\u9759\u6b62\u8cea\u91cf$M$\u306e\u7c92\u5b50\u304c\u8cea\u91cf$m_1$\u3068$m_2$\u306e\u4e8c\u3064\u306e\u7c92\u5b50\u306b\u5206\u88c2\u3059\u308b\u904e\u7a0b\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001$u^2$\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
$$\n\\begin{align*}\n u^2 &\\equiv \\eta_{\\mu\\nu}u^{\\mu}u^{\\nu} \\\\\n &=-(\\frac{dct(\\tau)}{d\\tau})^2+(\\frac{dx(\\tau)}{d\\tau})^2+(\\frac{dy(\\tau)}{d\\tau})^2+(\\frac{dz(\\tau)}{d\\tau})^2 \\\\\n &=(\\frac{ds}{d\\tau})^2 \\\\\n &=-c^2\n\\end{align*}\n$$<\/p>\n\n\n\n
4\u5143\u904b\u52d5\u91cf$p^{\\mu}=(\\gamma mc,\\gamma mv^i) \\equiv (\\frac{E}{c},p^i)$\u306b\u3064\u3044\u3066\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n p^2 &\\equiv \\eta_{\\mu\\nu}p^{\\mu}p^{\\nu} \\\\\n &=\\eta_{\\mu\\nu}(mu^{\\mu})(mu^{\\nu})=m^2\\eta_{\\mu\\nu}u^{\\mu}u^{\\nu} \\\\\n &=m^2u^2 \\\\\n &=-m^2c^2\n\\end{align*}\n$$<\/p>\n\n\n\n
\u3088\u308a\u3001<\/p>\n\n\n\n
$$\np^2=\\eta_{\\mu\\nu}p^{\\mu}p^{\\nu}=-\\frac{E^2}{c^2}+\\delta_{ij}p^ip^j=-m^2c^2\n$$<\/p>\n\n\n\n
\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n
$$\nE^2=(mc^2)^2+(cp^i)^2\n$$<\/p>\n\n\n\n
\u5404\u7c92\u5b50\u306e4\u5143\u904b\u52d5\u91cf\u3092\u305d\u308c\u305e\u308c$p_M^{\\mu}$\u3001$p_1^{\\mu}$\u3001$p_2^{\\mu}$\u3068\u3059\u308b\u3068\u304d\u30014\u5143\u904b\u52d5\u91cf\u306e\u4fdd\u5b58\u5247\u3088\u308a\u3001$p_M^{\\mu}=p_1^{\\mu}+p_2^{\\mu}$\u3068\u306a\u308b\u3002
\u3053\u306e\u5f0f\u3068\u3001\u521d\u671f\u72b6\u614b\u3067\u306f\u7c92\u5b50\u304c\u9759\u6b62\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u4eee\u5b9a\u3059\u308b\u3068\u3001<\/p>\n\n\n\n
$$\n\\begin{gather*}\n p_M^{\\mu}=\n \\begin{pmatrix}\n Mc \\\\\n 0\n \\end{pmatrix} \\\\\n p_1^{\\mu}=\n \\begin{pmatrix}\n \\frac{E_1}{c} \\\\\n p^i\n \\end{pmatrix} \\\\\n p_2^{\\mu}=\n \\begin{pmatrix}\n \\frac{E_2}{c} \\\\\n -p^i\n \\end{pmatrix} \\\\\n Mc^2=E_1+E_2\n\\end{gather*}\n$$<\/p>\n\n\n\n
\u7c92\u5b501\u30012\u306b\u3064\u3044\u3066\u95a2\u4fc2\u5f0f$E^2=(mc^2)^2+(cp^i)^2$\u3092\u7acb\u3066\u3066\u307f\u308b\u3068\u3001<\/p>\n\n\n\n
$$ \\left\\{ \\begin{aligned} E_1^{\\;2}&=(m_1c^2)^2+(cp^i)^2 \\\\ E_2^{\\;2}&=(m_2c^2)^2+(-cp^i)^2 \\end{aligned} \\right. $$<\/p>\n\n\n\n
\u3088\u308a\u3001<\/p>\n\n\n\n
$$\n\\begin{align*}\n (Mc^2)^2&=(E_1+E_2)^2 \\\\\n &=E_1^{\\;2}+E_2^{\\;2}+2E_1E_2 \\\\\n &=(m_1c^2)^2+(m_2c^2)^2+2(cp^i)^2 \\\\\n &\\qquad+2\\sqrt{(m_1c^2)^2+(cp^i)^2}\\sqrt{(m_2c^2)^2+(-cp^i)^2} \\\\\n &=(m_1c^2+m_2c^2)^2+\\frac{(m_1+m_2)^2}{m_1m_2}(cp^i)^2+\\mathcal{O}((cp^i)^4)\n\\end{align*}\n$$<\/p>\n\n\n\n
3\u884c\u76ee\u304b\u30894\u884c\u76ee\u3078\u306e\u5f0f\u5909\u63db\u306b\u306f$(cp^i)^2$\u306b\u3064\u3044\u3066\u306e\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3092\u7528\u3044\u307e\u3057\u305f\u3002<\/p>\n\n\n\n
\u3053\u306e\u5f0f\u304b\u3089\u308f\u304b\u308b\u3068\u304a\u308a\u3001\u5143\u306e\u7c92\u5b50\u306e\u9759\u6b62\u8cea\u91cf\u30a8\u30cd\u30eb\u30ae\u30fc$Mc^2$\u306b\u6bd4\u3079\u3066\u5d29\u58ca\u5f8c\u306e\u7c92\u5b50\u306e\u9759\u6b62\u8cea\u91cf\u30a8\u30cd\u30eb\u30ae\u30fc$m_1c^2,m_2c^2$\u306e\u548c\u306f\u5c0f\u3055\u304f\u306a\u3063\u3066\u304a\u308a\u3001\u305d\u306e\u5dee\u5206\u306b\u3064\u3044\u3066\u306f\u751f\u6210\u3055\u308c\u308b\u7c92\u5b50\u304c$p^i$\u306b\u6bd4\u4f8b\u3059\u308b\u901f\u5ea6\u3092\u3082\u3064\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u9759\u6b62\u8cea\u91cf$M$\u306e\u7c92\u5b50\u304c\u8cea\u91cf$m_1$\u3068$m_2$\u306e\u4e8c\u3064\u306e\u7c92\u5b50\u306b\u5206\u88c2\u3059\u308b\u904e\u7a0b\u3092\u8003\u3048\u307e\u3059\u3002 \u3053\u3053\u3067\u3001$u^2$\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002 $$ \\begin{align*} u^2 &\\equiv \\eta_{\\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-355","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\/355","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=355"}],"version-history":[{"count":5,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/355\/revisions"}],"predecessor-version":[{"id":1600,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/355\/revisions\/1600"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=355"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=355"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}