{"id":547,"date":"2023-01-14T18:05:30","date_gmt":"2023-01-14T09:05:30","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=547"},"modified":"2025-08-18T21:15:08","modified_gmt":"2025-08-18T12:15:08","slug":"post-547","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=547","title":{"rendered":"\u4e8b\u8c61\u306e\u5730\u5e73\u9762"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/daba-no-heya.com\/?p=547\/#%E3%82%B7%E3%83%A5%E3%83%AF%E3%83%AB%E3%83%84%E3%82%B7%E3%83%AB%E3%83%88%E8%A7%A3%E3%81%AE%E7%89%B9%E7%95%B0%E7%82%B9\" >\u30b7\u30e5\u30ef\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u306e\u7279\u7570\u70b9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/daba-no-heya.com\/?p=547\/#%E4%BA%8B%E8%B1%A1%E3%81%AE%E5%9C%B0%E5%B9%B3%E9%9D%A2\" >\u4e8b\u8c61\u306e\u5730\u5e73\u9762<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%82%B7%E3%83%A5%E3%83%AF%E3%83%AB%E3%83%84%E3%82%B7%E3%83%AB%E3%83%88%E8%A7%A3%E3%81%AE%E7%89%B9%E7%95%B0%E7%82%B9\"><\/span>\u30b7\u30e5\u30ef\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u306e\u7279\u7570\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u524d\u56de\u7d39\u4ecb\u3057\u305f\u3068\u304a\u308a\u3001\u30b7\u30e5\u30ef\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3059\u3002<\/p>\n\n\n\n<p>$$\nds^2=-(1-\\frac{r_s}{r})dt^2+\\frac{dr^2}{1-\\frac{r_s}{r}}+r^2d\\Omega_{\\rm{II}}^2\n$$<\/p>\n\n\n\n<p>$r_s=\\frac{2GM}{c^2}$\u306f\u30b7\u30e5\u30ef\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u30b7\u30e5\u30ef\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u306f$r=0$\u3068$r=r_s$\u3067\u3044\u304f\u3064\u304b\u306e\u9805\u304c\u767a\u6563\u3059\u308b\u305f\u3081\u3001\u4e8c\u3064\u306e\u7279\u7570\u70b9\u3092\u3082\u3064\u3088\u3046\u306b\u898b\u3048\u307e\u3059\u3002<br>$r=r_s$\u306b\u304a\u3051\u308b\u7279\u7570\u70b9\u306e\u3053\u3068\u3092<strong>\u5ea7\u6a19\u7279\u7570\u70b9<\/strong>(coordinate singularity)\u3068\u547c\u3073\u3001\u3053\u308c\u306f\u9069\u5207\u306a\u5ea7\u6a19\u3092\u9078\u3076\u3053\u3068\u306b\u3088\u3063\u3066\u53d6\u308a\u9664\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br>\u4e00\u65b9\u3001$r=0$\u306e\u7279\u7570\u70b9\u306f\u53d6\u308a\u9664\u304f\u3053\u3068\u304c\u3067\u304d\u305a\u3001\u3053\u306e\u3088\u3046\u306a\u7279\u7570\u70b9\u306e\u3053\u3068\u3092<strong>\u91cd\u529b\u306e\u7279\u7570\u70b9<\/strong>(gravitational singularity)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30ea\u30fc\u30de\u30f3\u30c6\u30f3\u30bd\u30eb\u304b\u3089\u69cb\u6210\u3067\u304d\u308b\u5ea7\u6a19\u4f9d\u5b58\u3057\u306a\u3044\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30b9\u30ab\u30e9\u30fc\u91cf(\u30af\u30ec\u30c3\u30c1\u30de\u30f3\u4e0d\u5909\u91cf)\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<br>(\u7ba1\u7406\u4eba\u306f\u8a08\u7b97\u3057\u3066\u306a\u3044\u3067\u3059\u3001Wikipedia\u306b\u66f8\u3044\u3066\u3042\u308a\u307e\u3057\u305f)<\/p>\n\n\n\n<p>$$\nR^{\\alpha\\beta\\gamma\\delta}R_{\\alpha\\beta\\gamma\\delta}=\\frac{12r_s^2}{r^6}=\\frac{48G^2M^2}{c^4r^6}\n$$<\/p>\n\n\n\n<p>\u3053\u3053\u304b\u3089\u3082\u308f\u304b\u308b\u3088\u3046\u306b\u3001$r=0$\u306b\u304a\u3044\u3066\u66f2\u7387\u304c\u7121\u9650\u5927\u306b\u767a\u6563\u3057\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E4%BA%8B%E8%B1%A1%E3%81%AE%E5%9C%B0%E5%B9%B3%E9%9D%A2\"><\/span>\u4e8b\u8c61\u306e\u5730\u5e73\u9762<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u66f2\u9762$r=r_s$\u3092<strong>\u4e8b\u8c61\u306e\u5730\u5e73\u9762<\/strong>(event horizon)\u3068\u547c\u3073\u307e\u3059\u3002<br>\u534a\u5f84$r$\u304c\u30b7\u30e5\u30ef\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84$r_s$\u3088\u308a\u3082\u5c0f\u3055\u304f\u306a\u3063\u305f\u7269\u4f53\u306f\u91cd\u529b\u5d29\u58ca\u3092\u8d77\u3053\u3057\u3066<strong>\u30d6\u30e9\u30c3\u30af\u30db\u30fc\u30eb<\/strong>(black hole)\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u3001$r=r_s$\u306b\u304a\u3051\u308b\u5ea7\u6a19\u7279\u7570\u70b9\u3092\u53d6\u308a\u9664\u304f\u305f\u3081\u306b\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5ea7\u6a19\u5909\u63db\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>$r&gt;r_s$\u306b\u3064\u3044\u3066\u3001<\/p>\n\n\n\n<p>$$ \\left\\{ \\begin{aligned} &amp;T=(\\frac{r}{r_s}-1)^{\\frac{1}{2}}e^{\\frac{r}{2r_s}}\\sinh(\\frac{t}{2r_s}) \\\\ &amp;X=(\\frac{r}{r_s}-1)^{\\frac{1}{2}}e^{\\frac{r}{2r_s}}\\cosh(\\frac{t}{2r_s}) \\end{aligned} \\right. $$<\/p>\n\n\n\n<p>$0&lt;r&lt;r_s$\u306b\u3064\u3044\u3066\u3001<\/p>\n\n\n\n<p>$$ \\left\\{ \\begin{aligned} &amp;T=(1-\\frac{r}{r_s})^{\\frac{1}{2}}e^{\\frac{r}{2r_s}}\\cosh(\\frac{t}{2r_s}) \\\\ &amp;X=(1-\\frac{r}{r_s})^{\\frac{1}{2}}e^{\\frac{r}{2r_s}}\\sinh(\\frac{t}{2r_s}) \\end{aligned} \\right. $$<\/p>\n\n\n\n<p>\u3053\u306e\u3088\u3046\u306b$T,X$\u3092\u8a2d\u5b9a\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>$$\nT^2-X^2=(1-\\frac{r}{r_s})e^{-\\frac{r}{r_s}}\n$$<\/p>\n\n\n\n<p>\u3068\u306a\u308b\u306e\u3067\u3001\u30b7\u30e5\u30ef\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84$r=r_s$\u306f\u66f2\u7dda$T^2-X^2=0$\u306b\u5bfe\u5fdc\u3057\u307e\u3059\u3002<br>\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u5c0e\u5165\u3055\u308c\u305f\u5ea7\u6a19$T,X$\u3092<strong>\u30af\u30eb\u30b9\u30ab\u30eb\u5ea7\u6a19<\/strong>(Kruskal coordinates)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u30af\u30eb\u30b9\u30ab\u30eb\u5ea7\u6a19\u3092\u7528\u3044\u3066\u30b7\u30e5\u30ef\u30eb\u30c4\u30b7\u30eb\u30c8\u8a08\u91cf\u3092\u8868\u3059\u3068\u3001<\/p>\n\n\n\n<p>$$\n\\begin{align*}\n\tds^2&amp;=-(1-\\frac{r_s}{r})dt^2+\\frac{dr^2}{1-\\frac{r_s}{r}}+r^2d\\Omega_{\\rm{II}}^2 \\\\\n\t&amp;=\\frac{4r_s^3e^{-\\frac{r}{r_s}}}{r}(-dT^2+dX^2)+r^2d\\Omega_{\\rm{II}}^2\n\\end{align*}\n$$<\/p>\n\n\n\n<p>\u3068\u306a\u308a\u3001$r=r_s$\u3067\u6b63\u5247\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u306e\u8a08\u7b97\u3092\u81ea\u5206\u3067\u3082\u3084\u3063\u3066\u307f\u305f\u306e\u3067\u3059\u304c\u3001\u7d50\u5c40\u3088\u304f\u308f\u304b\u308a\u307e\u305b\u3093\u3067\u3057\u305f\u3002<br>\u308f\u304b\u3089\u306a\u3044\u3068\u3053\u308d\u3092\u308f\u304b\u3089\u306a\u3044\u307e\u307e\u306b\u3057\u3066\u304a\u304f\u306e\u306f\u3061\u3087\u3063\u3068\u6c17\u6301\u3061\u60aa\u3044\u3067\u3059\u304c\u3001\u3053\u306e\u5148\u306e\u8b70\u8ad6\u306b\u7279\u306b\u5f71\u97ff\u304c\u3042\u308b\u308f\u3051\u3067\u3082\u306a\u3044\u306e\u3067\u3001\u4eca\u306f\u305d\u306e\u307e\u307e\u306b\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u52d5\u5f84$r$\u306e\u65b9\u5411\u306b\u98db\u3076\u5149\u306e\u7d4c\u8def($d\\Omega_{II}^2=0$)\u306b\u6ce8\u76ee\u3057\u307e\u3059\u3002<br>$ds^2=0$\u3068\u306a\u308b\u3088\u3046\u306a$(T,X)$\u304c\u5149\u306e\u7d4c\u8def\u306a\u306e\u3067\u3001$-dT^2+dX^2=0$\u3068\u306a\u308c\u3070\u3088\u3044\u3067\u3059\u3002<\/p>\n\n\n\n<p>$(T,X)=(T,X(T))$\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>$$\n-dT^2+dX^2=(-1+(\\frac{dX(T)}{dT})^2)dT^2=0\n$$<\/p>\n\n\n\n<p>\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$\n-1+(\\frac{dX(T)}{dT})^2=0\n$$<\/p>\n\n\n\n<p>\u3088\u308a\u3001<\/p>\n\n\n\n<p>$$\n\\frac{dX(T)}{dT}=\\pm 1\n$$<\/p>\n\n\n\n<p>\u3053\u308c\u3088\u308a\u3001$X$\u306e\u521d\u671f\u5024\u3092$X_0$\u3068\u3059\u308b\u3068\u3001$X(T)=X_0\\pm T$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u6642\u7a7a\u306e\u69cb\u9020\u3092\u8abf\u3079\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u4e09\u3064\u306e\u5149\u7dda\u306b\u6ce8\u76ee\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>$X=X_0+T\\;(X_0&gt;0)$<\/li>\n\n\n\n<li>$X=T$<\/li>\n\n\n\n<li>$X=T-T_0\\;(0&lt;T_0&lt;1)$<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>$X=X_0+T\\;(X_0&gt;0)$\u306e\u5834\u5408<\/p>\n\n\n\n<p>\u3053\u306e\u5149\u7dda\u306e\u8ecc\u9053\u306b\u3064\u3044\u3066\u305d\u306e\u4f4d\u7f6e$r(T)$\u304c\u6e80\u305f\u3059\u5f0f\u306f\u3001<\/p>\n\n\n\n<p>$$\n(\\frac{r(T)}{r_s}-1)e^{\\frac{r(T)}{r_s}}=X^2-T^2=X_0^2+2X_0T\n$$<\/p>\n\n\n\n<p>\u6642\u9593$T$\u304c\u7d4c\u3064\u306b\u3064\u308c\u3066$r(T)$\u304c\u7121\u9650\u5927\u306b\u5897\u5927\u3059\u308b\u3053\u3068\u304b\u3089\u3001\u3053\u306e\u9818\u57df\u306e\u5149\u7dda\u306f\u7121\u9650\u9060\u306b\u98db\u3073\u53bb\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$X=T$\u306e\u5834\u5408<\/p>\n\n\n\n<p>$$\n(\\frac{r(T)}{r_s}-1)e^{\\frac{r(T)}{r_s}}=X^2-T^2=0\\Leftrightarrow r(T)=r_s\n$$<\/p>\n\n\n\n<p>\u6642\u9593$T$\u304c\u7d4c\u904e\u3057\u3066\u3082\u5149\u7dda\u306f$r=r_s$\u306b\u3068\u3069\u307e\u308b\u305f\u3081\u3001$r=r_s$\u304b\u3089\u767a\u3057\u305f\u5149\u7dda\u306f\u7121\u9650\u9060\u306b\u5230\u9054\u3067\u304d\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>$X=T-T_0\\;(0&lt;T_0&lt;1)$\u306e\u5834\u5408<\/p>\n\n\n\n<p>$$\n(\\frac{r(T)}{r_s}-1)e^{\\frac{r(T)}{r_s}}=X^2-T^2=T_0(T_0-2T)\n$$<\/p>\n\n\n\n<p>\u5149\u7dda\u306f$(T,X)=(T_0,0)$\u304b\u3089\u30b9\u30bf\u30fc\u30c8\u3057\u305f\u5f8c\u3001\u66f2\u7387\u7279\u7570\u70b9$r=0\\Leftrightarrow X^2-T^2=-1$\u306b\u6709\u9650\u6642\u9593$T=\\frac{1}{2}(T_0+\\frac{1}{T_0})$\u3067\u5f53\u305f\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002<br>\u3064\u307e\u308a\u3001\u3053\u306e\u5834\u5408\u306b\u3082\u5149\u7dda\u306f\u7121\u9650\u9060\u306b\u306f\u5230\u9054\u3067\u304d\u307e\u305b\u3093\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\u4e0a\u306b\u793a\u3057\u305f\u3068\u304a\u308a\u3001$r=r_s$\u306f\u5149\u306b\u3088\u3063\u3066\u5341\u5206\u9060\u65b9\u306e\u89b3\u6e2c\u8005\u306b\u4fe1\u53f7\u3092\u9001\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3069\u3046\u304b\u306e\u5883\u754c\u3068\u306a\u3063\u3066\u304a\u308a\u3001$r=r_s$\u306e\u4f4d\u7f6e\u306b\u3042\u308b\u7403\u9762\u3092<strong>\u4e8b\u8c61\u306e\u5730\u5e73\u9762<\/strong>(event horizon)\u3068\u547c\u3073\u307e\u3059\u3002<br>\u4e8b\u8c61\u306e\u5730\u5e73\u9762\u306e\u5185\u90e8\u306e\u73fe\u8c61\u306f\u3001\u5916\u90e8\u304b\u3089\u5149\u3084\u305d\u306e\u4ed6\u306e\u3044\u304b\u306a\u308b\u624b\u6bb5\u3092\u3082\u3063\u3066\u3057\u3066\u3082\u76f4\u63a5\u89b3\u6e2c\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002<\/p>\n\n\n\n<p>\u7121\u9650\u9060\u3068\u56e0\u679c\u7684\u66f2\u7dda(\u30cc\u30eb(\u5149\u7684)\u304b\u6642\u9593\u7684\u65b9\u5411\u306e\u66f2\u7dda)\u3067\u7d50\u3076\u3053\u3068\u306e\u3067\u304d\u306a\u3044\u6642\u7a7a\u9818\u57df\u306e\u3053\u3068\u3092<strong>\u30d6\u30e9\u30c3\u30af\u30db\u30fc\u30eb<\/strong>(black 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