{"id":3062,"date":"2022-06-23T12:49:21","date_gmt":"2022-06-23T03:49:21","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=3062"},"modified":"2024-10-08T14:47:27","modified_gmt":"2024-10-08T05:47:27","slug":"%e5%85%89%e3%81%ae%e3%83%89%e3%83%83%e3%83%97%e3%83%a9%e3%83%bc%e5%8a%b9%e6%9e%9c%e3%81%ae%e3%81%8a%e3%81%95%e3%82%89%e3%81%84","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%81%ae%e7%b5%b1%e4%b8%80%e7%9a%84%e7%90%86%e8%a7%a3\/%e5%85%89%e3%81%ae%e3%83%89%e3%83%83%e3%83%97%e3%83%a9%e3%83%bc%e5%8a%b9%e6%9e%9c%e3%81%ae%e3%81%8a%e3%81%95%e3%82%89%e3%81%84\/","title":{"rendered":"\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u30fb\u5149\u884c\u5dee\u306e\u304a\u3055\u3089\u3044"},"content":{"rendered":"

\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u3067\u306f\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306f\u3069\u306e\u3088\u3046\u306b\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u304b\u3002\u4f8b\u3048\u3070\uff0c\u300c\u76f8\u5bfe\u6027\u7406\u8ad6\u300d\uff08<\/span><\/span>\u4e2d\u91ce\u8463\u592b\u8457\uff0c\u5ca9\u6ce2\u66f8\u5e97\uff09<\/a>\u306e p. 87 \u3042\u305f\u308a\u7b49\u3092\u53c2\u8003\u306b\u3057\u3066\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306b\u3082\u3068\u3065\u3044\u305f\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u8aac\u660e\u306b\u3064\u3044\u3066\uff0c\u304a\u3055\u3089\u3044\u3059\u308b\u3002\u7c21\u5358\u306e\u305f\u3081\u306b \\(c=1\\) \u3068\u3059\u308b\u3002<\/p>\n

\u5149\u306e\u4f1d\u64ad\u3092\u3042\u3089\u308f\u30594\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u89b3\u6e2c\u8005 \\(A\\) \u306e\u9759\u6b62\u6163\u6027\u7cfb\u3067\u3042\u308b \\(S\\) \u7cfb\u3067\u306f\uff0c\u5149\u306e4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306f<\/p>\n

$$k^{\\mu} = (k^0, k^1, k^2, k^3) = (\\omega, k_x, k_y, k_z)$$<\/p>\n

\\(A\\) \u306b\u5bfe\u3057\u3066\u901f\u3055 \\(V\\) \u3067 \\(x\\) \u65b9\u5411\u306b\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u306e\u9759\u6b62\u6163\u6027\u7cfb\u3067\u3042\u308b \\(S’\\) \u7cfb\u3067\u306f<\/p>\n

$$k^{\\mu’} = (k^{0^{\\prime}}, k^{1^{\\prime}}, k^{2^{\\prime}}, k^{3^{\\prime}}) = (\\omega’, k’_x, k’_y, k’_z)$$<\/p>\n

\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db<\/h3>\n

\u3042\u3089\u304b\u3058\u3081\u5ff5\u306e\u305f\u3081\u306b\u8a00\u3063\u3066\u304a\u304f\u304c\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306b\u57fa\u3065\u3044\u305f\u4ee5\u4e0b\u306e\u8aac\u660e\u306f\uff0c\u91cd\u529b\u304c\u306a\u3044\u7279\u6b8a\u76f8\u5bfe\u8ad6\u7684\u72b6\u6cc1\u3067\u306e\u307f\u6709\u52b9\u3067\u3042\u308b\u3002<\/span>\u91cd\u529b\u304c\u3042\u308b\u5834\u5408\u306f\uff0c\u305d\u3082\u305d\u3082\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u304c\u4f7f\u3048\u306a\u3044\u305f\u3081\uff0c\u5225\u9014\u7406\u5c48\u3092\u8003\u3048\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/strong><\/span><\/p>\n

\\(S\\) \u7cfb\u306e\u5ea7\u6a19 \\(x^{\\mu}\\) \u3068 \\(S’\\) \u7cfb\u306e\u5ea7\u6a19 \\(x^{\\mu’}\\) \u306e\u9593\u306e\u5ea7\u6a19\u5909\u63db\u306f\uff0c\u4ee5\u4e0b\u306e\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002<\/p>\n

\\begin{eqnarray}
t’ &=& \\frac{t -V x}{\\sqrt{1-V^2}} \\\\
x’ &=& \\frac{x -V t}{\\sqrt{1-V^2}} \\\\
y’ &=& y \\\\
z’ &=& z
\\end{eqnarray}<\/p>\n

\u3053\u306e\u9006\u5909\u63db\uff08\\(S\\) \u7cfb\u306e\u5ea7\u6a19 \\(x^{\\mu}\\) \u3092 \\(S’\\) \u7cfb\u306e\u5ea7\u6a19 \\(x^{\\mu’}\\) \u3067\u8868\u3059\u3053\u3068\uff09\u306f<\/p>\n

\\begin{eqnarray}
t &=& \\frac{t’ + V x’}{\\sqrt{1-V^2}} \\\\
x &=& \\frac{x’ + V t’}{\\sqrt{1-V^2}}
\\end{eqnarray}<\/p>\n

\u5149\u306e\u4f4d\u76f8\u306f\u5ea7\u6a19\u7cfb\u306b\u3088\u3089\u306a\u3044\u4e0d\u5909\u91cf\u3067\u3042\u308b<\/p>\n

\\begin{eqnarray}
\n\\omega t -\\boldsymbol{k}\\cdot\\boldsymbol{r} &=& \\omega t -k_x x -k_y y -k_z z \\\\
\n&=& \\omega’ t’-k_x’ x’ -k_y’ y’ -k_z’ z’
\n\\end{eqnarray}<\/p>\n

\u3068\u3044\u3046\u3053\u3068\u304b\u3089\uff0c4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3082\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3067\u5909\u63db\u3055\u308c\u308b\u3053\u3068\u304c\u5c0e\u304b\u308c\uff0c<\/p>\n

\\begin{eqnarray}
\\omega’ &=& \\frac{\\omega -V k_x}{\\sqrt{1-V^2}} \\tag{1}\\\\
k_x’ &=& \\frac{k_x -V \\omega}{\\sqrt{1-V^2}} \\tag{2}\\\\
k_y’ &=& k_y \\\\
k_z’ &=& k_z
\\end{eqnarray}<\/p>\n

\u9006\u5909\u63db\u306f<\/p>\n

\\begin{eqnarray}
\\omega &=& \\frac{\\omega’ + V k_x’}{\\sqrt{1-V^2}} \\tag{3}\\\\
k_x &=& \\frac{k_x’ + V \\omega’}{\\sqrt{1-V^2}} \\tag{4}\\\\
\\end{eqnarray}<\/p>\n

4\u6b21\u5143\u6dfb\u5b57\u8868\u8a18\u306e\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db<\/h4>\n

\u3061\u306a\u307f\u306b\uff0c\u5ea7\u6a19\u3092 $x^{\\mu} = \\left(x^0, x^1, x^2, x3\\right)$ \u306e\u3088\u3046\u306b4\u6b21\u5143\u306e\u6dfb\u5b57\u3067\u8868\u3057\u305f\u3068\u304d\u306e\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306f \\(c = 1\\) \u3068\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nx^{0^{\\prime}} &=& \\frac{x^0 -V x^1}{\\sqrt{1-V^2}} \\\\
x^{1^{\\prime}} &=& \\frac{x^1 -V x^0}{\\sqrt{1-V^2}} \\\\
\nx^{2^{\\prime}} &=& x^{2} \\\\
\nx^{3^{\\prime}} &=& x^{3}
\n\\end{eqnarray}<\/p>\n

\u5149\u306e4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u3082\u540c\u69d8\u306b\uff0c\u6210\u5206\u3092 $k^{\\mu} = \\left(k^0, k^1, k^2, k^3\\right)$ \u306e\u3088\u3046\u306b4\u6b21\u5143\u306e\u6dfb\u5b57\u3067\u8868\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nk^{0^{\\prime}} &=& \\frac{k^0 -V k^1}{\\sqrt{1-V^2}} \\\\
k^{1^{\\prime}} &=& \\frac{k^1 -V k^0}{\\sqrt{1-V^2}} \\\\
\nk^{2^{\\prime}} &=& k^{2} \\\\
\nk^{3^{\\prime}} &=& k^{3}
\n\\end{eqnarray}<\/p>\n

\u9006\u5909\u63db\u306f<\/p>\n

\\begin{eqnarray}
\nk^{0} &=& \\frac{k^{0^{\\prime}} +V k^{1^{\\prime}}}{\\sqrt{1-V^2}} \\\\
k^{1} &=& \\frac{k^{1^{\\prime}} +V k^{0^{\\prime}}}{\\sqrt{1-V^2}}
\n\\end{eqnarray}<\/p>\n

\u5149\u306e\u30cc\u30eb\u6761\u4ef6<\/h3>\n

\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c\u5149\u306f \\(xy\\) \u5e73\u9762\u4e0a\u3092\u4f1d\u64ad\u3059\u308b\u3068\u4eee\u5b9a\u3057\u3066\uff0c\\(k_z = k_z’ = 0\\) \u3068\u3059\u308b\u3002<\/p>\n

\u30cc\u30eb\u6761\u4ef6 \\(\\displaystyle \\eta_{\\mu\\nu} k^{\\mu} k^{\\nu} = \\eta_{\\mu\\nu} k^{\\mu’} k^{\\nu’} = 0\\) \u3088\u308a<\/p>\n

\\begin{eqnarray}
-\\omega^2 + k_x^2 + k_y^2 &=& 0 \\\\
-\\left(\\omega’ \\right)^2 + \\left(k_x’ \\right)^2 +\\left(k_y’ \\right)^2 &=&0
\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u5149\u306e4\u5143\u30d9\u30af\u30c8\u30eb\u306e\u7a7a\u9593\u6210\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\\begin{eqnarray}
k_x = \\omega \\cos\\theta, &\\quad& k_y = \\omega \\sin \\theta\\\\
k_x’ = \\omega’ \\cos\\theta’, &\\quad& k_y’ = \\omega’ \\sin \\theta’
\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067 \\(\\theta\\) \u306f\u89b3\u6e2c\u8005 \\(A\\) \u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u9032\u884c\u65b9\u5411\u3068 \\(x\\) \u65b9\u5411\u3068\u306e\u306a\u3059\u89d2\uff0c\\(\\theta’\\) \u306f\u89b3\u6e2c\u8005 \\(B\\) \u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u9032\u884c\u65b9\u5411\u3068 \\(x’\\) \u65b9\u5411\u3068\u306e\u306a\u3059\u89d2\u3067\u3042\u308b\u3002<\/p>\n

\\(A\\) \u3068\u3068\u3082\u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u5149\u6e90\u304b\u3089\u306e\u5149\u3092\u904b\u52d5\u3059\u308b \\(B\\) \u304c\u89b3\u6e2c\u3059\u308b\u5834\u5408<\/h3>\n

\\((3)\\) \u5f0f\u304b\u3089<\/p>\n

\\begin{eqnarray}
\\omega &=& \\frac{\\omega’ + V k_x’}{\\sqrt{1-V^2}} \\\\
&=& \\frac{\\omega’ + V \\omega’ \\cos\\theta’}{\\sqrt{1-V^2}} \\\\
&=& \\omega’ \\frac{1 + V \\cos \\theta’}{\\sqrt{1-V^2}} \\\\
\\therefore\\ \\ \\omega’ &=& \\omega\\frac{\\sqrt{1-V^2}}{1 + V \\cos \\theta’} \\tag{5}
\\end{eqnarray}<\/p>\n

\u5149\u6e90\u9759\u6b62\u7cfb\u3067\u306e\u632f\u52d5\u6570 \\(\\omega\\) \u3092\u3042\u3089\u305f\u3081\u3066 \\(\\omega_0\\) \uff0c\u89b3\u6e2c\u8005 \\(B\\) \u306e\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570 \\(\\omega’\\) \u3092\u3042\u3089\u305f\u3081\u3066 \\(\\omega_{\\rm obs}\\) \u3068\u3059\u308b\u3068<\/p>\n

$$\\omega_{\\rm obs}= \\omega_0\\frac{\\sqrt{1-V^2}}{1 + V \\cos \\theta’} $$<\/p>\n

\u7279\u306b\uff0c\u89b3\u6e2c\u8005\u304c\u5149\u6e90\u304b\u3089\u9060\u3056\u304b\u308b\u5834\u5408\u306f \\(\\theta’ = 0\\) \u3068\u3059\u308b\u3068<\/p>\n

\"\"<\/p>\n

$$\\omega_{\\rm obs}= \\omega_0\\frac{\\sqrt{1-V^2}}{1 + V} = \\omega_0 \\sqrt{\\frac{1-V}{1+V} } < \\omega_0$$<\/p>\n

\u3068\u306a\u308a\uff0c\u632f\u52d5\u6570\u306f\u5c0f\u3055\u304f\uff0c\u3057\u305f\u304c\u3063\u3066\u6ce2\u9577\u304c\u4f38\u3073\u3066\u89b3\u6e2c\u3055\u308c\u308b\u3002<\/p>\n

\u307e\u305f\uff0c\\((4)\\) \u5f0f\u304b\u3089\uff08\\((5)\\) \u5f0f\u3082\u4f7f\u3063\u3066… \uff09<\/p>\n

\\begin{eqnarray}
\nk_x = \\omega \\cos\\theta &=& \\frac{k’_x + V \\omega’}{\\sqrt{1-V^2}} \\\\
\n&=& \\frac{\\omega’ \\cos\\theta’ + V \\omega’}{\\sqrt{1-V^2}} \\\\
\n&=& \\omega’ \\frac{\\cos\\theta’ + V}{\\sqrt{1-V^2}} \\\\
\n\\therefore\\ \\ \\cos\\theta &=& \\frac{\\omega’}{\\omega} \\frac{\\cos\\theta’ + V}{\\sqrt{1-V^2}} \\\\
\n&=& \\frac{\\sqrt{1-V^2}}{1 + V \\cos \\theta’}\\frac{\\cos\\theta’ + V}{\\sqrt{1-V^2}} \\\\
\n&=& \\frac{\\cos\\theta’ + V}{1 + V \\cos \\theta’} \\tag{6}
\n\\end{eqnarray}<\/p>\n

\\(B\\) \u3068\u3068\u3082\u306b\u904b\u52d5\u3057\u3066\u3044\u308b\u5149\u6e90\u304b\u3089\u306e\u5149\u3092\u9759\u6b62\u3057\u3066\u3044\u308b \\(A\\) \u304c\u89b3\u6e2c\u3059\u308b\u5834\u5408<\/h3>\n

\\((1)\\) \u5f0f\u304b\u3089<\/p>\n

\\begin{eqnarray}
\n\\omega’ &=& \\frac{\\omega -V k_x}{\\sqrt{1-V^2}} \\\\
\n&=& \\frac{\\omega -V \\omega \\cos\\theta}{\\sqrt{1-V^2}} \\\\
\n&=& \\omega \\frac{1 -V \\cos \\theta}{\\sqrt{1-V^2}} \\\\
\n\\therefore\\ \\ \\omega &=& \\omega’\\frac{\\sqrt{1-V^2}}{1 -V \\cos \\theta} \\tag{7}
\n\\end{eqnarray}<\/p>\n

\u5149\u6e90\u9759\u6b62\u7cfb\u3067\u306e\u632f\u52d5\u6570 \\(\\omega’\\) \u3092\u3042\u3089\u305f\u3081\u3066 \\(\\omega_0\\) \uff0c\u89b3\u6e2c\u8005 \\(A\\) \u306e\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570 \\(\\omega\\) \u3092\u3042\u3089\u305f\u3081\u3066 \\(\\omega_{\\rm obs}\\) \u3068\u3059\u308b\u3068<\/p>\n

$$\\omega_{\\rm obs}= \\omega_0\\frac{\\sqrt{1-V^2}}{1 -V \\cos \\theta} $$<\/p>\n

\u7279\u306b\uff0c\u5149\u6e90\u304c\u89b3\u6e2c\u8005\u304b\u3089\u9060\u3056\u304b\u308b\u5834\u5408\u306f \\(\\theta = \\pi\\) \u3067\u3042\u308a\uff0c<\/p>\n

\"\"<\/p>\n

$$\\omega_{\\rm obs}= \\omega_0\\frac{\\sqrt{1-V^2}}{1 + V} = \\omega_0 \\sqrt{\\frac{1-V}{1+V} } < \\omega_0$$<\/p>\n

\u3068\u306a\u308a\uff0c\u632f\u52d5\u6570\u306f\u5c0f\u3055\u304f\uff0c\u3057\u305f\u304c\u3063\u3066\u6ce2\u9577\u304c\u4f38\u3073\u3066\u89b3\u6e2c\u3055\u308c\u308b\u3002<\/p>\n

\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089\uff0c\u5149\u6e90\u304c\u52d5\u304f\u304b\u89b3\u6e2c\u8005\u304c\u52d5\u304f\u304b\u306b\u304b\u304b\u308f\u3089\u305a\uff0c\u4e92\u3044\u306e\u9593\u306e\u76f8\u5bfe\u901f\u5ea6\u306b\u3088\u3063\u3066\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u304c\u8d77\u3053\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u307e\u305f\uff0c\\((2)\\) \u5f0f\u304b\u3089\uff08\\((7)\\) \u5f0f\u3082\u4f7f\u3063\u3066… \uff09<\/p>\n

\\begin{eqnarray}
\nk’_x = \\omega’ \\cos\\theta’ &=& \\frac{k_x -V \\omega}{\\sqrt{1-V^2}} \\\\
\n&=& \\frac{\\omega \\cos\\theta -V \\omega’}{\\sqrt{1-V^2}} \\\\
\n&=& \\omega \\frac{\\cos\\theta -V}{\\sqrt{1-V^2}} \\\\
\n\\therefore\\ \\ \\cos\\theta’ &=& \\frac{\\omega}{\\omega’} \\frac{\\cos\\theta -V}{\\sqrt{1-V^2}} \\\\
\n&=& \\frac{\\sqrt{1-V^2}}{1 -V \\cos \\theta}\\frac{\\cos\\theta -V}{\\sqrt{1-V^2}} \\\\
\n&=& \\frac{\\cos\\theta -V}{1 -V \\cos \\theta} \\tag{8}
\n\\end{eqnarray}<\/p>\n

\\((6)\\) \u5f0f\u304a\u3088\u3073 \\((8)\\) \u5f0f\u304c\uff0c\u9759\u6b62\u89b3\u6e2c\u8005\u306e\u89b3\u6e2c\u3059\u308b\u5149\u306e\u5165\u5c04\u89d2 $\\theta$ \u3068\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306e\u89b3\u6e2c\u3059\u308b\u5149\u306e\u5165\u5c04\u89d2 $\\theta’$ \u306e\u95a2\u4fc2\u3092\u8868\u3059\u3002\u89b3\u6e2c\u8005\u306e\u904b\u52d5\u306b\u3088\u3063\u3066 $\\theta’ \\neq \\theta$ \u3068\u306a\u308b\u73fe\u8c61\u304c\u5149\u884c\u5dee<\/strong><\/span>\u3067\u3042\u308b\u3002<\/p>\n

 <\/p>\n

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\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u3067\u306f\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306f\u3069\u306e\u3088\u3046\u306b\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u304b\u3002\u4f8b\u3048\u3070\uff0c\u300c\u76f8\u5bfe\u6027\u7406\u8ad6\u300d\uff08\u4e2d\u91ce\u8463\u592b\u8457\uff0c\u5ca9\u6ce2\u66f8\u5e97\uff09\u306e p. 87 \u3042\u305f\u308a\u7b49\u3092\u53c2\u8003\u306b\u3057\u3066\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306b\u3082\u3068\u3065\u3044\u305f\u5149\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u8aac\u660e\u306b\u3064\u3044\u3066\uff0c\u304a\u3055\u3089\u3044\u3059\u308b\u3002<\/p>

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