{"id":83619,"date":"2023-11-27T23:24:30","date_gmt":"2023-11-27T23:24:30","guid":{"rendered":"https:\/\/science-hub.click\/%E7%9B%B8%E5%AF%BE%E8%AB%96%E7%9A%84%E8%A8%88%E7%AE%97%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC\/"},"modified":"2023-11-27T23:24:30","modified_gmt":"2023-11-27T23:24:30","slug":"%E7%9B%B8%E5%AF%BE%E8%AB%96%E7%9A%84%E8%A8%88%E7%AE%97%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=83619","title":{"rendered":"\u76f8\u5bfe\u8ad6\u7684\u8a08\u7b97\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac"},"content":{"rendered":"

\u3053\u306e\u8a18\u4e8b\u306f\u3001\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u8a08\u7b97\u3059\u308b<\/i>\u3053\u3068\u306b\u3088\u3063\u3066\u672c\u8cea\u7684\u306a\u3053\u3068\u304c\u63a8\u5b9a<\/strong>\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3059\u305f\u3081\u306b\u3001\u610f\u56f3\u7684\u306b\u8a08\u7b97\u7684\u3067\u3042\u308b\u3053\u3068\u3092\u76ee\u7684\u3068\u3057\u3066\u3044\u307e\u3059\u3002 <\/p>
\n\"\u30a2\u30eb\u30d0\u30fc\u30c8\u30fb\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\"<\/figure><\/td><\/tr>
\u3053\u306e\u7269\u7406\u5b66\u306e\u8a18\u4e8b\u3067\u306f\u3001\u76f8\u5bfe\u6027\u7406\u8ad6<\/strong>\u30b7\u30ea\u30fc\u30ba\u306e\u4e00\u90e8<\/small><\/td><\/tr>
\u57fa\u672c<\/i><\/td><\/tr>
\u6b74\u53f2 –\u7406\u8ad6<\/a><\/span><\/small><\/td><\/tr>
\u30ed\u30fc\u30ec\u30f3\u30c4 – \u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3 – \u30de\u30c3\u30cf<\/small><\/td><\/tr>
\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db<\/a><\/span><\/strong><\/small><\/td><\/tr>
\u30d5\u30a1\u30a4\u30f3\u30de\u30f3 – \u30dd\u30a2\u30f3\u30ab\u30ec – \u30de\u30a4\u30b1\u30eb\u30bd\u30f3<\/small><\/td><\/tr>
\u6642\u7a7a-c – E=mc\u00b2 – t<\/small><\/td><\/tr>
EQR<\/small><\/td><\/tr>
\u4f8b:\u30de\u30a4\u30b1\u30eb\u30bd\u30f3\u3068\u30e2\u30fc\u30ea\u30fc<\/small><\/td><\/tr>
exp:\u8003\u3048\u305f?-\u30a8\u30fc\u30c6\u30eb<\/small><\/td><\/tr>
\u53cc\u5b50\u3092\u8a13\u7df4\u3059\u308b<\/small><\/td><\/tr>
\u7279\u6b8a\u4e00\u822c\u76f8\u5bfe\u6027\u7406\u8ad6<\/small><\/td><\/tr>
\u76f8\u5bfe\u6027\u7406\u8ad6<\/a><\/span><\/small><\/td><\/tr>
\u6b74\u53f2\u8ad6\u4e89<\/small><\/td><\/tr>
\u30c6\u30af\u30cb\u30c3\u30af<\/i><\/td><\/tr>
\u30b5\u30a4\u30af\u30ed\u30c8\u30ed\u30f3<\/a><\/span><\/small><\/td><\/tr>
\u7c92\u5b50\u52a0\u901f\u5668<\/a><\/span><\/small><\/td><\/tr>
\u30e1\u30bf<\/i><\/td><\/tr>
\u8a18\u4e8b<\/small><\/td><\/tr>
\u7269\u7406<\/a><\/span>\u30ea\u30f3\u30af<\/small><\/td><\/tr>
\u5f62\u72b6<\/small><\/td><\/tr><\/table>

\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db<\/span><\/h2>

2 \u3064\u306e\u57fa\u6e96\u67a0\u304c\u3042\u308b\u3068\u3057\u307e\u3059

$$ {\\mathbb{R}} $$<\/div>\u305d\u3057\u3066
$$ {\\mathbb R’} $$<\/div>\u8ef8\uff2f\uff58\u306b\u6cbf\u3063\u305f\u76f8\u5bfe\u901f\u5ea6\uff56\u3067\u3001\u5e73\u884c\u8ef8\u4e0a\u3067\u76f8\u4e92\u306b\u76f4\u7dda\u79fb\u52d5\u3059\u308b\u3002\u6700\u521d\u306e\u53c2\u7167\u30d5\u30ec\u30fc\u30e0\u304b\u3089 2 \u756a\u76ee\u306e\u53c2\u7167\u30d5\u30ec\u30fc\u30e0\u306b\u79fb\u52d5\u3059\u308b\u3068\u3001\u5ea7\u6a19\u306f\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306b\u3088\u3063\u3066\u30ea\u30f3\u30af\u3055\u308c\u307e\u3059\u3002 <\/p>
$$ {x’ = \\gamma (x – vt) \\qquad y’ = y \\qquad z’ = z \\qquad t’ = \\gamma (t – \\frac{vx}{c^2})} $$<\/div>\u3068
$$ {\\gamma = \\frac{1}{\\sqrt{1- \\frac{v^2}{c^2}}} = \\frac{1}{\\sqrt{1- \\beta^2}}} $$<\/div>\u305d\u3057\u3066
$$ {\\beta = \\frac{v}{c}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u9006\u5909\u63db

$$ {\\mathbb R’ \\rightarrow \\mathbb R} $$<\/div> R’ \u306e\u5ea7\u6a19\u306e\u95a2\u6570\u3068\u3057\u3066 R \u306e\u5ea7\u6a19\u3092\u4e0e\u3048\u308b\u3082\u306e\u306f\u30012 \u3064\u306e\u672a\u77e5\u6570\u3092\u6301\u3064 2 \u3064\u306e\u65b9\u7a0b\u5f0f\u7cfb\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u3063\u3066\u63a8\u5b9a\u3055\u308c\u307e\u3059\u3002
$$ {x’ = \\gamma (x – vt) \\qquad t’ = \\gamma (t – \\frac{vx}{c^2})} $$<\/div>\u901f\u5ea6\u306e\u7b26\u53f7\u3092\u5909\u3048\u308b\u3060\u3051\u3067\u3059\u3002 <\/p>
$$ {x’ = \\gamma (x – vt) \\qquad y’ = y \\qquad z’ = z \\qquad t’ = \\gamma (t – \\frac{vx}{c^2})} $$<\/div><\/dd>
$$ {\\qquad x = \\gamma (x’ + vt’) \\qquad y = y’ \\qquad z = z’ \\qquad t = \\gamma (t’ + \\frac{vx’}{c^2})} $$<\/div><\/dd><\/dl><\/dd><\/dl>\u3057\u305f\u304c\u3063\u3066\u3001\u884c\u5217\u3092\u66f8\u304f\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
$$ {\\begin{pmatrix} ct’\\\\x’\\\\y’\\\\z’\\\\\\end{pmatrix} = \\begin{bmatrix} \\gamma & -\\gamma\\beta& 0 & 0\\\\ -\\gamma\\beta & \\gamma & 0& 0\\\\ 0 & 0 & 1& 0\\\\ 0 & 0 & 0& 1\\end{bmatrix}\\begin{pmatrix} ct\\\\x\\\\y\\\\z\\\\\\end{pmatrix}} $$<\/div><\/dd><\/dl><\/dd><\/dl>\u305d\u3057\u3066\u305d\u306e\u9006\u3082\u540c\u69d8\u3067\u3059:
$$ {\\begin{pmatrix} ct\\\\x\\\\y\\\\z\\\\\\end{pmatrix} = \\begin{bmatrix} \\gamma & +\\gamma\\beta& 0 & 0\\\\ +\\gamma\\beta & \\gamma & 0& 0\\\\ 0 & 0 & 1& 0\\\\ 0 & 0 & 0& 1\\end{bmatrix}\\begin{pmatrix} ct’\\\\x’\\\\y’\\\\z’\\\\\\end{pmatrix}} $$<\/div>\u3053\u308c\u306f\u3001\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u8a18\u8ff0\u3059\u308b\u884c\u5217\u306e\u65b9\u6cd5\u3067\u3059\u3002<\/blockquote>

x’ \u3068 t’ \u3092\u3001x \u3068 t \u306b\u95a2\u3059\u308b\u4e0a\u8a18\u306e\u5f0f\u306b\u7f6e\u304d\u63db\u3048\u308b\u3068\u3001x = x \u304a\u3088\u3073 t = t \u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u4e0a\u8a18\u306e\u5f0f\u304c\u4e92\u3044\u306b\u9006\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u6570\u5b66\u8005\u306f\u3053\u308c\u3092\u3001\u3053\u306e\u6027\u8cea\u306f\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u304c\u7fa4\u3092\u5f62\u6210\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u6027\u8cea\u306e 1 \u3064\u3067\u3042\u308a\u3001\u305d\u306e\u4e3b\u306a\u7d50\u679c\u306f 2 \u3064\u306e\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306e\u5408\u6210\u304c\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3067\u3042\u308b\u3068\u8868\u73fe\u3057\u3066\u8868\u73fe\u3057\u307e\u3059\u3002\u901f\u5ea6\u306e\u69cb\u6210\u306b\u95a2\u9023\u3059\u308b\u6bb5\u843d\u3067\u3001\u30ed\u30fc\u30ec\u30f3\u30c4\u7fa4\u306e\u4f7f\u7528\u3092\u898b\u3066\u3044\u304d\u307e\u3059\u3002<\/p>

\u307b\u3068\u3093\u3069\u3059\u3079\u3066\u304c<\/a><\/span>\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u304b\u3089\u63a8\u5b9a\u3055\u308c\u308b\u305f\u3081\u3001\u7279\u6b8a\u76f8\u5bfe\u6027\u7406\u8ad6<\/a><\/span>\u3067\u306f\u3001\u300c\u611a\u304b\u306a\u300d\u63a8\u8ad6\u3092\u5b9f\u884c\u3059\u308b\u3088\u308a\u3082\u8a08\u7b97\u7d50\u679c\u3092\u4fe1\u983c\u3059\u308b\u65b9\u304c\u826f\u3044\u3068\u8a00\u3048\u307e\u3059\u3002\u4e0a\u8a18\u306e\u5909\u63db\u306f\u3001\u901a\u5e38\u306e\u901f\u5ea6\u306b\u5bfe\u3059\u308bGalileo<\/a><\/span>\u306e\u5909\u63db\u3068\u540c\u3058\u3067\u3059\u304c\u3001\u57fa\u6e96\u7cfb\u306b\u4f9d\u5b58\u3059\u308b\u305f\u3081\u3001\u4e16\u754c\u6642<\/a><\/span>\u306e\u6982\u5ff5\u304c\u7834\u58ca\u3055\u308c\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u3064\u307e\u308a\u3001\u4f4d\u7f6e\u304c\u5909\u5316\u3059\u308b\u3068\u6642\u9593<\/a><\/span>\u3082\u5909\u5316\u3057\u307e\u3059\u3002\u6642\u9593\u306f\u7269\u7406\u6cd5\u5247\u3092\u8868\u3059\u5ea7\u6a19\u3067\u3042\u308a\u3001\u3053\u308c\u3089\u306f 4 \u6b21\u5143\u7a7a\u9593 ( ct<\/i> , x<\/i> , y<\/i> , z<\/i> ) \u3067\u8868\u3059\u3068\u5171\u5909\u306b\u306a\u308a\u307e\u3059\u3002\u6ce8: \u3053\u3053\u3067 t \u3092 ct \u306b\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u306f\u3001\u6642\u9593\u5358\u4f4d\u3092\u79d2\u3067\u306f\u306a\u304f\u30e1\u30fc\u30c8\u30eb\u306b\u306a\u308b\u3088\u3046\u306b\u5909\u63db\u3057\u3066\u3044\u308b\u3060\u3051\u3067\u3059\u3002<\/p>

\u4ee5\u4e0b\u306b\u3001\u7d50\u679c\u3092\u63a8\u5b9a\u3067\u304d\u308b\u8a08\u7b97\u306e\u30da\u30fc\u30b8\u3092\u4f5c\u6210\u3057\u307e\u3059\u3002<\/p>

\u64ec\u4f3c\u6a19\u6e96<\/span><\/h3>

\u79c1\u305f\u3061\u306f\u3001\u6642\u7a7a\u9593\u306e 4 \u6b21\u5143\u5ea7\u6a19\u7cfb\u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb<\/a><\/span>\u306e\u5ea7\u6a19\u306b\u3088\u3063\u3066\u30a4\u30d9\u30f3\u30c8\u3092\u8b58\u5225\u3057\u307e\u3059\u3002

$$ {\\mathbf{r}=( ct, \\vec r) = (ct,x,y,z)} $$<\/div><\/p>

\u6b21\u306e\u3053\u3068\u304c\u7c21\u5358\u306b\u308f\u304b\u308a\u307e\u3059\u3002 <\/p>

$$ {\\mathbf{r^2}=c^2t^2 – \\vec r^2=c^2t^2 – (x^2+y^2+z^2)=c^2t’^2 – (x’^2+y’^2+z’^2)=c^2t’^2 – \\vec{r’}^2=\\mathbf{r’^2}} $$<\/div><\/dd><\/dl>

\u78ba\u304b\u306b \uff1a <\/p>

$$ {\\left(\\gamma(ct’+\\beta x’)\\right)^2 – (\\left(\\gamma(x’+\\beta ct’)\\right)^2+y’^2+z’^2)=(\\left(\\gamma^2(1-\\beta^2)\\right) c^2t’^2 – ((\\left(\\gamma^2(1-\\beta^2)\\right)x’^2+y’^2+z’^2)=c^2t’^2 – (x’^2+y’^2+z’^2)} $$<\/div><\/dd><\/dl>

4\u6b21\u5143<\/a><\/span>\u30d9\u30af\u30c8\u30eb\u307e\u305f\u306f\u30af\u30a2\u30c9\u30ea\u30d9\u30af\u30bf<\/span>\u306e\u64ec\u4f3c\u30ce\u30eb\u30e0\u3092\u592a\u5b57\u3067\u793a\u3057\u307e\u3059\u3002

$$ {\\mathbf{r^2}=c^2t^2 – \\vec r^2} $$<\/div><\/p>

\u3053\u308c\u306f\u30014 \u6b21\u5143\u6642\u7a7a\u3067\u7279\u5b9a\u3055\u308c\u308b\u30a4\u30d9\u30f3\u30c8\u306e 4 \u30d9\u30af\u30c8\u30eb\u4f4d\u7f6e\u306e\u64ec\u4f3c\u30ce\u30eb\u30e0<\/a><\/span>\u3068\u547c\u3070\u308c\u307e\u3059\u3002\u6211\u3005\u306f\u3001\u3053\u306e\u91cf\u304c<\/a><\/span>\u57fa\u6e96\u7cfb\u306b\u4f9d\u5b58\u305b\u305a\u3001\u3057\u305f\u304c\u3063\u3066\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306b\u5bfe\u3059\u308b\u4e0d\u5909\u91cf\u3092\u69cb\u6210\u3059\u308b\u3053\u3068\u3092\u6570\u5b66\u7684\u306b\u793a\u3057\u307e\u3057\u305f\u3002<\/p>

\u671f\u9593\u306e\u62e1\u5927<\/span><\/span><\/h3>

\u3059\u3067\u306b\u3001\u6642\u9593 t’ \u306f\u7121\u9650\u306e\u6642\u9593\u306b\u5bfe\u5fdc\u3059\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002

$$ {\\mathbb{R}} $$<\/div> :
$$ {ct = \\frac{(ct’+\\frac{vx’}{c})}{\\sqrt{1 – {v^2 \\over c^2}}}} $$<\/div><\/p>

\u5358\u7d14\u5316\u3059\u308b\u305f\u3081\u306b\u3001t’=0 \u3068\u3057\u307e\u3059\u3002

$$ {ct = \\frac{(\\frac{vx’}{c})}{\\sqrt{1 – {v^2 \\over c^2}}}} $$<\/div><\/p>

\u540c\u3058\u6642\u9593 t’=0 \u306f\u3001\u5c06\u6765\u306e\u6b63\u306e x \u306e\u6642\u9593 t \u306b<\/strong>\u5bfe\u5fdc\u3057\u307e\u3059\u3002

$$ {\\mathbb{R}} $$<\/div>\u904e\u53bb\u306e\u8ca0\u306e x \u306b\u3064\u3044\u3066\u306f\u3001
$$ {\\mathbb{R}} $$<\/div> \uff01<\/p>

2 \u3064\u306e\u53c2\u7167\u30d5\u30ec\u30fc\u30e0\u304c\u3042\u308b\u3068\u3057\u307e\u3059\u3002

$$ {\\mathbb{R}} $$<\/div> \u3001 \u305d\u3057\u3066
$$ {\\mathbb{R’}} $$<\/div>\u901f\u5ea6 v \u3067\u6b63\u306e x \u8ef8\u306b\u6cbf\u3063\u305f\u6700\u521d\u306e\u57fa\u6e96\u5ea7\u6a19\u7cfb\u306b\u5bfe\u3057\u3066\u5747\u4e00\u306a\u76f4\u7dda\u79fb\u52d5\u3057\u307e\u3059\u3002<\/p>

\u3067\u6b62\u307e\u3063\u3066\u3044\u308b\u6642\u8a08

$$ {\\mathbb{R’}} $$<\/div>\u70b9M<\/i> ‘( x’o<\/i> ,<\/i><\/sub> y’o<\/i> ,<\/i><\/sub> z’o<\/i> )<\/i><\/sub><\/span>\u3067\u306f\u3001\u6b21\u306e 2 \u3064\u306e\u30a4\u30d9\u30f3\u30c8\u304c\u6e2c\u5b9a\u3055\u308c\u307e\u3059\u3002
$$ {\\mathbb{R’}} $$<\/div> : (<\/i> ct’1<\/i> ,<\/i> x’o<\/i> ,<\/i> y’o<\/i><\/sub> ,<\/sub> z’o<\/i><\/sub> )<\/i><\/sub><\/span>\u304a\u3088\u3073(<\/i> ct’2<\/sub> ,<\/i> x’o<\/i> ,<\/i> y’o<\/i><\/sub> ,<\/i> z’o<\/i><\/sub> )<\/i><\/sub><\/span>\u306f\u3001\u540c\u3058\u5834\u6240\u3067\u7570\u306a\u308b\u6642\u9593\u306b\u767a\u751f\u3057\u307e\u3059\u3002<\/p>

\u30d5\u30ec\u30fc\u30e0\u5909\u63db\u306e\u5909\u5316\u306b\u5f93\u3063\u3066\u3001 x<\/i> 1<\/sub> = \u03b3( v<\/i> t<\/i> ‘ 1<\/sub> + x<\/i> ‘ o<\/i><\/sub> )<\/span>\u304a\u3088\u3073x<\/i> 2<\/sub> = \u03b3( v<\/i> t<\/i> ‘ 2<\/sub> + x<\/i> ‘ o<\/i><\/sub> )<\/span>\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u30ea\u30dd\u30b8\u30c8\u30ea\u5185\u306e 2 \u3064\u306e\u30a4\u30d9\u30f3\u30c8\u9593\u306e\u671f\u9593

$$ {\\mathbb{R’}} $$<\/div> M<\/i> ‘( x<\/i> ‘ o<\/i><\/sub> , y<\/i> ‘ o<\/i><\/sub> , z<\/i> ‘ o<\/i><\/sub> )<\/span>\u3067\u767a\u751f\u3059\u308b\u6642\u9593\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059: t<\/i> ‘ 1<\/sub> \u2212 t<\/i> ‘ 2<\/sub><\/span>\u57fa\u6e96\u30d5\u30ec\u30fc\u30e0\u5185\u306e\u3053\u308c\u3089 2 \u3064\u306e\u30a4\u30d9\u30f3\u30c8\u9593\u306e\u7d99\u7d9a\u6642\u9593
$$ {\\mathbb{R}} $$<\/div>\u6771 \uff1a
$$ {t_1-t_2=\\gamma(t’_1+\\frac{vx’_o}{c^2}-t’_2-\\frac{vx’_o}{c^2})=\\gamma(t’_1-t’_2)} $$<\/div><\/p>

\u03c4 0<\/sub> = t<\/i> ‘ 1<\/sub> \u2212 t<\/i> ‘ 2<\/sub><\/span>\u3092\u9759\u6b62\u6642\u306e\u6301\u7d9a\u6642\u9593\u3001 \u03c4 = t<\/i> 1<\/sub> \u2212 t<\/i> 2 \u3092<\/sub><\/span>\u57fa\u6e96\u7cfb\u3067\u89b3\u6e2c\u3055\u308c\u308b\u6301\u7d9a\u6642\u9593\u3092\u8a2d\u5b9a\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001

$$ {\\mathbb{R}} $$<\/div> \u3001\u3044\u308f\u3086\u308b\u6301\u7d9a\u6642\u9593\u62e1\u5f35\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059\u3002 <\/p>
$$ {\\tau=\\frac{\\tau_0}{\\sqrt{1-\\frac{v^2}{c^2}}}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3057\u305f\u304c\u3063\u3066\u3001\u30ea\u30d5\u30a1\u30ec\u30f3\u30b9\u304b\u3089\u898b\u308b\u3068\u3001

$$ {\\mathbb{R}} $$<\/div> x<\/i> 1<\/sub> = \u03b3( v<\/i> t<\/i> ‘ 1<\/sub> + x<\/i> ‘ o<\/i><\/sub> )<\/span>\u304a\u3088\u3073x<\/i> 2<\/sub> = \u03b3( v<\/i> t<\/i> ‘ 2<\/sub> + x<\/i> ‘ o<\/i><\/sub> )<\/span>\u306b\u4f4d\u7f6e\u3057\u3001\u57fa\u6e96\u30d5\u30ec\u30fc\u30e0\u5185\u3067\u30af\u30ed\u30c3\u30af\u3092\u540c\u671f\u3057\u3066\u3044\u308b 2 \u4eba\u306e\u89b3\u6e2c\u8005\u306b\u3088\u308b
$$ {\\mathbb{R}} $$<\/div>\u6642\u9593\u9593\u9694\u306e\u6e2c\u5b9a\u5024\u306f\u3001 M<\/i><\/sub> ‘<\/i> ( x’o<\/i> ,<\/i><\/sub> y’o<\/i> ,<\/i><\/sub> z’o<\/i> )<\/span>\u306b\u4f4d\u7f6e\u3059\u308b\u9759\u6b62\u3057\u305f\u89b3\u6e2c\u8005\u306b\u3088\u3063\u3066\u6e2c\u5b9a\u3055\u308c\u305f\u3082\u306e\u3068\u7b49\u3057\u304f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>

\u91cf\u03c4 0<\/sub><\/span>\u306f\u56fa\u6709\u6642\u9593<\/a><\/span><\/strong>\u3068\u547c\u3070\u308c\u307e\u3059\u3002\u3068\u3057\u3066

$$ {\\tau_0 = \\tau \\sqrt{1 – {v^2 \\over c^2}}} $$<\/div> \u3001\u91cf\u304c\u308f\u304b\u308a\u307e\u3059\u3002
$$ {\\tau \\sqrt{1 – {v^2 \\over c^2}}} $$<\/div>\u306f\u53c2\u7167\u30d5\u30ec\u30fc\u30e0\u306b\u4f9d\u5b58\u3057\u306a\u3044\u76f8\u5bfe\u8ad6\u7684\u4e0d\u5909\u91cf\u3067\u3059
$$ {\\mathbb R} $$<\/div>\u305d\u308c\u3092\u6e2c\u5b9a\u3059\u308b\u305f\u3081\u306b\u9078\u629e\u3055\u308c\u307e\u3057\u305f\u3002<\/p>

\u90f5\u4fbf\u5c4b\u3055\u3093

$$ {\\gamma = {1 \\over \\sqrt{1 – v^2\/c^2}}} $$<\/div>\u6301\u7d9a\u6642\u9593\u306e\u62e1\u5f35\u306b\u95a2\u4e0e\u3059\u308b\u3053\u306e\u6642\u9593\u306f\u3001\u98db\u884c\u6a5f\u306e\u5834\u5408\u30011 + v\u00b2\/2c\u00b2 \u307e\u305f\u306f 1+10^(-10) =1.0000000001 \u306e\u8fd1\u4f3c\u5024\u306b\u306a\u308a\u307e\u3059\u30021 \u5e74\u9593\u306e\u8d85\u97f3\u901f\u98db\u884c\u3067\u6570\u30de\u30a4\u30af\u30ed\u79d2\u3067\u3059\u3002\u5730\u4e0a\u306b\u6b8b\u3055\u308c\u305f\u6642\u8a08\u306e\u8868\u793a\u3068\u98db\u884c\u6a5f\u306b\u642d\u8f09\u3055\u308c\u3066\u3044\u308b\u6642\u8a08\u306e\u8868\u793a\u3092\u6bd4\u8f03\u3057\u3066\u6642\u9593\u306e\u9045\u308c\u3092\u6e2c\u5b9a\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u4fe1\u3058\u304c\u305f\u3044\u3053\u3068\u3067\u3059\uff08\u3053\u306e\u5b9f\u9a13\u306f 1972 \u5e74\u306b\u30a2\u30e1\u30ea\u30ab\u3067\u884c\u308f\u308c\u307e\u3057\u305f\u304c\u3001\u6c7a\u5b9a\u7684\u306a\u3082\u306e\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3067\u3057\u305f\uff09\u3002\u3057\u304b\u3057\u3001\u30b7\u30f3\u30af\u30ed\u30c8\u30ed\u30f3\u3067\u751f\u6210\u3055\u308c\u305f\u7c92\u5b50\u3092\u7814\u7a76\u3057\u3066\u3044\u308b\u7814\u7a76\u8005\u306f\u3001\u6642\u9593\u306e\u9045\u308c T= \u03b3 T’ \u306e\u5f71\u97ff\u3092\u65e5\u5e38\u7684\u306b\u7d4c\u9a13\u3057\u3066\u3044\u307e\u3059\u3002<\/p>

\u305d\u3057\u3066\u4eca\u65e5\u3001GPS \u30b7\u30b9\u30c6\u30e0\u306e\u885b\u661f\u306b\u642d\u8f09\u3055\u308c\u305f\u539f\u5b50\u6642\u8a08\u306f\u3001\u305d\u306e\u8868\u793a\u304c\u5730\u7403<\/a><\/span>\u4e0a\u306b\u6b8b\u3063\u3066\u3044\u308b\u6642\u8a08\u3068\u4e92\u63db\u6027\u304c\u3042\u308b\u3088\u3046\u306b\u6821\u6b63\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>

\u885b\u661f\u306f<\/a><\/span>\u5730\u7403\u306e\u57fa\u6e96\u5ea7\u6a19\u7cfb\u306b\u5bfe\u3057\u3066\u5e73\u884c\u79fb\u52d5\u3057\u307e\u305b\u3093\u3002<\/p>

\u9577\u3055\u306e\u53ce\u7e2e<\/a><\/span><\/span><\/h3>

\u79c1\u305f\u3061\u306f\u3001\u524d\u306e\u6bb5\u843d\u3067\u8ff0\u3079\u305f\u72b6\u6cc1\u306b\u8eab\u3092\u7f6e\u3044\u3066\u3044\u307e\u3059\u3002\u9577\u3055<\/a><\/span>M 1<\/sub> M 2<\/sub>\u3092\u6e2c\u5b9a\u3059\u308b\u3053\u3068\u306f\u3001\u5ea7\u6a19\u7cfb<\/a><\/span>\u3067 2 \u3064\u306e\u7aef M 1<\/sub>\u3068 M 2 \u306e<\/sub>\u4f4d\u7f6e\u3092\u7279\u5b9a\u3059\u308b\u3053\u3068\u306b\u76f8\u5f53\u3057\u307e\u3059\u3002\u6642\u9593\u304c\u7d4c\u3063\u3066\u3082\u52d5\u304b\u306a\u3051\u308c\u3070\u554f\u984c\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u4e00\u65b9\u3001\u305d\u308c\u3089\u304c\u540c\u3058\u901f\u5ea6 v \u3067\u79fb\u52d5\u3059\u308b\u5834\u5408\u3001\u3053\u308c\u3089 2 \u3064\u306e\u7aef\u3092\u540c\u6642\u306b\u898b\u3064\u3051\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u9759\u6b62\u72b6\u614b\u306e\u30eb\u30fc\u30eb\u3092\u8003\u616e\u3057\u307e\u3059\u3002

$$ {\\mathbb{R’}} $$<\/div> \u3001\u9759\u6b62\u72b6\u614b\u3067\u306f\u9577\u3055L<\/i> 0<\/sub><\/span>\u3067\u3059\u3002\u305d\u306e\u7aef\u306e\u5ea7\u6a19\u306fx<\/i> ‘ 1<\/sub><\/span>\u3068x<\/i> ‘ 2<\/sub><\/span>\u3067\u3059\u3002\u305d\u306e\u4e21\u7aef\u306e\u30a4\u30d9\u30f3\u30c8\u306f( t<\/i> ‘, x<\/i> ‘ 1,0,0<\/sub> )<\/span>\u304a\u3088\u3073( t<\/i> ‘, x<\/i> ‘ 2,0,0<\/sub> )<\/span>\u3067\u3059\u3002\u3053\u308c\u3089\u306e\u30a4\u30d9\u30f3\u30c8\u306f\u540c\u6642\u306b\u89b3\u6e2c\u3055\u308c\u308b\u5fc5\u8981\u304c\u3042\u308b\u305f\u3081\u3067\u3059\u3002
$$ {\\mathbb{R’}} $$<\/div> \u3002<\/p>

\u3053\u3053\u3067\u3001\u6b21\u306e\u3088\u3046\u306b\u5f97\u3089\u308c\u308b\u30a4\u30d9\u30f3\u30c8( t<\/i> ‘ 1<\/sub> , x<\/i> ‘ 1,0,0<\/sub> )<\/span>\u3068( t<\/i> ‘ 2<\/sub> , x<\/i> ‘ 2,0,0<\/sub> )<\/span>\u3092\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002

$$ {\\mathbb{R}} $$<\/div> : <\/p>
$$ {\\begin{matrix}\\left\\{\\begin{matrix} x_1=\\gamma(x’_1+vt’_1)\\\\ t_1=\\gamma(t’_1+\\frac{v}{c^2}x’_1) \\end{matrix}\\right. &\\left\\{\\begin{matrix} x_2=\\gamma(x’_2+vt’_2)\\\\ t_2=\\gamma(t’_2+\\frac{v}{c^2}x’_2) \\end{matrix}\\right.\\end{matrix}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3053\u308c\u3089\u306e\u30a4\u30d9\u30f3\u30c8\u304c\u540c\u6642\u306b\u767a\u751f\u3059\u308b\u3088\u3046\u306bt<\/i> ‘ 2<\/sub> \u2212 t<\/i> ‘ 1<\/sub><\/span>\u3092\u6c7a\u5b9a\u3057\u307e\u3057\u3087\u3046\u3002

$$ {\\mathbb{R}} $$<\/div> \u3001\u6b21\u306e\u3053\u3068\u304c\u5fc5\u8981\u3067\u3059\u3002 <\/p>
$$ {t_2-t_1=\\gamma(t’_2-t’_1+\\frac{v}{c^2}(x’_2-x’_1))=0} $$<\/div>\u3069\u3061\u3089\u304b \uff1a <\/dd>
$$ {t’_2-t’_1=-\\frac{v}{c^2}(x’_2-x’_1)} $$<\/div><\/dd>
$$ {x_2-x_1=\\gamma(x’_2-x’_1)+\\gamma v(t’_2-t’_1)=\\frac{1}{\\gamma}(x’_2-x’_1)} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u30ea\u30dd\u30b8\u30c8\u30ea\u5185\u3067\u89b3\u5bdf\u3055\u308c\u308b\u30eb\u30fc\u30eb\u306e\u9577\u3055

$$ {\\mathbb{R}} $$<\/div>\u305d\u308c\u81ea\u4f53\u3092\u8868\u73fe\u3057\u307e\u3059: <\/p>
$$ {L=L_{0}\\sqrt{1-\\frac{v^2}{c^2}}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3057\u305f\u304c\u3063\u3066\u3001\u30ea\u30dd\u30b8\u30c8\u30ea\u5185\u306e\u30eb\u30fc\u30eb\u306f\u77ed\u304f\u306a\u308a\u307e\u3059

$$ {\\mathbb{R}} $$<\/div>\u30ea\u30dd\u30b8\u30c8\u30ea\u5185\u306e\u305d\u308c
$$ {\\mathbb{R}’} $$<\/div> : \u904b\u52d5\u4e2d\u306e\u898f\u5247 M 1<\/sub> M 2<\/sub>\u306f\u3001M 1<\/sub> M 2<\/sub>\u304c\u904b\u52d5\u3057\u3066\u3044\u308b\u57fa\u6e96\u7cfb\u3067\u305d\u306e\u9577\u3055\u306e\u6e2c\u5b9a\u304c\u884c\u308f\u308c\u308b\u5834\u5408\u306b\u77ed\u304f\u306a\u308a\u307e\u3059\u3002<\/p>

\u3057\u305f\u304c\u3063\u3066\u3001100 \u30e1\u30fc\u30c8\u30eb\u306e\u30e9\u30f3\u30ca\u30fc R’ \u306f\u3001R \u306e L 0<\/sub> = 100 \u30e1\u30fc\u30c8\u30eb\u306e\u30c8\u30e9\u30c3\u30af\u4e0a\u3067 T’ 0<\/sub> =10 \u79d2\u306e\u9069\u5207\u306a\u30bf\u30a4\u30e0\u3092\u6642\u8a08\u3067\u81ea\u5df1\u8a08\u6e2c\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u30e9\u30f3\u30ca\u30fc\u306b\u3068\u3063\u3066\u3001\u81ea\u5206\u306e\u30b9\u30c8\u30e9\u30a4\u30c9\u306e\u901f\u5ea6 v \u3067\u901a\u904e\u3059\u308b\u30c8\u30e9\u30c3\u30af\u306f\u3001\u306f 100 m \u3067\u306f\u306a\u304f\u3001\u7e2e\u7d04\u3055\u308c\u307e\u3059\u3002 L’=L 0<\/sub> \/\u03b3\u3002\u4e00\u65b9\u3001\u30c8\u30e9\u30c3\u30af\u30b8\u30e3\u30c3\u30b8<\/span>\u306b\u3068\u3063\u3066\u30c8\u30e9\u30c3\u30af\u306f\u5f7c\u306b\u5bfe\u3057\u3066\u52d5\u304b\u306a\u3044\u3002\u9069\u5207\u306a\u9577\u3055\u306f L 0<\/sub> = 100m\u3001\u6642\u9593\u306f T=\u03b3T’ 0\u3001<\/sub>\u3064\u307e\u308a\u62e1\u5f35\u3055\u308c\u307e\u3059\u3002\u30e9\u30f3\u30ca\u30fc\u3068\u5be9\u5224\u306f\u6642\u9593\u3084\u8ddd\u96e2\u306b\u3064\u3044\u3066\u306f\u540c\u610f\u3057\u307e\u305b\u3093\u304c\u3001\u901f\u5ea6 v = L’\/T’ 0<\/sub> = L 0<\/sub> \/T \u306b\u3064\u3044\u3066\u306f\u540c\u610f\u3057\u307e\u3059\u3002\u3082\u3061\u308d\u3093\u3001100 \u30e1\u30fc\u30c8\u30eb\u8d70\u8005\u306e\u30b9\u30d4\u30fc\u30c9\u3067\u306f\u3001\u3053\u308c\u3089\u306e\u9055\u3044\u306f\u3059\u3079\u3066\u77e5\u899a\u3067\u304d\u307e\u305b\u3093\u3002\u76f8\u5bfe\u8ad6\u7684\u52b9\u679c\u306f\u3001\u6838<\/a><\/span>\u307e\u305f\u306f\u9280\u6cb3\u306e\u30b9\u30b1\u30fc\u30eb\u3067\u306e\u307f\u8a8d\u8b58\u3055\u308c\u307e\u3059\u3002<\/p>

\u901f\u5ea6\u306e\u69cb\u6210<\/span><\/h3>

\u79c1\u305f\u3061\u306f\u65e5\u5e38\u751f\u6d3b<\/a><\/span>\u306e\u4e2d\u3067\u3001\u901f\u5ea6\u304c\u52a0\u7b97\u3055\u308c\u308b\u3053\u3068\u3092\u77e5\u3063\u3066\u3044\u307e\u3059\u3002\u5177\u4f53\u7684\u306a\u4f8b\u3092\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u5730\u4e0b\u9244<\/a><\/span>\u306b\u4e57\u308a\u3001\u30c8\u30ec\u30c3\u30c9\u30df\u30eb\u3092\u6642\u901f 5 km \u3067\u540c\u3058\u65b9\u5411<\/a><\/span>\u306b\u6642\u901f 4 km \u3067\u6b69\u304d\u307e\u3059<\/a><\/span>\u3002\u79c1\u306e\u5bfe\u5730\u901f\u5ea6\u306f\u6642\u901f9\u30ad\u30ed\u3067\u3059\u3002\u30ac\u30ea\u30ec\u30aa\u5f0f\u3001\u305d\u3057\u3066\u76f8\u5bfe\u8ad6\u7684\u306a\u901f\u5ea6\u306e\u5408\u6210\u5f0f\u3092\u53d6\u5f97\u3059\u308b\u65b9\u6cd5\u3092\u898b\u3066\u3044\u304d\u307e\u3059\u3002\u3053\u306e\u6bb5\u843d\u3067\u306f\u3001\u3059\u3079\u3066\u306e\u52d5\u304d\u304c\u540c\u3058\u8ef8\u306b\u5e73\u884c\u306b\u884c\u308f\u308c\u308b\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002<\/p>

\u30ac\u30ea\u30e9\u30e4\u4e8b\u4ef6<\/span><\/h4>

\u30ac\u30ea\u30ec\u30aa\u306e\u5909\u63db\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 <\/p>

$$ {\\left\\{ \\begin{matrix} x=x’+vt’\\\\ t=t’ \\end{matrix} \\right.} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u5fae\u5206\u3059\u308b\u3068\u6b21\u304c\u5f97\u3089\u308c\u307e\u3059\u3002 <\/p>

$$ {\\begin{matrix} dx=dx’+vdt’=(\\frac{dx’}{dt’}+v)dt’\\\\ dt=dt’ \\end{matrix}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u5546\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059: u<\/i> = u<\/i> ‘ + v<\/i><\/span> \u3001

$$ {u = {dx \\over dt}} $$<\/div>\u305d\u3057\u3066
$$ {u’ = {dx’ \\over dt’}} $$<\/div> \u3001\u3053\u308c\u306f\u53e4\u5178\u7684\u306a\u5408\u6210\u6cd5\u5247\u3067\u3059\u3002<\/p>

\u76f8\u5bfe\u8ad6\u7684\u306a\u5834\u5408<\/span><\/h4>

\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 <\/p>

$$ {\\left\\{ \\begin{matrix} x=\\gamma(x’+vt’)\\\\ t=\\gamma(t’+\\frac{v}{c^2}x’) \\end{matrix} \\right.} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u5fae\u5206\u3059\u308b\u3068\u6b21\u304c\u5f97\u3089\u308c\u307e\u3059\u3002 <\/p>

$$ {\\begin{matrix} dx=\\gamma(dx’+vdt’)=\\gamma(\\frac{dx’}{dt’}+v)dt’\\\\ dt=\\gamma(dt’+\\frac{v}{c^2}dx’)=\\gamma(1+\\frac{v}{c^2}\\frac{dx’}{dt’})dt’ \\end{matrix}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u5546\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {\\frac{dx}{dt}=\\frac{\\frac{dx’}{dt’}+v}{1+\\frac{v}{c^2}\\frac{dx’}{dt’}}} $$<\/div>\u3069\u3061\u3089\u304b \uff1a
$$ {u=\\frac{u’+v}{1+\\frac{u’v}{c^2}}} $$<\/div>\u901f\u5ea6\u306e\u76f8\u5bfe\u8ad6\u7684\u5408\u6210\u6cd5\u5247:\u305d\u308c\u3089\u306f\u52a0\u7b97\u3055\u308c\u306a\u3044<\/strong><\/dd><\/dl><\/dd><\/dl>

u’<\/i> = c<\/i>\u306e\u5834\u5408\u3001 u<\/i> = c<\/i>\u304c\u5f97\u3089\u308c\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u5149\u306e\u901f\u5ea6\u306f<\/a><\/span>\u4e21\u65b9\u306e\u5ea7\u6a19\u7cfb\u3067\u540c\u3058\u3067\u3059\u3002<\/p>

\n\"\u76f8\u5bfe\u8ad6\u7684\u8a08\u7b97\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u30ed\u30fc\u30ec\u30f3\u30c4\u7fa4\u306e\u4f7f\u7528<\/span><\/h2>

L(v) \u3092\u884c\u5217\u3068\u3057\u307e\u3059<\/p>

$$ {\\begin{bmatrix} \\gamma & \\gamma\\beta& 0 & 0\\\\ \\gamma\\beta & \\gamma & 0& 0\\\\ 0 & 0 & 1& 0\\\\ 0 & 0 & 0& 1\\end{bmatrix}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3068

$$ {\\beta = \\beta(v) = {v \\over c}} $$<\/div>\u305d\u3057\u3066
$$ {\\gamma = \\gamma(v) = {1 \\over \\sqrt{1 – \\frac{v^2}{c^2}}}} $$<\/div> \u3002\u3053\u306e\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306f\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8\u306b\u9069\u7528\u3055\u308c\u307e\u3059
$$ {{\\mathbf{X’}}} $$<\/div>\u30ea\u30dd\u30b8\u30c8\u30ea\u5185\u3067
$$ {\\mathbb R’} $$<\/div>\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8\u3092\u4e0e\u3048\u308b
$$ {{\\mathbf{X}}} $$<\/div>\u30ea\u30dd\u30b8\u30c8\u30ea\u5185\u3067
$$ {\\mathbb R} $$<\/div>\u3069\u308c\u3068\u6bd4\u3079\u3066
$$ {\\mathbb R’} $$<\/div>\u901f\u5ea6 v \u3067\u79fb\u52d5\u3057\u307e\u3059: <\/p>
$$ {\\mathbf{X} = L(v)\\mathbf{X’}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u53c2\u8003\u306b\u306a\u308b\u3082\u306e\u304c\u3042\u308c\u3070

$$ {\\mathbb R”} $$<\/div>\u57fa\u6e96\u30d5\u30ec\u30fc\u30e0\u306b\u5bfe\u3057\u3066\u76f8\u5bfe\u7684\u306b\u79fb\u52d5\u3059\u308b
$$ {\\mathbb R’} $$<\/div>\u901f\u5ea6 u’ \u3067\u306e\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8\u9593\u306e\u95a2\u4fc2
$$ {\\mathbf{X”}} $$<\/div>\u30ea\u30dd\u30b8\u30c8\u30ea\u5185\u3067
$$ {\\mathbb R”} $$<\/div>\u305d\u3057\u3066\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8
$$ {\\mathbf{X’}} $$<\/div>\u30ea\u30dd\u30b8\u30c8\u30ea\u5185\u3067
$$ {\\mathbb R’} $$<\/div>\u306f\u6b21\u306e\u3088\u3046\u306b\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002 <\/p>
$$ {\\mathbf{X’} = L(u’)\\mathbf{X”}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3057\u305f\u304c\u3063\u3066\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {\\mathbf{X} = L(v)\\mathbf{X’} = L(v)L(u’)\\mathbf{X”}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u7a4d\u884c\u5217L<\/i> ( v<\/i> ) L<\/i> ( u<\/i> ‘) \u306f<\/span>\u884c\u5217L<\/i> ( u<\/i> )<\/span>\u306b\u4ed6\u306a\u308a\u307e\u305b\u3093\u3002\u3053\u3053\u3067u \u306f<\/i>\u30d5\u30ec\u30fc\u30e0\u306e\u901f\u5ea6\u3067\u3059

$$ {\\mathbb R”} $$<\/div>\u30ea\u30d5\u30a1\u30ec\u30f3\u30b9\u3068\u6bd4\u8f03\u3057\u3066
$$ {\\mathbb R} $$<\/div> \u3002\u305d\u308c\u3092\u78ba\u8a8d\u3055\u305b\u3066\u3044\u305f\u3060\u304d\u307e\u3059
$$ {L(v)L(u’) = L({u’+v \\over 1+u’v\/c^2})} $$<\/div> \u3001\u3057\u305f\u304c\u3063\u3066\u3001
$$ {u=\\frac{u’+v}{1+\\frac{u’v}{c^2}}} $$<\/div> \u3001\u4e0a\u8a18\u306e\u901a\u308a\u3002<\/p>

\u901f\u5ea6 u \u3068 v \u304c\u540c\u4e00\u76f4\u7dda\u4e0a\u306b\u306a\u3044\u3001\u3088\u308a\u4e00\u822c\u7684\u306a\u30b1\u30fc\u30b9\u306b\u3064\u3044\u3066\u306f\u3001\u4ee5\u4e0b\u306e\u6bb5\u843d\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>

\u30b9\u30d4\u30fc\u30c9\u30af\u30ef\u30c3\u30c9\u30c9\u30e9\u30a4\u30d0\u30fc<\/span><\/h3>

\u901f\u5ea6\u306e\u5909\u63db<\/span><\/h3>

\u6642\u9593\u3068\u8ddd\u96e2\u306e\u6bd4\u7387\u3092\u8a08\u7b97\u3059\u308b<\/a><\/span>\u3053\u3068\u3067\u901f\u5ea6\u3092\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {\\frac{V_x}{c}= \\frac{x}{ct} = \\frac{\\frac{x’}{ct’} + \\beta}{1 +\\beta\\frac{x’}{ct’}} \\qquad \\frac{V_y}{c}= \\frac{y}{ct} = \\frac{\\frac{y’}{ct’}}{\\gamma (1 +\\beta\\frac{x’}{ct’})} \\qquad \\frac{V_z}{c}= \\frac{z}{ct} = \\frac{\\frac{z’}{ct’}}{\\gamma (1 +\\beta\\frac{x’}{ct’})} \\qquad} $$<\/div><\/dd><\/dl><\/dd><\/dl>
$$ {\\frac{V_x}{c}= \\frac{\\frac{V_x’}{c} + \\beta}{1 + \\beta{\\frac{V_x’}{c}} } \\qquad \\frac{V_y}{c}= \\frac{\\frac{V’_y}{c} }{\\gamma (1 + \\beta{\\frac{V_x’}{c}})} \\qquad \\frac{V_z}{c}= \\frac{\\frac{V’_z}{c} }{\\gamma (1 + \\beta{\\frac{V_x’}{c}})} \\qquad} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3053\u308c\u3089\u306f\u901f\u5ea6\u306b\u95a2\u3059\u308b\u5909\u63db\u3067\u3042\u308a\u3001\u901f\u5ea6\u304c\u52a0\u7b97\u3055\u308c\u3066\u3044\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u4e00\u65b9\u3001\u3053\u308c\u3089\u306e\u95a2\u4fc2\u3092\u300c\u52a0\u7b97<\/a><\/span>\u300d\u3068\u3044\u3046\u8a00\u8449\u3092\u4f7f\u7528\u3057\u3066\u547c\u3076\u3079\u304d\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>

\u3053\u308c\u3089\u306e\u95a2\u4fc2\u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u3068\u5225\u306e\u65b9\u6cd5\u3067\u8a18\u8ff0\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {1-\\frac{V^2}{c^2}=1-\\frac{V_x^2+V_y^2+V_z^2}{c^2}=1- \\left(\\frac{\\frac{V_x’}{c} +\\beta}{1 + \\beta{\\frac{V_x’}{c}} }\\right)^2- (1-\\frac{v^2}{c^2})\\left(\\left(\\frac{\\frac{V_y’}{c} }{1 + \\beta{\\frac{V_x’}{c}} }\\right)^2 +\\left(\\frac{\\frac{V_z’}{c} }{1 + \\beta{\\frac{V_x’}{c}} }\\right)^2\\right)} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3069\u3061\u3089\u304b\uff1a

$$ {(1-\\frac{V^2}{c^2})(1 + \\beta\\frac{V_x’}{c})^2=(1-\\frac{V’^2}{c^2})(1-\\frac{v^2}{c^2})} $$<\/div><\/p>

\u30dd\u30fc\u30ba\u3092\u3068\u308b\u3053\u3068\u3067

$$ {\\Gamma (V) = \\frac {1}{\\sqrt{1-\\frac{V^2}{c^2}}}= \\frac {1}{\\sqrt{1-(\\frac{x}{ct} )^2-(\\frac{y}{ct} )^2-(\\frac{z}{ct} )^2}}=\\Gamma_V} $$<\/div>\u305d\u3057\u3066
$$ {\\Gamma (V’) = \\frac {1}{\\sqrt{1-\\frac{V’^2}{c^2}}}= \\frac {1}{\\sqrt{1-(\\frac{x’}{ct’} )^2-(\\frac{y’}{ct’} )^2-(\\frac{z’}{ct’} )^2}}=\\Gamma_{V’}} $$<\/div>\u305d\u3057\u3066
$$ {\\gamma = \\gamma_v = \\frac {1}{\\sqrt{1-\\frac{v^2}{c^2}}}} $$<\/div> \u3002<\/p>

\u66f8\u304b\u308c\u3066\u3044\u307e\u3059

$$ {(\\Gamma_Vc) = \\gamma_v\\left( (\\Gamma_{V’}c) + \\beta(\\Gamma_{V’}V_x’) \\right)} $$<\/div> \u3002\u3053\u308c\u306f\u30014 \u3064\u306e\u30d9\u30af\u30c8\u30eb\u3092\u8003\u616e\u3057\u305f\u5834\u5408\u306e\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306e 1 \u3064\u3067\u3059\u3002 <\/p>
$$ {(\\Gamma_V c, \\Gamma_V\\vec V )= \\frac {(c,\\vec V)}{\\sqrt{1-\\frac{V^2}{c^2}}}} $$<\/div>\u306f 4 \u30d9\u30af\u30c8\u30eb\u901f\u5ea6\u3067\u3059\u3002<\/dd><\/dl>\u540c\u3058\u304f \uff1a
$$ {\\Gamma_V c = \\gamma_v ( \\Gamma_{V’} c + \\beta \\Gamma_{V’} V_x’)\\qquad \\Gamma_V V_x = \\gamma_v ( \\Gamma_{V’} V_x’ + \\beta \\Gamma_{V’}c)} $$<\/div><\/dd>
$$ {\\qquad \\Gamma_V V_y = \\Gamma_{V’} V_y’ \\qquad \\Gamma_V V_z = \\Gamma_{V’} V_z’} $$<\/div><\/dd><\/dl><\/dd><\/dl>
$$ {\\begin{pmatrix} \\gamma (V) c\\\\ \\gamma (V) V_x\\\\ \\gamma (V) V_y\\\\ \\gamma (V) V_z\\\\\\end{pmatrix} = \\begin{bmatrix} \\gamma & \\gamma\\beta& 0 & 0\\\\ \\gamma\\beta & \\gamma & 0& 0\\\\ 0 & 0 & 1& 0\\\\ 0 & 0 & 0& 1\\end{bmatrix}\\begin{pmatrix} \\gamma (V’) c\\\\ \\gamma (V’) V_x’\\\\ \\gamma (V’) V_y’\\\\ \\gamma (V’) V_z’\\\\\\end{pmatrix}} $$<\/div><\/dd><\/dl><\/dd><\/dl><\/dd><\/dl>\u3053\u308c\u306f\u3001\u901f\u5ea6\u306b\u95a2\u3059\u308b\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u884c\u5217\u3067\u8a18\u8ff0\u3059\u308b\u65b9\u6cd5\u3067\u3059\u3002\u307e\u305f\u306f \uff1a
$$ {(\\gamma (V) c, \\gamma (V)\\vec V )= \\frac {(c,\\vec V)}{\\sqrt{1-\\frac{V^2}{c^2}}}} $$<\/div>\u306f 4 \u30d9\u30af\u30c8\u30eb\u901f\u5ea6\u3067\u3059\u3002<\/dd><\/dl><\/dd><\/dl>\u3053\u308c\u306f\u30c0\u30a4\u30ca\u30df\u30af\u30b9<\/a><\/span>\u306b\u591a\u5927\u306a\u5f71\u97ff\u3092\u4e0e\u3048\u308b\u3067\u3057\u3087\u3046\u3002<\/blockquote>
\n\"\u76f8\u5bfe\u8ad6\u7684\u8a08\u7b97\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u4e0d\u5909\u6761\u4ef6\u306e\u4f7f\u7528: \u64ec\u4f3c\u6a19\u6e96\u3068\u56fa\u6709\u6642\u9593<\/span><\/h3>

\u8003\u616e\u3059\u308b\uff1a\uff1a

$$ {ds=\\sqrt{c^2dt^2-dx^2-dy^2-dz^2}} $$<\/div>\u305d\u3057\u3066
$$ {d\\tau=dt\\sqrt{1-\\frac{V^2}{c^2}}} $$<\/div>\u3069\u3061\u3089\u3082\u4e0d\u5909\u3067\u3059\u3002<\/p>
c<\/i> d<\/i> \u03c4 = d<\/i> s<\/i><\/span><\/dd><\/dl>

\u7279\u6b8a\u76f8\u5bfe\u6027\u7406\u8ad6\u3067\u306f\u3001\u6b21\u5143<\/a><\/span>\u304c\u9577\u3055 (\u3053\u308c\u3092\u9069\u6b63\u9577\u3068\u547c\u3076) \u3067\u3042\u308b\u4e0d\u5909\u91cf\u304c\u3042\u308a\u307e\u3059\u3002\u9069\u6b63\u6642\u9593\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002 <\/p>

$$ {\\frac{d}{d\\tau}=\\gamma(V)\\frac{d}{dt}} $$<\/div><\/dd><\/dl><\/dd><\/dl>

d<\/i> \u03c4 \u306f\u3001<\/span>\u57fa\u6e96\u30d5\u30ec\u30fc\u30e0\u306b\u5bfe\u3057\u3066\u901f\u5ea6 V \u3067\u79fb\u52d5\u3059\u308b\u57fa\u6e96\u30d5\u30ec\u30fc\u30e0\u3067\u6e2c\u5b9a\u3055\u308c\u305f\u6642\u9593\u306e\u5897\u52a0\u3067\u3059\u3002

$$ {\\mathbb R} $$<\/div>\u3053\u3053\u3067\u3001\u6642\u9593\u306e\u5897\u52a0\u306f dt \u3067\u3059\u3002\u6570\u91cf
$$ {dt\\sqrt{1-\\frac{V^2}{c^2}}} $$<\/div>\u30ea\u30dd\u30b8\u30c8\u30ea\u306b\u4f9d\u5b58\u3057\u306a\u3044
$$ {\\mathbb R} $$<\/div>\u9078\u629e\u3055\u308c\u307e\u3057\u305f\u3002\u305d\u308c\u306f\u76f8\u5bfe\u8ad6\u7684\u4e0d\u5909\u91cf\u3067\u3059\u3002<\/p>

\u6b21\u306b\u3001\u81ea\u7136\u306b quadrivelocity \u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002

$$ {u^\\alpha=\\frac{dx^\\alpha}{d\\tau}} $$<\/div><\/p>
$$ {u^\\alpha=({\\gamma(V)}c,\\gamma(V)\\vec{V})} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u64ec\u4f3c\u30ce\u30eb\u30e0\u304c 1 \u306b\u7b49\u3057\u3044<\/p>

\u4ed6\u4eba\u306e\u672a\u6765\u3078\u306e\u65c5<\/a><\/span><\/span><\/h3>

\u3042\u308b\u3044\u306f\u53cc\u5b50\u306e\u30d1\u30e9\u30c9\u30c3\u30af\u30b9<\/a><\/span><\/strong>:<\/p>