{"id":8731,"date":"2024-05-20T12:40:56","date_gmt":"2024-05-20T03:40:56","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=8731"},"modified":"2024-08-06T17:02:32","modified_gmt":"2024-08-06T08:02:32","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%82%92%e4%bd%bf%e3%81%a3%e3%81%a63%e6%ac%a1%e5%85%83%e9%80%9f%e5%ba%a6%e3%81%ae%e5%90%88%e6%88%90%e5%89%87","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%81%ab%e3%82%88%e3%82%89%e3%81%aa%e3%81%84%e7%9b%b8%e5%af%be%e8%ab%96%e3%81%ae%e7%90%86%e8%a7%a3\/%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%81%ab%e3%82%88%e3%82%89%e3%81%aa%e3%81%84%e7%89%b9%e6%ae%8a%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%8a%b9%e6%9e%9c%e3%81%ae%e7%90%86-2\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%82%92%e4%bd%bf%e3%81%a3%e3%81%a63%e6%ac%a1%e5%85%83%e9%80%9f%e5%ba%a6%e3%81%ae%e5%90%88%e6%88%90%e5%89%87\/","title":{"rendered":"\u88dc\u8db3\uff1a\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u4f7f\u3063\u30663\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u3092\u5c0e\u304f"},"content":{"rendered":"

\u7df4\u7fd2\u554f\u984c 1<\/h3>\n

\u7279\u6b8a\u76f8\u5bfe\u6027\u7406\u8ad6\u306b\u304a\u3051\u308b\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3068\u306f\uff0c\u3069\u306e\u3088\u3046\u306a\u5ea7\u6a19\u5909\u63db\u304b\u3002\u6163\u6027\u7cfb \\(S\\) \u306b\u5bfe\u3057\u3066\u901f\u3055 \\(V\\) \u3067 \\(x\\) \u65b9\u5411\u306b\u904b\u52d5\u3059\u308b\u5225\u306e\u6163\u6027\u7cfb \\(S^{\\prime}\\) \u306b\u3064\u3044\u3066\u306e\u5ea7\u6a19\u5909\u63db\u306e\u5f0f\u3067\u793a\u305b\u3002<\/p>\n

\u7df4\u7fd2\u554f\u984c 2<\/h3>\n

\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u4f7f\u3063\u30663\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u3092\u5c0e\u3051\u3002<\/p>\n

\n

\u7df4\u7fd2\u554f\u984c 1 \u306e\u89e3\u7b54\u4f8b<\/h3>\n

$S$ \u7cfb\u306e\u5ea7\u6a19\u3092 $(t, x, y, z)$\uff0c$S$ \u7cfb\u306b\u5bfe\u3057\u3066\u901f\u3055 $V$ \u3067 $x$ \u65b9\u5411\u306b\u904b\u52d5\u3059\u308b $S’$ \u7cfb\u306e\u5ea7\u6a19\u3092 $(t’, x’, y’, z’)$ \u3068\u3059\u308b\u3068\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306f<\/p>\n

\\begin{eqnarray}
\nt’ &=& \\frac{t -\\frac{V}{c^2} x}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2}} \\tag{1}\\\\
\nx’ &=& \\frac{x -V\\, t}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2}} \\tag{2}\\\\
\ny’ &=& y \\\\
\nz’ &=& z
\n\\end{eqnarray}<\/p>\n

\u53c2\u8003\u307e\u3067\u306b\u9006\u5909\u63db\u3092\u66f8\u3044\u3066\u304a\u304f\u3068<\/p>\n

\\begin{eqnarray}
\nt &=& \\frac{t’ + \\frac{V}{c^2} x’}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2}} \\\\
\nx &=& \\frac{x’ + V\\, t’}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2}}\\\\
\ny &=& y’ \\\\
\nz &=& z’
\n\\end{eqnarray}<\/p>\n

\u7df4\u7fd2\u554f\u984c\u306e\u89e3\u7b54\u4f8b 2<\/h3>\n

\u89e3\u7b54\u4f8b I<\/h4>\n

$\\displaystyle v_x \\equiv \\frac{dx}{dt}$ \u3068\u3059\u308b\u3002\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306e\u9006\u5909\u63db\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\ndt &=& \\frac{dt’ + \\frac{V}{c^2} dx’}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2}} \\\\
\ndx &=& \\frac{dx’ + V\\, dt’}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2}}\\\\
\n\\therefore\\ \\ v_x \\equiv \\frac{dx}{dt} &=& \\frac{dx’ + V\\, dt’}{dt’ + \\frac{V}{c^2} dx’} \\\\
\n&=& \\frac{\\frac{dx’}{dt’} + V}{1 + \\frac{V}{c^2} \\frac{dx’}{dt’}} \\\\
\n&=& \\frac{v’_x + V}{1 + \\frac{V v’_x}{c^2}}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c$\\displaystyle v’_x \\equiv \\frac{dx’}{dt’} $\u3002<\/p>\n

\u3064\u307e\u308a\uff0c$S$ \u7cfb\u306b\u5bfe\u3057\u3066\u901f\u3055 $V$ \u3067\u904b\u52d5\u3059\u308b $S’$ \u7cfb\u3067\u898b\u305f\u3068\u304d\u306e\u901f\u3055 $v’_x$ \u3092 $S$ \u304b\u3089\u307f\u308b\u3068\u901f\u3055 $v_x$ \u3067\u898b\u3048\u308b\u3002\u30ac\u30a4\u30ec\u30a4\u5909\u63db\u306b\u3088\u308b\u306a\u3089\u3070 $v_x = v’_x + V$ \u3068\u306a\u308b\u3079\u304d\u3068\u3053\u308d\u3060\u304c\uff0c\u7279\u6b8a\u76f8\u5bfe\u8ad6\u3067\u306f\u4e0a\u8a18\u306e\u3088\u3046\u306b\u306a\u308b\uff0c\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n

\u89e3\u7b54\u4f8b II<\/h4>\n

$S’$ \u7cfb\u306b\u5bfe\u3057\u3066\u901f\u3055 $W$ \u3067 $x’$ \u65b9\u5411\u306b\u904b\u52d5\u3059\u308b$S^{\\prime\\prime} $ \u7cfb\u306e\u5ea7\u6a19\u3092 $(t^{\\prime\\prime} , x^{\\prime\\prime} , y^{\\prime\\prime} , z^{\\prime\\prime} )$ \u3068\u3059\u308b\u3068\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306b\u3088\u308a<\/p>\n

\\begin{eqnarray}
\nt^{\\prime\\prime} &=& \\frac{t’ -\\frac{W}{c^2} x’}{\\sqrt{1 – \\left(\\frac{W}{c}\\right)^2}} \\tag{3}\\\\
\nx^{\\prime\\prime} &=& \\frac{x’ -W\\, t’}{\\sqrt{1 – \\left(\\frac{W}{c}\\right)^2}} \\tag{4}\\\\
\ny^{\\prime\\prime} &=& y’ \\\\
\nz^{\\prime\\prime} &=& z’
\n\\end{eqnarray}<\/p>\n

$(1), (2), (3), (4)$ \u5f0f\u3088\u308a\uff0c$t’, x’$ \u3092\u6d88\u53bb\u3057\u3066 $t^{\\prime\\prime}$ \u3092\u76f4\u63a5 $t, x$ \u3067\u3042\u3089\u308f\u3059\u3068<\/p>\n

\\begin{eqnarray}
\nt^{\\prime\\prime} &=&
\n\\frac{1}{\\sqrt{1 -\\left(\\frac{W}{c}\\right)^2}} \\left\\{\\frac{t -\\frac{V}{c^2} x}{\\sqrt{1 -\\left(\\frac{V}{c}\\right)^2}}
\n-\\frac{W}{c^2}\\frac{x -V\\, t}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2}}\\right\\}\\\\
\n&=& \\frac{1 + \\frac{V W}{c^2}}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2} \\sqrt{1 -\\left(\\frac{W}{c}\\right)^2}}
\n\\left\\{t -\\frac{1}{c^2} \\frac{V + W}{1 + \\frac{V W}{c^2}} x\\right\\}\\\\
\n&=& \\frac{1}{\\sqrt{1 -\\left(\\frac{U}{c}\\right)^2}}\\left( t -\\frac{U}{c^2} x\\right)
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c$\\displaystyle U \\equiv \\frac{V + W}{1 + \\frac{V W}{c^2}}$\u3002<\/p>\n

$x^{\\prime\\prime}$ \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b<\/p>\n

\\begin{eqnarray}
\nx^{\\prime\\prime} &=&
\n\\frac{1}{\\sqrt{1 -\\left(\\frac{W}{c}\\right)^2}} \\left\\{\\frac{x -{V} t}{\\sqrt{1 -\\left(\\frac{V}{c}\\right)^2}}
\n– W\\frac{t\u00a0 -\\frac{V}{c^2} x}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2}}\\right\\}\\\\
\n&=& \\frac{1 + \\frac{V W}{c^2}}{\\sqrt{1 – \\left(\\frac{V}{c}\\right)^2} \\sqrt{1 -\\left(\\frac{W}{c}\\right)^2}}
\n\\left\\{x -\\frac{V + W}{1 + \\frac{V W}{c^2}} t\\right\\}\\\\
\n&=& \\frac{1}{\\sqrt{1 -\\left(\\frac{U}{c}\\right)^2}}\\left( x -{U} t\\right)
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c$S^{\\prime\\prime}$ \u7cfb\u306f\uff0c$S$ \u7cfb\u306b\u5bfe\u3057\u3066\u901f\u3055 $U$ \u3067\u904b\u52d5\u3059\u308b\u7cfb\u3067\u3042\u308a\uff0c$S$ \u7cfb\u306b\u5bfe\u3059\u308b $S’$ \u7cfb\u306e\u901f\u3055 $V$ \u3068\uff0c$S’$ \u7cfb\u306b\u5bfe\u3059\u308b $S^{\\prime\\prime}$ \u7cfb\u306e\u901f\u3055 $W$ \u3068\uff0c$U$ \u3068\u306e\u95a2\u4fc2\u306f3\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247<\/p>\n

$$U = \\frac{V + W}{1 + \\frac{V W}{c^2}}$$<\/p>\n

\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u7df4\u7fd2\u554f\u984c 1 <\/p>\n

\u7279\u6b8a\u76f8\u5bfe\u6027\u7406\u8ad6\u306b\u304a\u3051\u308b\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3068\u306f\uff0c\u3069\u306e\u3088\u3046\u306a\u5ea7\u6a19\u5909\u63db\u304b\u3002\u6163\u6027\u7cfb \\(S\\) \u306b\u5bfe\u3057\u3066\u901f\u3055 \\(V\\) \u3067 \\(x\\) \u65b9\u5411\u306b\u904b\u52d5\u3059\u308b\u5225\u306e\u6163\u6027\u7cfb \\(S^{\\prime}\\) \u306b\u3064\u3044\u3066\u306e\u5ea7\u6a19\u5909\u63db\u306e\u5f0f\u3067\u793a\u305b\u3002<\/p>\n

\u7df4\u7fd2\u554f\u984c 2 <\/p>\n

\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u4f7f\u3063\u30663\u6b21\u5143\u901f\u5ea6\u306e\u5408\u6210\u5247\u3092\u5c0e\u3051\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":185,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-8731","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t