
dx’ \\\\
dy’
\\end{pmatrix} \\\\
&=
\\begin{pmatrix}
dx & dy
\\end{pmatrix}
\\begin{pmatrix}
s & u \\\\
t & v
\\end{pmatrix}
\\begin{pmatrix}
X & 0 \\\\
0 & Y
\\end{pmatrix}
\\begin{pmatrix}
s & t \\\\
u & v
\\end{pmatrix}
\\begin{pmatrix}
dx \\\\
dy
\\end{pmatrix}
\\end{align*}
$$<\/p>\n\n\n\n
\u3088\u308a\u3001<\/p>\n\n\n\n
$$\n\\begin{pmatrix}\n 1+A & B \\\\\n B & 1-A\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n s^2X+u^2Y & stX+uvY \\\\\n stX+uvY & t^2X+v^2Y\n\\end{pmatrix}\n$$<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001$X=Y=1$\u3068\u3057\u3066\u3057\u307e\u3046\u3068\u3001<\/p>\n\n\n\n
$$ \\left\\{ \\begin{aligned} &s^2+u^2=1+A \\\\ &t^2+v^2=1-A \\\\ &st+uv=B \\end{aligned} \\right. $$<\/p>\n\n\n\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u3082\u3057$A=B=0$\u306a\u3089\u3001$s,t,u,v$\u306e\u884c\u5217\u306f\u5358\u4f4d\u884c\u5217\u306b\u306a\u308b\u306e\u3067\u3001$s=v=1$\u3001$t=u=0$\u3068\u306a\u308a\u307e\u3059\u3002
$A$\u3068$B$\u304c\u5909\u5316\u3059\u308b\u3068$s,t,u,v$\u306e\u5024\u306f\u305d\u3053\u304b\u3089\u5c11\u3057\u305a\u308c\u308b\u306e\u3067\u3001\u5fae\u5c0f\u91cf$s’,v’$\u3092\u7528\u3044\u3066\u3001<\/p>\n\n\n\n
$$ \\left\\{ \\begin{aligned} s&=1+s’ \\\\ v&=1+v’ \\end{aligned} \\right. $$<\/p>\n\n\n\n
\u3068\u304a\u304d\u307e\u3059\u3002
$t$\u3068$u$\u306e\u305a\u308c\u3082\u308f\u305a\u304b\u3060\u3068\u8003\u3048\u3089\u308c\u308b\u306e\u3067\u3001$s’,t,u,v’$\u306f\u3059\u3079\u3066\u5fae\u5c0f\u91cf\u3067\u3059\u3002<\/p>\n\n\n\n
\u3053\u306e\u3088\u3046\u306b\u304a\u304f\u3068\u3001\u5148\u7a0b\u306e\u5f0f\u306f\u3001<\/p>\n\n\n\n
$$ \\left\\{ \\begin{aligned} &(1+2s’+s’^2)+u^2=1+A \\\\ &t^2+(1+2v’+v’^2)=1-A \\\\ &(1+s’)t+u(1+v’)=B \\end{aligned} \\right. $$<\/p>\n\n\n\n
2\u6b21\u306e\u5fae\u5c0f\u91cf\u3092\u7121\u8996\u3057\u3066\u6574\u7406\u3059\u308b\u3068\u3001<\/p>\n\n\n\n
$$ \\left\\{ \\begin{aligned} &s’=\\frac{A}{2} \\\\ &v’=-\\frac{A}{2} \\\\ &t+u=B \\end{aligned} \\right. $$<\/p>\n\n\n\n
$B$\u304c\u5897\u6e1b\u3059\u308b\u3053\u3068\u306b\u3088\u308b\u52b9\u679c\u306f$dx’$\u3068$dy’$\u304c\u7b49\u3057\u304f\u534a\u5206\u305a\u3064\u53d7\u3051\u308b\u3068\u3057\u3066\u3001$t=u=\\frac{B}{2}$\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u4ee5\u4e0a\u3088\u308a\u3001<\/p>\n\n\n\n
$$\n\\begin{pmatrix}\n dx’ \\\\\n dy’\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n s & t \\\\\n u & v\n\\end{pmatrix}\n\\begin{pmatrix}\n dx \\\\\n dy\n\\end{pmatrix}\n\\approx\n\\begin{pmatrix}\n 1+\\frac{A}{2} & \\frac{B}{2} \\\\\n \\frac{B}{2} & 1-\\frac{A}{2}\n\\end{pmatrix}\n\\begin{pmatrix}\n dx \\\\\n dy\n\\end{pmatrix}\n$$<\/p>\n\n\n\n
\n\n\n\n\u91cd\u529b\u6ce2\u304c\u6765\u305f\u3068\u304d\u306b\u539f\u70b9\u304b\u3089\u5186\u4e0a\u306b\u3042\u308b\u5404\u70b9\u307e\u3067\u306e\u8ddd\u96e2\u304c\u3069\u306e\u3088\u3046\u306b\u5909\u5316\u3059\u308b\u306e\u304b\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3057\u3066\u307f\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n
\u5148\u7a0b\u306e\u5f0f\u306b$dx=\\cos\\theta$\u3001$dy=\\sin\\theta$\u3092\u4ee3\u5165\u3057\u3066\u3001<\/p>\n\n\n\n
$$\n\\begin{pmatrix}\n dx’ \\\\\n dy’\n\\end{pmatrix}\n\\approx\n\\begin{pmatrix}\n 1+\\frac{a\\sin(\\omega_at+\\delta_a)}{2} & \\frac{b\\sin(\\omega_bt+\\delta_b)}{2} \\\\\n \\frac{b\\sin(\\omega_bt+\\delta_b)}{2} & 1-\\frac{a\\sin(\\omega_at+\\delta_a)}{2}\n\\end{pmatrix}\n\\begin{pmatrix}\n \\cos\\theta \\\\\n \\sin\\theta\n\\end{pmatrix}\n$$<\/p>\n\n\n\n
$A$\u3060\u3051\u304c\u5909\u5316\u3059\u308b\u5834\u5408\u3068$B$\u3060\u3051\u304c\u5909\u5316\u3059\u308b\u5834\u5408\u3092\u5206\u3051\u3066\u8003\u3048\u3066\u307f\u307e\u3059\u3002
\u3053\u3053\u3067\u306f\u3001$\\omega=\\omega_a=\\omega_b$\u3001$\\delta_a=\\delta_b=0$\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
$B=0$\u3068\u3057\u3066$A$\u3060\u3051\u3092\u5909\u5316\u3055\u305b\u308b\u3068\u3001\u5909\u5316\u306e\u69d8\u5b50\u304c+\u8a18\u53f7\u306e\u3088\u3046\u306b\u898b\u3048\u308b\u306e\u3067\u3001\u3053\u308c\u3092\u30d7\u30e9\u30b9\u504f\u6975\u30e2\u30fc\u30c9\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n