{"id":353,"date":"2022-12-29T14:17:19","date_gmt":"2022-12-29T05:17:19","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=353"},"modified":"2024-04-27T08:38:03","modified_gmt":"2024-04-26T23:38:03","slug":"post-353","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=353","title":{"rendered":"\u9759\u6b62\u8cea\u91cf\u30a8\u30cd\u30eb\u30ae\u30fc"},"content":{"rendered":"\n

\u56fa\u6709\u6642\u9593\u3092\u4f7f\u3063\u3066\u6e2c\u3063\u305f\u901f\u5ea6$u^{\\mu} \\equiv \\frac{dx^{\\mu}}{d\\tau}$\u30924\u5143\u901f\u5ea6<\/strong>(four-velocity)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n

$$\nu^{\\mu} \\equiv \\frac{dx^{\\mu}}{d\\tau}=(\\frac{dct(\\tau)}{d\\tau},\\frac{dx(\\tau)}{d\\tau},\\frac{dy(\\tau)}{d\\tau},\\frac{dz(\\tau)}{d\\tau})\n$$<\/p>\n\n\n\n

4\u5143\u901f\u5ea6\u306b\u7c92\u5b50\u306e\u9759\u6b62\u8cea\u91cf$m$\u3092\u304b\u3051\u305f\u91cf<\/p>\n\n\n\n

$$\np^{\\mu} \\equiv mu^{\\mu}=(mc\\frac{dt(\\tau)}{d\\tau},m\\frac{dx^i(\\tau)}{d\\tau})\n$$<\/p>\n\n\n\n

\u30924\u5143\u904b\u52d5\u91cf<\/strong>(four-momentum)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n

\u5148\u306e\u8a18\u4e8b\u3067\u793a\u3057\u305f$d\\tau=\\frac{dt}{\\gamma}$\u3068\u3044\u3046\u95a2\u4fc2\u5f0f\u3088\u308a\u3001$\\frac{dt}{d\\tau}=\\gamma$\u3068\u306a\u308a\u307e\u3059\u3002
\u3057\u305f\u304c\u3063\u3066\u30014\u5143\u901f\u5ea6\u306e\u6642\u9593\u6210\u5206\u306f$u^t=c\\frac{dt}{d\\tau}=\\gamma c$\u3001\u7a7a\u9593\u6210\u5206\u306f$u^i=\\frac{dx^i}{d\\tau}=\\frac{dt}{d\\tau}\\frac{dx^i}{dt}=\\gamma v^i$\u3068\u306a\u308a\u307e\u3059\u3002
\u307e\u305f\u30014\u5143\u904b\u52d5\u91cf\u306e\u6642\u9593\u6210\u5206\u306f$p^t=mc\\frac{dt}{d\\tau}=\\gamma mc$\u3001\u7a7a\u9593\u6210\u5206\u306f$p^i=m\\frac{dx^i}{d\\tau}=mu^i=\\gamma mv^i$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

4\u5143\u904b\u52d5\u91cf\u306e\u7a7a\u9593\u6210\u5206\u306f\u3001$v \\ll c$\u306e\u5834\u5408\u306b\u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u904b\u52d5\u91cf$mv^i$\u306b\u4e00\u81f4\u3057\u307e\u3059\u3002
\u3053\u3053\u3067\u306f\u30014\u5143\u904b\u52d5\u91cf\u306e\u6642\u9593\u6210\u5206$p^t$\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u898b\u3066\u3044\u304f\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

$p^t$\u3092$v$\u306b\u3064\u3044\u3066\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n\n\n\n

$$\n\\begin{align*}\n p^t&=\\frac{mc}{\\sqrt{1-(\\frac{v}{c})^2}} \\\\\n &=mc+\\frac{m}{2c}v^2+\\cdots \\\\\n &=\\frac{1}{c}(mc^2+\\frac{1}{2}mv^2+\\cdots)\n\\end{align*}\n$$<\/p>\n\n\n\n

\u5c55\u958b\u5f8c\u306b\u73fe\u308c\u308b$\\frac{1}{2}mv^2$\u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5f0f\u306b\u4e00\u81f4\u3059\u308b\u3053\u3068\u304b\u3089\u3001$cp^t$\u306f\u4f55\u3089\u304b\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8868\u3057\u3066\u3044\u308b\u3082\u306e\u3068\u8003\u3048\u3089\u308c\u307e\u3059\u3002
\u305d\u3046\u3059\u308b\u3068\u3001$mc^2$\u306b\u3064\u3044\u3066\u3082\u4f55\u3089\u304b\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8868\u3059\u3082\u306e\u3060\u3068\u8003\u3048\u3089\u308c\u307e\u3059\u3002
\u3053\u306e\u30a8\u30cd\u30eb\u30ae\u30fc$E=mc^2$\u306f\u7c92\u5b50\u304c\u9759\u6b62\u3057\u3066\u3044\u308b\u5834\u5408\u306b\u3082\u6b8b\u308b\u5b9a\u6570\u9805\u3067\u3001\u3053\u308c\u3092\u9759\u6b62\u8cea\u91cf\u30a8\u30cd\u30eb\u30ae\u30fc<\/strong>(rest energy)\u3068\u547c\u3073\u307e\u3059\u3002
\u3053\u306e\u6bb5\u968e\u3067\u306f$mc^2$\u304c\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8868\u3057\u3066\u3044\u308b\u3068\u3044\u3046\u306e\u306f\u5358\u306a\u308b\u4eee\u8aac\u3067\u3059\u304c\u3001\u3053\u306e\u4eee\u8aac\u304c\u6b63\u3057\u3044\u3053\u3068\u306f\u8cea\u91cf\u6b20\u640d\u3084\u6838\u53cd\u5fdc\u30fb\u5bfe\u6d88\u6ec5\u306b\u4f34\u3046\u30a8\u30cd\u30eb\u30ae\u30fc\u653e\u51fa\u30fb\u5438\u53ce\u304b\u3089\u78ba\u304b\u3081\u3089\u308c\u3066\u3044\u307e\u3059\u3002
\u3057\u305f\u304c\u3063\u3066\u3001\u8cea\u91cf$m$\u3092\u3082\u3064\u3053\u3068\u3068\u30a8\u30cd\u30eb\u30ae\u30fc$mc^2$\u3092\u3082\u3064\u3053\u3068\u306f\u7b49\u4fa1\u3067\u3042\u308b\u3068\u8a00\u3048\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u3053\u3067\u3001\u5149\u901f\u3067\u79fb\u52d5\u3059\u308b\u6709\u9650\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u3082\u3063\u305f\u7c92\u5b50\u3092\u8003\u3048\u307e\u3059\u3002
\u3053\u306e\u3068\u304d\u3001$cp^t=\\gamma mc^2$\u306e$\\gamma$\u304c\u7121\u9650\u5927\u306b\u767a\u6563\u3057\u3066\u3057\u307e\u3046\u306e\u3067\u3001$m=0$\u3067\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093\u3002($\\infty \\times 0$\u306e\u4e0d\u5b9a\u5f62)
\u3053\u306e\u9006\u3082\u6210\u7acb\u3059\u308b\u305f\u3081\u3001\u8cea\u91cf\u3092\u3082\u305f\u305a\u306b\u6709\u9650\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u3082\u3064\u7c92\u5b50\u306f\u3001\u5149\u901f\u3067\u904b\u52d5\u3057\u3066\u3044\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002
\u6211\u3005\u4eba\u9593\u304c\u5149\u901f\u3067\u79fb\u52d5\u3059\u308b\u305f\u3081\u306b\u306f\u3001\u307e\u305a\u3053\u306e\u8089\u4f53\u3092\u6368\u3066\u53bb\u308b\u5fc5\u8981\u304c\u3042\u308b\u306e\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u306d\u3002(\u9069\u5f53)<\/p>\n","protected":false},"excerpt":{"rendered":"

\u56fa\u6709\u6642\u9593\u3092\u4f7f\u3063\u3066\u6e2c\u3063\u305f\u901f\u5ea6$u^{\\mu} \\equiv \\frac{dx^{\\mu}}{d\\tau}$\u30924\u5143\u901f\u5ea6(four-velocity)\u3068\u547c\u3073\u307e\u3059\u3002 $$ u^{\\mu} \\equiv \\frac{dx^{\\mu […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,10],"tags":[],"class_list":["post-353","post","type-post","status-publish","format-standard","hentry","category-9","category-10"],"_links":{"self":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/353","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=353"}],"version-history":[{"count":4,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/353\/revisions"}],"predecessor-version":[{"id":1164,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/353\/revisions\/1164"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=353"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=353"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=353"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}