<\/p>\n
\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306e\u7403\u5bfe\u79f0<\/strong><\/span>\u304b\u3064\u771f\u7a7a<\/strong><\/span> (\\(T^{\\mu\\nu} = 0\\)) \u89e3\u304c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3<\/strong><\/span>\u3067\u3042\u3063\u305f\u3002\u5c0e\u51fa\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u3082\u53c2\u7167\uff1a<\/p>\n \u7dda\u7d20<\/strong><\/span>\u306f\uff0c<\/p>\n $$ ds^2 = -\\left(1-\\dfrac{r_g}{r}\\right) c^2 dt^2 + \\frac{dr^2} {1-\\dfrac{r_g}{r}} + r^2(d\\theta^2 + \\sin^2\\theta d\\phi^2)$$ \u4e0a\u8a18\u306b\u3088\u3063\u3066\uff0c\u5ea7\u6a19\u7cfb\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3068\u3063\u3066\u3044\u308b\u3053\u3068 \u4fdd\u5b58\u91cf\u304c\u308f\u304b\u308a\u3084\u3059\u3044\u4ee5\u4e0b\u306e\u5f62\u3092\u4f7f\u3046\u3002 \u3053\u306e\u5f0f\u304b\u3089\uff0c\u4e00\u822c\u306b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span>\u306e\u6210\u5206 \\(g_{\\lambda\\mu} \\) \u304c \\(x^{\\nu} \\) \u4f9d\u5b58\u6027\u3092\u3082\u305f\u306a\u3044\u5834\u5408\u306f\uff0c \u307e\u305f4\u5143\u901f\u5ea6<\/strong><\/span>\u306e\u898f\u683c\u5316\u6761\u4ef6<\/strong><\/span>\u306f\uff0c<\/p>\n $$ \\boldsymbol{u}\\cdot\\boldsymbol{u} = g_{\\mu\\nu} u^{\\mu} u^{\\nu} =\u00a0 -c^2 $$ \u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u8a08\u91cf<\/strong><\/span>\u306e\u6210\u5206\u306f \\(x^0 = t\\) \u3092\u542b\u307e\u306a\u3044\u306e\u3067\uff0c\\(u_0\\) \u304c\u4fdd\u5b58\u91cf\u3068\u306a\u308b\u3002\u3053\u306e\u91cf\u3092 \\(-\\epsilon\\) \u3068\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \u307e\u305f\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u8a08\u91cf<\/strong><\/span>\u306e\u6210\u5206\u306f \\( x^3 = \\phi\\) \u3082\u542b\u307e\u306a\u3044\u306e\u3067\uff0c\\(u_3\\) \u304c\u4fdd\u5b58\u91cf\u3068\u306a\u308b\u3002\u3053\u306e\u91cf\u3092 \\(\\ell\\) \u3068\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \\( \\displaystyle u^2 = \\frac{d x^2}{d\\tau} = \\frac{d\\theta}{d\\tau} \\) \u306b\u3064\u3044\u3066\u306f\uff0c<\/p>\n $$ \\frac{du_{2}}{d\\tau} =\\frac{d}{d\\tau} \\left( g_{22} \\frac{d\\theta}{d\\tau}\\right)\u00a0 = \\frac{d}{d\\tau} \\left( r^2 \\frac{d\\theta}{d\\tau}\\right) = \\frac{1}{2} g_{33, 2} u^3 u^3$$ \u3088\u308a<\/p>\n $$ \\frac{d}{d\\tau}\\frac{d\\theta}{d\\tau} + \\frac{2}{r} \\frac{dr}{d\\tau} \\frac{d\\theta}{d\\tau} = \\frac{\\ell^2 \\cos\\theta}{r^2 \\sin^3\\theta}$$\u3092\u5f97\u308b\u3002<\/p>\n \u521d\u671f\u6761\u4ef6<\/strong><\/span>\u3068\u3057\u3066\u56fa\u6709\u6642\u9593<\/strong><\/span> \\(\\tau \\) \u304c\u3042\u308b\u5024 \\(\\tau=0\\) \u306e\u3068\u304d\uff0c \u7403\u5bfe\u79f0\u6027\u306b\u3088\u308a\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f \u3053\u3068\u3092\u521d\u671f\u5024\u554f\u984c\u3068\u3057\u3066\u793a\u3057\u305f\u3082\u306e\u3067\u3042\u308b\u3002<\/p>\n \u3055\u3066\uff0c\u3053\u308c\u307e\u3067\u306e\u3068\u3053\u308d\uff0c\u308f\u304b\u3063\u305f\u306e\u306f<\/p>\n $$ u^0 = \\frac{\\epsilon c}{1 – \\dfrac{r_g}{r}}, \\quad u^2 = 0\u00a0 \\ \\left( \\theta =\u00a0 \\frac{\\pi}{2}\\right), \\quad \u6b8b\u308a\u306e \\(\\displaystyle u^1 = \\frac{dr}{d\\tau} \\) \u306b\u3064\u3044\u3066\u306f\uff0c\u898f\u683c\u5316\u6761\u4ef6<\/strong><\/span>\u3088\u308a<\/p>\n \\begin{eqnarray} \u5186\u904b\u52d5\u306e\u5834\u5408\u3084\u52d5\u7cfb\u65b9\u5411\u306e\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u306e\u3088\u3046\u306a\uff0c\u7c21\u5358\u306a\u5834\u5408\u306f\u3053\u306e\u65b9\u7a0b\u5f0f\u304c\u4f7f\u3048\u308b\u3002\u305d\u308c\u4ee5\u5916\u306e\u4e00\u822c\u7684\u306a\u904b\u52d5\uff08\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u3067\u306f\u4e00\u822c\u7684\u306a\u6955\u5186\u904b\u52d5\u306b\u3042\u305f\u308b\uff09\u306e\u5834\u5408\u306f\uff0c\u3053\u308c\u3092\u3082\u3046\u5c11\u3057\u5909\u5f62\u3057\u3066\u89e3\u304d\u3084\u3059\u3044\u5f62\u306b\u3057\u3066\u304a\u3044\u305f\u307b\u3046\u304c\u3088\u3044\u3060\u308d\u3046\u3002<\/p>\n $$\\frac{dr}{d\\tau} = \\frac{d\\phi}{d\\tau} \\frac{dr}{d\\phi} = \\frac{\\ell}{r^2} \\frac{dr}{d\\phi}$$ \u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n $$\\left( \\frac{1}{r^2} \\frac{dr}{d\\phi} \\right)^2 =\u00a0 \\frac{\\epsilon^2 c^2 -c^2}{\\ell^2} + \\frac{2GM}{\\ell^2} \\frac{1}{r} -\\frac{1}{r^2}+\u00a0 r_g \\frac{1}{r^3} $$<\/p>\n \u6c42\u3081\u305f\u3044\u5909\u6570 \\(r\\) \u304c\u5206\u6bcd\u306b\u3070\u3063\u304b\u308a\u3042\u3089\u308f\u308c\u308b\u306e\u3067\uff0c\u3044\u3063\u305d\u306e\u3053\u3068<\/p>\n $$\\frac{1}{r} \\equiv s, \\quad -\\frac{1}{r^2} \\frac{dr}{d\\phi} = \\frac{ds}{d\\phi}$$ \u3068\u5909\u6570\u5909\u63db\u3057\u3066\u3084\u308b\u3068\uff08\u306a\u305c\u5149\u306e\u3068\u304d\u306e\u3088\u3046\u306b \\(\\displaystyle \\frac{1}{r} \\equiv u\\) \u3068\u3057\u306a\u3044\u306e\u304b\u306f\u5f8c\u3067\u308f\u304b\u3063\u3066\u304f\u308b… \uff09\uff0c<\/p>\n \\begin{eqnarray} \u4ee5\u4e0a\u3092\u307e\u3068\u3081\u308b\u3068\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u3068\u3057\u3066\u7528\u9014\u306b\u5fdc\u3058\u30662\u3064\u306e\u8868\u793a\u3092\u6e96\u5099\u3057\u3066\u304a\u304f\u3053\u3068\u3068\u3059\u308b\u3002<\/p>\n \u4e00\u3064\u76ee\u306f\uff0c\\begin{eqnarray} \u3082\u3046\u4e00\u3064\u306f\uff0c\u3053\u308c\u3092$$\\frac{dr}{d\\tau} = \\frac{d\\phi}{d\\tau} \\frac{dr}{d\\phi} = \\frac{\\ell}{r^2} \\frac{dr}{d\\phi}, \\quad\\frac{1}{r} \\equiv s$$ \u3069\u3061\u3089\u306e\u5f0f\u3082\uff0c\u7403\u5bfe\u79f0\u6027\u304b\u3089\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u8d64\u9053\u9762\u4e0a\u306b\u904b\u52d5\u3092\u5236\u9650\u3067\u304d\u308b\u3053\u3068\u3092\u5229\u7528\u3057\u3066\uff0c\n
\n\u3053\u3053\u3067 \\(\\displaystyle r_g \\equiv \\frac{2 G M}{c^2} \\) \u306f\u539f\u70b9\u306b\u3042\u308b\u4e2d\u5fc3\u5929\u4f53\u306e\u8cea\u91cf $M$ \u306b\u3088\u3063\u3066\u5b9a\u307e\u308b\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84<\/strong><\/span>\u307e\u305f\u306f\u91cd\u529b\u534a\u5f84<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u5b9a\u6570\u3067\u3042\u308a\uff0c\u4e2d\u5fc3\u5929\u4f53\u304b\u3089\u306e\u8ddd\u96e2 $r$ \u304c\u91cd\u529b\u534a\u5f84 $r_g$ \u306b\u8fd1\u304f\u306a\u308b\u3068\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u91cd\u529b\u306e\u52b9\u679c\u304c\u6975\u3081\u3066\u9855\u8457\u306b\u306a\u3063\u3066\u304f\u308b\uff0c\u305d\u3093\u306a\u9577\u3055\u3092\u3042\u3089\u308f\u3059\u3093\u3067\u3059\u3088\u3002<\/p>\n
\n$$x^{\\nu} = (x^0, x^1, x^2, x^3) = (c t, r, \\theta, \\phi)$$
\n\u304a\u3088\u3073\uff0c\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb<\/strong><\/span> \\(g_{\\mu\\nu}\\) \u306e\u30bc\u30ed\u3067\u306a\u3044\u6210\u5206\u306f
\n\\begin{eqnarray}
\ng_{00} &=& -\\left(1-\\frac{r_g}{r}\\right) \\\\
\ng_{11} &=& \\frac{1} {1-\\dfrac{r_g}{r}}\\\\
\ng_{22} &=& r^2\\\\
\ng_{33} &=& r^2 \\sin^2\\theta
\n\\end{eqnarray}
\n\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/h3>\n
\n$$\\frac{d u_{\\nu}}{d\\tau} = \\frac{1}{2} g_{\\lambda\\mu, \\nu} u^{\\lambda} u^{\\mu}$$<\/p>\n
\n$$ g_{\\lambda\\mu, \\nu} = 0 \\quad\\Rightarrow\\quad \\frac{d u_{\\nu}}{d\\tau} = 0 \\quad\\Rightarrow\\quad\u00a0 u_{\\nu} = \\mbox{const.} $$\u3068\u306a\u308a \\(u_{\\nu} \\) \u6210\u5206\u304c\u4fdd\u5b58\u91cf\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n
\n\u3067\u3042\u3063\u305f\u3002<\/p>\n\\(u^0\\) \u306e\u89e3<\/h4>\n
u_0 = g_{0\\mu} u^{\\mu} &=& g_{00} u^0 = \\mbox{const.} \\equiv -\\epsilon c\\\\
\\therefore \\ \\ u^0 &=& \\frac{c\\,dt}{d\\tau} = \\frac{-\\epsilon c}{g_{00}} = \\frac{\\epsilon c}{1 – \\dfrac{r_g}{r}} \\\\
\n\\therefore \\ \\ \\frac{dt}{d\\tau} &=& \\frac{\\epsilon}{1 – \\dfrac{r_g}{r}}\u00a0
\\end{eqnarray}<\/p>\n\\(u^3\\) \u306e\u89e3<\/h4>\n
u_3 = g_{3\\mu} u^{\\mu} &=& g_{33} u^3 = \\mbox{const.} \\equiv \\ell \\\\
\\therefore \\ \\ u^3 &=& \\frac{\\ell}{g_{33}} = \\frac{\\ell}{r^2 \\sin^2\\theta}
\\end{eqnarray}<\/p>\n\\(u^2\\) \u306f\u521d\u671f\u6761\u4ef6\u304b\u3089<\/h4>\n
\n$$\\displaystyle \\theta(0) = \\frac{\\pi}{2}, \\quad \\frac{d\\theta}{d\\tau}\\Biggr|_0 = 0 $$
\n\u3068\u3059\u308b\u3068\uff0c\\(\\displaystyle \\frac{d}{d\\tau}\\frac{d\\theta}{d\\tau}\\Biggr|_{0} = 0 \\) \u3068\u306a\u308a\uff0c\u5e38\u306b \\( \\displaystyle \\frac{d\\theta}{d\\tau} = 0 \\) \u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\uff0c\u3053\u306e\u521d\u671f\u6761\u4ef6<\/strong><\/span>\u3092\u63a1\u7528\u3057\uff0c
\n$$ \\theta = \\frac{\\pi}{2}, \\quad u^2 = \\frac{d\\theta}{d\\tau} = 0$$\u3068\u3059\u308b\u3002\u3053\u306e\u3053\u3068\u306f<\/p>\n
\n<\/strong>\u8d64\u9053\u9762\u4e0a \\( \\displaystyle \\theta = \\frac{\\pi}{2}\\) \u306b\u904b\u52d5\u3092\u5236\u9650\u3067\u304d\u308b<\/strong><\/span><\/p>\n\\(u^1\\) \u306f\u898f\u683c\u5316\u6761\u4ef6\u304b\u3089<\/h3>\n
u^3 = \\frac{\\ell}{r^2 \\sin^2\\theta} = \\frac{\\ell}{r^2}$$<\/p>\n
-c^2 &=& g_{\\mu\\nu} u^{\\mu} u^{\\nu} \\\\
&=& g_{00} \\left(u^0\\right)^2 + g_{11} \\left(u^1\\right)^2 + g_{33} \\left(u^3\\right)^2 \\\\
&=& – \\left(1-\\dfrac{r_g}{r}\\right) \\left( \\dfrac{\\epsilon c}{1 -\\dfrac{r_g}{r}}\\right)^2 + \\frac{1}{1 -\\dfrac{r_g}{r}}\\left( \\frac{dr}{d\\tau}\\right)^2 + r^2 \\left( \\frac{\\ell}{r^2}\\right)^2 \\\\ \\ \\\\
\n\\therefore\\ \\left( \\frac{dr}{d\\tau} \\right)^2 &=& \\epsilon^2 c^2 -c^2\u00a0 + \\frac{r_g c^2}{r}\u00a0 -\\frac{\\ell^2}{r^2}
\n+r_g\\frac{\\ell^2}{r^3}\\\\
\n&=& \\epsilon^2 c^2 -c^2\u00a0 + \\frac{2 G M}{r}\u00a0 -\\frac{\\ell^2}{r^2}
\n+r_g\\frac{\\ell^2}{r^3}
\\end{eqnarray}<\/p>\n
\n\\left( \\frac{ds}{d\\phi} \\right)^2 &=& \\frac{\\epsilon^2 c^2 -c^2}{\\ell^2} + \\frac{2GM}{\\ell^2} s -s^2 + r_g\\, s^3\\\\
\n&=&\\frac{\\epsilon^2 c^2 -c^2}{\\ell^2} +\u00a0 \\left(\\frac{GM}{\\ell^2}\\right)^2 -\\left(s -\\frac{GM}{\\ell^2} \\right)^2+ r_g\\, s^3
\n\\end{eqnarray}<\/p>\n\u307e\u3068\u3081\uff1a\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f<\/h3>\n
\n\\left( \\frac{dr}{d\\tau} \\right)^2 &=&\\epsilon^2 c^2 -c^2\u00a0 + \\frac{2 G M}{r}\u00a0 -\\frac{\\ell^2}{r^2}
\n+r_g\\frac{\\ell^2}{r^3}
\n\\end{eqnarray}<\/p>\n
\n\u3068\u3057\u3066\u5909\u5f62\u3057\u305f\u5f0f
\n$$\\left( \\frac{ds}{d\\phi} \\right)^2 =\\frac{1}{b^2}\u00a0 -\\left(s -\\frac{GM}{\\ell^2} \\right)^2+ r_g\\, s^3$$
\n\u3053\u3053\u3067\uff0c
\n$$\\frac{1}{b^2} \\equiv \\frac{\\epsilon^2 c^2 -c^2}{\\ell^2} +\u00a0 \\left(\\frac{GM}{\\ell^2}\\right)^2$$
\n\u3068\u304a\u3044\u305f\u3002\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f<\/a>\u3068\u306e\u985e\u4f3c\u6027\u306b\u522e\u76ee\u305b\u3088\u3002<\/p>\n
\n$$\\theta = \\frac{\\pi}{2}$$\u3068\u3057\u3066\u3044\u308b\u3002\u307e\u305f\uff0c\u65b9\u7a0b\u5f0f\u306e\u4e2d\u306b\u73fe\u308c\u308b\u5b9a\u6570 \\(\\epsilon, \\ell\\) \u306b\u3064\u3044\u3066\u306f\u65e2\u306b\u4e0a\u3067\u66f8\u3044\u3066\u3044\u308b\u3088\u3046\u306b
\n$$u^0 = \\frac{c dt}{d\\tau} = \\frac{\\epsilon c}{1 – \\dfrac{r_g}{r}}, \\quad u^3 = \\frac{d\\phi}{d\\tau} = \\frac{\\ell}{r^2}$$<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":85,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1012","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1012","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=1012"}],"version-history":[{"count":32,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1012\/revisions"}],"predecessor-version":[{"id":10209,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1012\/revisions\/10209"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/85"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=1012"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}