\u7d50\u5c40\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u3092\u904b\u52d5\u3059\u308b\u6642\u8a08\u304c\u793a\u3059\u6642\u9593\u306e\u9045\u308c\u306f\u5168\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7d71\u4e00\u7684\u306b\u7406\u89e3\u3067\u304d\u308b\u3002\u6955\u5186\u8ecc\u9053\u4e0a\u3092\u904b\u52d5\u3059\u308b\u5834\u5408\u306e\u6642\u9593\u306e\u9045\u308c\u306b\u3064\u3044\u3066\u3082\u307e\u3068\u3081\u3066\u307f\u305f\u3002<\/p>\n
<\/p>\n
\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306e\u7403\u5bfe\u79f0\u771f\u7a7a\u89e3\u3067\u3042\u308b\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/strong><\/span>\u306e\u7dda\u7d20\u306f<\/p>\n $$ds^2 = -c^2 d\\tau^2 = -\\left(1-\\frac{r_g}{r}\\right) c^2 dt^2 + \\frac{dr^2}{1-\\frac{r_g}{r}} \u3053\u3053\u3067\uff0c$\\tau$ \u306f\u56fa\u6709\u6642\u9593<\/strong><\/span>\uff0c\u3057\u305f\u304c\u3063\u3066 $d\\tau$ \u306f\u7d4c\u904e\u56fa\u6709\u6642\u9593\uff0c$\\displaystyle r_g \\equiv \\frac{2GM}{c^2}$ \u306f\u91cd\u529b\u534a\u5f84<\/strong><\/span>\uff08\u307e\u305f\u306f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84<\/strong><\/span>\uff09\u3002\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3\u304c\u3069\u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u3089\u308c\u308b\u304b\u306f\uff0c\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\uff1a<\/p>\n \u4e16\u306b\u3042\u307e\u305f\u3042\u308b\u6559\u79d1\u66f8\u3067\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u6642\u9593\u306e\u9045\u308c\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u306b\u3082\u66f8\u3044\u3066\u3044\u308b\u3002\uff08\u5186\u8ecc\u9053\u306e\u5834\u5408\u306e\u307f\u3002\uff09<\/p>\n \u304b\u3044\u3064\u307e\u3093\u3067\u307e\u3068\u3081\u308b\u3068\uff0c$d\\tau$ \u304c\u91cd\u529b\u5834\u4e2d\u3092\u904b\u52d5\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u7d4c\u904e\u56fa\u6709\u6642\u9593\u3067\u3042\u308a\uff0c\u3053\u308c\u3068\u7d4c\u904e\u5ea7\u6a19\u6642\u9593 $dt$ \uff08\u3053\u308c\u306f\u307e\u305f\u9060\u65b9\u89b3\u6e2c\u8005\u306e\u7d4c\u904e\u6642\u9593\u3068\u547c\u3093\u3067\u3082\u3088\u3044\uff09\u3068\u306e\u95a2\u4fc2\u306f $dr = 0, \\theta = \\frac{\\pi}{2}, d\\theta = 0$ \u306e\u5834\u5408\u306b<\/p>\n $$d\\tau = dt\\, \\sqrt{1 -\\frac{r_g}{r} -\\frac{1}{c^2} \\left(r \\frac{d\\phi}{dt}\\right)^2}$$<\/p>\n \u3068\u306a\u308b\u3002\u6642\u8a08\u304c\u9759\u6b62\u3057\u3066\u3044\u308b\u5834\u5408\u306b\u306f $d\\phi = 0$ \u3060\u304b\u3089<\/p>\n $$d\\tau = dt \\sqrt{1 -\\frac{r_g}{r} }$$<\/p>\n \u307e\u305f\uff0c\u5186\u8ecc\u9053\u4e0a\u3092\u904b\u52d5\u3057\u3066\u3044\u308b\u5834\u5408\u306b\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u3068\u9060\u5fc3\u529b\u3068\u306e\u91e3\u308a\u5408\u3044\u306e\u5f0f<\/p>\n \\begin{eqnarray} \u3092\u4f7f\u3063\u3066\uff0c<\/p>\n \\begin{eqnarray} \u3068\u306a\u308b\u3002\u3057\u304b\u3057\uff0c\u3053\u306e\u8aac\u660e\u65b9\u6cd5\u306f\u7c21\u5358\u3067\u306f\u3042\u308b\u304c\uff0c\u3088\u308a\u4e00\u822c\u7684\u306a\u904b\u52d5\u3092\u3057\u3066\u3044\u308b\u5834\u5408\uff0c\u4f8b\u3048\u3070\u6955\u5186\u8ecc\u9053\u306e\u5834\u5408\u306b\u306f\u305d\u306e\u307e\u307e\u3067\u306f\u4f7f\u3048\u306a\u3044\uff08\u53b3\u5bc6\u306b\u306f\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u306b\u306f\u9589\u3058\u305f\u6955\u5186\u8ecc\u9053\u3068\u306f\u306a\u3089\u306a\u3044\u306e\u3067\u3042\u308b\u304c… \uff09\u3002\u305d\u3053\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u91cd\u529b\u5834\u4e2d\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u4efb\u610f\u306e\u8ecc\u9053\u904b\u52d5\u306e\u5834\u5408\u306b\u3082\u6642\u9593\u306e\u9045\u308c\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306a\u7d71\u4e00\u7684\u306a\u65b9\u6cd5\u3092\u4ee5\u4e0b\u306b\u307e\u3068\u3081\u308b\u3002<\/p>\n \u91cd\u529b\u4ee5\u5916\u306e\u529b\u3092\u53d7\u3051\u305a\u306b\u91cd\u529b\u5834\u4e2d\u3092\u904b\u52d5\u3059\u308b\u7269\u4f53\u306e\u8ecc\u9053\uff08\u4e16\u754c\u7dda\uff09\u306f\uff0c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u7269\u4f53\u306e\u4e16\u754c\u7dda\u3092 $x^{\\mu}(\\tau)$\uff0c\u305d\u306e\u63a5\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b4\u5143\u901f\u5ea6<\/strong><\/span>\u3092 $u^{\\mu}$ \u3068\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f<\/p>\n $$\\frac{d u^{\\lambda}}{d\\tau} + \\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\nu} u^{\\mu} u^{\\nu} = 0$$<\/p>\n 4\u5143\u901f\u5ea6\u306e\u300c\u5171\u5909\u300d\u6210\u5206 $u_{\\mu} \\equiv g_{\\mu\\nu} u^{\\nu}$ \u306b\u5bfe\u3057\u3066\u306f\uff0c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n $$\\frac{d u_{\\mu}}{d\\tau} = \\frac{1}{2} \\frac{\\partial g_{\\alpha\\beta}}{\\partial x^{\\mu}} u^{\\alpha} u^{\\beta} \u3053\u306e\u3078\u3093\u306e\u4e8b\u60c5\u306f\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u306b\u307e\u3068\u3081\u3066\u3042\u308b\u3002<\/p>\n \u6642\u7a7a\u306e\u5bfe\u79f0\u6027\u306b\u3088\u308a\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u8ecc\u9053\u3092\u300c\u8d64\u9053\u9762\u300d\u4e0a\uff0c\u3059\u306a\u308f\u3061 $\\displaystyle \\theta = \\frac{\\pi}{2}$ \u306b\u5236\u9650\u3067\u304d\u308b<\/strong><\/span>\u306e\u3067\uff0c\u4ee5\u5f8c\u306f<\/p>\n $$\\theta = \\frac{\\pi}{2}, \\quad u^2 = \\frac{d\\theta}{d\\tau} = 0$$<\/p>\n \u3068\u3059\u308b\u3002\u306a\u3093\u3067\u305d\u3046\u306a\u308b\u304b\uff0c\u3069\u3046\u3057\u3066\u3082\u77e5\u308a\u305f\u3044\u306e\u306a\u3089\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\uff1a<\/p>\n <\/p>\n \u30e1\u30c8\u30ea\u30c3\u30af\u306f $x^0 = c t$ \u3092\u542b\u307e\u306a\u3044\u306e\u3067\uff0c\u305f\u3060\u3061\u306b<\/p>\n \\begin{eqnarray} \u3053\u306e $(1)$ \u5f0f\u304c\uff0c\u91cd\u529b\u5834\u4e2d\u306e\u6642\u8a08\u306e\u56fa\u6709\u6642\u9593\u306e\u7d4c\u904e $d\\tau$ \u3068\uff0c\u5ea7\u6a19\u6642\u9593\u306e\u7d4c\u904e $dt$ \u3068\u306e\u95a2\u4fc2\u3092\u4e0e\u3048\u308b\u3002<\/p>\n \u30e1\u30c8\u30ea\u30c3\u30af\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u5341\u5206\u9060\u65b9 $r \\gg r_g$ \u3067\u306f<\/p>\n $$ds^2 \\simeq -c^2 dt^2 + dr^2 + r^2 \\left( d\\theta^2 + \\sin^2 \\theta\\,d\\phi^2\\right)$$<\/p>\n \u306a\u306e\u3067\uff0c\u5ea7\u6a19\u6642\u9593\u306e\u7d4c\u904e $dt$ \u306f\u9060\u65b9\u89b3\u6e2c\u8005\u306b\u3068\u3063\u3066\u306e\u7d4c\u904e\u6642\u9593\u3068\u547c\u3093\u3067\u3082\u3088\u3044\u3002<\/p>\n \u6b21\u306b\uff0c\u30e1\u30c8\u30ea\u30c3\u30af\u306f $x^3 = \\phi$ \u3082\u542b\u307e\u306a\u3044\u306e\u3067<\/p>\n \\begin{eqnarray} \u3053\u3053\u3067 $\\displaystyle \\theta = \\frac{\\pi}{2}$ \u3067\u3042\u308b\u304b\u3089 $g_{33} = r^2 \\sin^2\\theta = r^2$ \u3068\u3057\u305f\u3002<\/p>\n \u6700\u5f8c\u306b\uff0c4\u5143\u901f\u5ea6\u306e\u898f\u683c\u5316\u6761\u4ef6\u304b\u3089<\/p>\n \\begin{eqnarray} \u4ee5\u4e0a\u306e\u7d50\u679c\u3092\u3042\u3089\u305f\u3081\u3066\u307e\u3068\u3081\u308b\u3068\uff0c\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58\u30fb\u89d2\u904b\u52d5\u91cf\u4fdd\u5b58\u3092\u3042\u3089\u308f\u3059\u904b\u52d5\u306e\u5b9a\u6570 $\\epsilon, \\ \\ell$ \u3092\u4f7f\u3063\u3066\uff0c<\/p>\n \\begin{eqnarray} \u3042\u308b\u77ac\u9593\u306b\u91cd\u529b\u5834\u4e2d\u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u6642\u8a08\u3092\u8003\u3048\u308b\u3068\uff0c\u305d\u306e\u6642\u523b\u306b\u306f<\/p>\n $$\\frac{dr}{d\\tau} = 0, \\quad \\frac{d\\phi}{d\\tau} = 0, \\ \\therefore\\ \\ \\ell = 0$$<\/p>\n \u3057\u305f\u304c\u3063\u3066 $(3)$ \u5f0f\u3088\u308a<\/p>\n \\begin{eqnarray} \u3053\u308c\u3092 $(1)$ \u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n $$d\\tau = dt\\, \\sqrt{1 -\\frac{r_g}{r}}$$<\/p>\n \u3053\u308c\u304c\uff0c\u91cd\u529b\u5834\u4e2d\u306e\u52d5\u5f84\u5ea7\u6a19 $r$ \u306e\u4f4d\u7f6e\u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u7d4c\u904e\u56fa\u6709\u6642\u9593 $d\\tau$ \u3068\u9060\u65b9\u89b3\u6e2c\u8005\u306e\u7d4c\u904e\u6642\u9593 $dt$ \u3068\u306e\u95a2\u4fc2\u3092\u3042\u3089\u308f\u3059\u3002\u3053\u306e\u5f0f\u306f\u91cd\u529b\u8d64\u65b9\u504f\u79fb<\/strong><\/span>\u3068\u3057\u3066\u3088\u304f\u5f15\u7528\u3055\u308c\u308b\u3002<\/p>\n \u3057\u305f\u304c\u3063\u3066\uff0c\u7570\u306a\u308b\u5730\u70b9 $r = r_1$ \u3068 $r = r_2$ \u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u9032\u307f\u306e\u6bd4\u306f<\/p>\n $$\\frac{d\\tau_2}{d\\tau_1} = \\frac{\\sqrt{1 -\\frac{r_g}{r_2}}}{\\sqrt{1 -\\frac{r_g}{r_1}}}$$<\/p>\n \u3068\u306a\u308a\uff0c\u300c\u91cd\u529b\u5834\u4e2d\u306e\u7570\u306a\u308b\u5730\u70b9\u3067\u306e\u6642\u9593\u306e\u9032\u307f\u65b9<\/a>\u300d\u306e\u7d50\u679c\u3092\u518d\u73fe\u3057\u3066\u3044\u308b\u3002<\/p>\n \u52d5\u5f84\u65b9\u5411\u306e\u904b\u52d5\u306b\u9650\u308b\u306e\u3067<\/p>\n $$\\frac{d\\phi}{d\\tau} = 0, \\quad \\therefore\\ \\ \\ell = 0$$<\/p>\n \u3053\u308c\u3092\u4ee3\u5165\u3057\uff0c\u521d\u671f\u6761\u4ef6\u3092 $r=r_i$ \u3067\u521d\u901f\u5ea6\u3092\u30bc\u30ed\u3068\u3059\u308b\u3068\uff0c$(3)$ \u5f0f\u304b\u3089<\/p>\n \\begin{eqnarray} $r=r_i$ \u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u7d4c\u904e\u56fa\u6709\u6642\u9593 $d\\tau_i$ \u306f<\/p>\n $$d\\tau_i = dt\\,\\sqrt{1 -\\frac{r_g}{r_i}}$$<\/p>\n \u3067\u3042\u308b\u304b\u3089\uff0c\u6bd4\u3092\u53d6\u308b\u3068<\/p>\n $$\\frac{d\\tau}{\\, d\\tau_i} = \\frac{1 -\\frac{r_g}{r}}{1 -\\frac{r_g}{r_i} }$$<\/p>\n \u3068\u306a\u308a\uff0c\u300c\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b\u3059\u308b\u89b3\u6e2c\u8005\u306e\u6642\u9593\u306e\u9032\u307f\u65b9<\/a>\u300d\u306e\u7d50\u679c\u3092\u518d\u73fe\u3057\u3066\u3044\u308b\u3002<\/p>\n \u6709\u754c\u306a\u904b\u52d5\u3067\u3042\u308b\u3053\u3068\u304b\u3089<\/p>\n $$r_g < r_{\\rm min} \\leq r \\leq r_{\\rm max}$$<\/p>\n $r=r_{\\rm min}$ \u304a\u3088\u3073 $r=r_{\\rm max}$ \u3067 $r$ \u304c\u6975\u5024\u3092\u3068\u308b\u3053\u3068\u304b\u3089\uff0c$(3)$ \u5f0f\u3088\u308a<\/p>\n \\begin{eqnarray} \u3053\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092 $\\epsilon^2$ \u3068 $\\ell^2$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068<\/p>\n \\begin{eqnarray} $r_{\\rm min}, \\ r_{\\rm max}$ \u306e\u304b\u308f\u308a\u306b\uff0c<\/p>\n $$r_{\\rm max} \\equiv a (1+e), \\quad r_{\\rm min}\\equiv a (1-e)$$<\/p>\n \u3059\u306a\u308f\u3061<\/p>\n $$ a \\equiv \\frac{1}{2} \\left( r_{\\rm max} + r_{\\rm min}\\right) , \\quad \u3067\u5b9a\u7fa9\u3055\u308c\u308b $a, \\ e$ \u3092\u4f7f\u3063\u3066\u3042\u3089\u308f\u3059\u3068<\/p>\n \\begin{eqnarray} \u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6\u306e\u6955\u5186\u8ecc\u9053\u306e\u5834\u5408\u306b\u306f\uff0c$a$ \u306f\u8ecc\u9053\u9577\u534a\u5f84\uff0c$e$ \u306f\u96e2\u5fc3\u7387\u3068\u547c\u3070\u308c\u308b\u3002\u4e00\u822c\u76f8\u5bfe\u8ad6\u3067\u306f\uff0c\u8ecc\u9053\u306f\u9589\u3058\u305f\u6955\u5186\u8ecc\u9053\u3068\u306a\u3089\u306a\u3044\u306e\u3067\uff0c$a$ \u3084 $e$ \u306f\u8ecc\u9053\u9577\u534a\u5f84\u3084\u96e2\u5fc3\u7387\u3068\u547c\u3076\u3053\u3068\u306f\u306a\u3044\uff08\u304c\uff0c\u3064\u3044\u305d\u3046\u547c\u3093\u3067\u3057\u307e\u3046\u304b\u3082\u3057\u308c\u306a\u3044\uff09\u3002<\/p>\n $e=0, \\ a \\Rightarrow r$ \u3068\u3059\u308b\u3068\uff0c\u534a\u5f84 $r$ \u306e\u5186\u8ecc\u9053\u4e0a\u3092\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u7d4c\u904e\u56fa\u6709\u6642\u9593 $d\\tau$ \u306f<\/p>\n \\begin{eqnarray} $r = r_1$ \u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u89b3\u6e2c\u8005\u306e\u7d4c\u904e\u56fa\u6709\u6642\u9593 $d\\tau_1$ \u306f<\/p>\n $$d\\tau_1 = dt\\, \\sqrt{1 -\\frac{r_g}{r_1}}$$<\/p>\n \u3067\u3042\u308b\u304b\u3089\uff0c\u6bd4\u3092\u3068\u308b\u3068<\/p>\n $$\\frac{d\\tau}{d\\tau_1} = \\frac{\\sqrt{1 -\\frac{3}{2} \\frac{r_g}{r}}}{\\sqrt{1 -\\frac{r_g}{r_1}}}$$<\/p>\n \u3068\u306a\u308a\uff0c\u300c\u5186\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306e\u6642\u9593\u306e\u9032\u307f\u65b9<\/a>\u300d\u306e\u7d50\u679c\u3092\u518d\u73fe\u3057\u3066\u3044\u308b\u3002<\/p>\n \u5186\u8ecc\u9053\u3067\u306a\u3044\u4e00\u822c\u7684\u306a\u675f\u7e1b\u8ecc\u9053\u4e0a\u3092\u904b\u52d5\u3057\u3066\u3044\u308b\u5834\u5408\u3067\u3082\uff0c\u6642\u9593\u306e\u9045\u308c\u306f\u8fd1\u4f3c\u7121\u3057\u306e\u53b3\u5bc6\u89e3\u3068\u3057\u3066<\/p>\n \\begin{eqnarray} \u3068\u66f8\u3051\u308b\u3002\u91cd\u529b\u5834\u4e2d\u3092\u904b\u52d5\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u9045\u308c $d\\tau$ \u306f\u6642\u3005\u523b\u3005\u5909\u5316\u3059\u308b $r$ \u3068\u3044\u3046\u5834\u6240\u306e\u95a2\u6570\u3068\u306a\u308b\u3002\u307e\u305f\uff0c\u6955\u5186\u8ecc\u9053\u3067\u3044\u3046\u3068\u3053\u308d\u306e\u8ecc\u9053\u9577\u534a\u5f84 $a$ \u3060\u3051\u3067\u306a\u304f\uff0c\u96e2\u5fc3\u7387\u306b\u76f8\u5f53\u3059\u308b $e$ \u306b\u3082\u4e00\u822c\u306b\u4f9d\u5b58\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n \u4e0a\u8a18\u306e\u3088\u3046\u306b\uff0c\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u306a\u304f\u53b3\u5bc6\u89e3\u304c\u6c42\u307e\u308b\u306e\u306f\u7d50\u69cb\u306a\u3053\u3068\u3060\u304c\uff0c$\\displaystyle 0 < \\frac{r_g}{a} \\ll 1$ \u3068\u3057\u3066 $\\epsilon$ \u3092\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3059\u308b\u3068<\/p>\n \\begin{eqnarray} $\\displaystyle \\left(\\frac{\\,r_g}{a}\\right)^2$ \u307e\u3067\u306e\u8fd1\u4f3c\u3067\u306f\uff0c$\\epsilon$ \u306f $e$ \u306b\u3088\u3089\u305a $a$ \u3060\u3051\u3067\u66f8\u3051\u308b\u3053\u3068\u306f\u5b9f\u306b\u8208\u5473\u6df1\u3044\u3002<\/p>\n \u4e00\u65b9\uff0c\u5186\u8ecc\u9053\u306e\u5834\u5408\u306e\u53b3\u5bc6\u89e3\u3082\u540c\u69d8\u306b\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \u3067\u3042\u308b\u304b\u3089\uff0c$\\displaystyle 0 < \\frac{r_g}{a} \\ll 1$ \u306e\u5834\u5408\u306b\u306f\uff0c$\\displaystyle \\left(\\frac{\\,r_g}{a}\\right)^2$ \u307e\u3067\u306e\u8fd1\u4f3c\u3067<\/p>\n $$\\epsilon \\simeq \\frac{1 -\\frac{\\,r_g}{a}}{\\sqrt{1 -\\frac{3}{2} \\frac{\\,r_g}{a}}}$$<\/p>\n \u3068\u3057\u3066\u3088\u3044\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n \u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u5186\u8ecc\u9053\u3067\u306a\u3044\u5834\u5408\u306e\u6642\u9593\u306e\u9045\u308c\u306f\uff0c$\\displaystyle 0 < \\frac{r_g}{a} \\ll 1$ \u3067\u3042\u308c\u3070<\/p>\n $$d\\tau = \\frac{1 -\\frac{\\,r_g}{r}}{\\epsilon} \\simeq dt\\,\u00a0 \\frac{1 -\\frac{\\,r_g}{r}}{1 -\\frac{\\,r_g}{a}}\\ \\sqrt{1 -\\frac{3}{2} \\frac{\\,r_g}{a}} $$<\/p>\n \u3068\u3057\u3066\u3088\u3044\u3053\u3068\u306b\u306a\u308b\u3002\u3053\u306e\u5f0f\u306f $\\displaystyle \\left(\\frac{\\,r_g}{a}\\right)^2$ \u307e\u3067\u306e\u7cbe\u5ea6\u3067\u6b63\u3057\u3044\u3002<\/p>\n \u5186\u8ecc\u9053\u306e\u5834\u5408\u3068\u7570\u306a\u308a\uff0c\u4e00\u822c\u306b\u5186\u8ecc\u9053\u3067\u306a\u3044\u5834\u5408\u306f $r$ \u306f\u6642\u3005\u523b\u3005\u3068\u5909\u5316\u3059\u308b\u305f\u3081\uff0c\u6642\u9593\u306e\u9045\u308c\u3082\u6642\u3005\u523b\u3005\u3068\u5909\u5316\u3059\u308b\u3002\u3057\u304b\u3057\uff0c$r_g$ \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067\u3088\u3044\u306e\u3067\u3042\u308c\u3070\uff0c\u6642\u9593\u306e\u9045\u308c\u306e\u5f0f\u306b\u306f $r_g\/r$ \u306e\u5f62\u3067\u65e2\u306b $r_g$ \u306e1\u6b21\u306e\u9805\u304b\u3089\u306f\u3058\u307e\u3063\u3066\u3044\u308b\u306e\u3067\uff0c$r$ \u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6\u306e\u6955\u5186\u8ecc\u9053<\/p>\n $$r_{\\scriptscriptstyle\\rm N} = \\frac{a\\, (1 -e^2)}{1 + e\\,\\cos\\phi}$$<\/p>\n \u3068\u3057\u3066\u3088\u3044\u3002\u3053\u306e\u5834\u5408\u306f\uff0c$\\displaystyle \\frac{1}{\\ r_{\\scriptscriptstyle\\rm N}}$ \u306e1\u5468\u671f $T$ \u3042\u305f\u308a\u306e\u6642\u9593\u5e73\u5747\u304c<\/p>\n \\begin{eqnarray} \u3068\u306a\u308b\u3053\u3068\u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u306a\u3093\u3067\u5e73\u5747\u304c\u3053\u3046\u306a\u308b\u304b\u306f\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\uff1a<\/p>\n
\n+ r^2 \\left( d\\theta^2 + \\sin^2 \\theta\\,d\\phi^2\\right)$$<\/p>\n\n
\u4e16\u306e\u4e2d\u306e\u6559\u79d1\u66f8\u3067\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u6642\u9593\u306e\u9045\u308c<\/h3>\n
\n
\nv &\\equiv& r \\frac{d\\phi}{dt} \\\\
\nm \\frac{v^2}{r} &=& \\frac{G M m}{r^2} \\\\
\n\\therefore\\ \\\u00a0 \\left(r \\frac{d\\phi}{dt} \\right)^2 &=& \\frac{G M}{r}
\n\\end{eqnarray}<\/p>\n
\nd\\tau &=& dt\\, \\sqrt{1 -\\frac{r_g}{r} -\\frac{1}{c^2} \\left(r \\frac{d\\phi}{dt}\\right)^2} \\\\
\n&=& dt\\, \\sqrt{1 -\\frac{2 G M}{c^2 r} -\\frac{1}{c^2}\\frac{G M}{r} } \\\\
\n&=& dt \\, \\sqrt{1 -\\frac{3}{2} \\frac{r_g}{r}}
\n\\end{eqnarray}<\/p>\n\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/h3>\n
\nu^{\\mu} = \\frac{dx^{\\mu}}{d\\tau} &=&
\n\\left(\\frac{\\,dx^{0}}{d\\tau}, \\frac{dx^{1}}{d\\tau}, \\frac{dx^{2}}{d\\tau}, \\frac{dx^{3}}{d\\tau} \\right) \\\\
\n&=& \\left(\\frac{c\\, dt}{d\\tau}, \\frac{dr}{d\\tau}, \\frac{d\\theta}{d\\tau}, \\frac{d\\phi}{d\\tau} \\right)
\n\\end{eqnarray}<\/p>\n
\n\\equiv\\frac{1}{2}g_{\\alpha\\beta, \\mu} u^{\\alpha} u^{\\beta} $$<\/p>\n\n
\n
\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306e\u89e3<\/h3>\n
\n
\n\\frac{d u_0}{d\\tau} &=& 0 \\\\
\n\\therefore\\ \\ u_0 &=& g_{00} u^0 \\\\
\n&=& -\\left(1-\\frac{r_g}{r}\\right) \\frac{c\\,dt}{d\\tau} = \\mbox{const.} \\equiv -\\epsilon \\,c \\\\
\n\\therefore\\ \\ d\\tau &=& dt \\,\\frac{1-\\frac{r_g}{r}}{\\epsilon} \\tag{1}
\n\\end{eqnarray}<\/p>\n
\n\\frac{d u_3}{d\\tau} &=& 0 \\\\
\n\\therefore\\ \\ u_3 &=& g_{33} u^3 \\\\
\n&=& r^2 \\frac{d\\phi}{d\\tau} = \\mbox{const.} \\equiv \\ell \\tag{2}
\n\\end{eqnarray}<\/p>\n
\n-c^2 &=& g_{\\mu\\nu} u^{\\mu} u^{\\nu} \\\\
\n&=& g_{00} \\left(u^0\\right)^2 + g_{11} \\left(u^1\\right)^2 + g_{33} \\left( u^3 \\right)^2 \\\\
\n&=& \\frac{1}{g_{00} }\\left(u_0\\right)^2 + g_{11} \\left(u^1\\right)^2 + \\frac{1}{g_{33}}\\left( u_3 \\right)^2\\\\
\n&=& -\\frac{1}{1-\\frac{r_g}{r}} \\left( -\\epsilon \\,c\\right)^2 + \\frac{1}{1-\\frac{r_g}{r}} \\left(\\frac{dr}{d\\tau}\\right)^2 + \\frac{\\ell^2}{r^2} \\\\
\n\\therefore\\ \\\u00a0 \\left(\\frac{dr}{d\\tau}\\right)^2 &=& \\epsilon^2 c^2 -\\left( 1-\\frac{r_g}{r}\\right)\\left(c^2 + \\frac{\\ell^2}{r^2} \\right) \\tag{3}
\n\\end{eqnarray}<\/p>\n\u307e\u3068\u3081<\/h4>\n
\nd\\tau &=& dt \\,\\frac{1 -\\frac{r_g}{r}}{\\epsilon} \\tag{1} \\\\
\nr^2 \\frac{d\\phi}{d\\tau} &=& \\ell\u00a0 \\tag{2} \\\\
\n\\left(\\frac{dr}{d\\tau}\\right)^2 &=& \\epsilon^2 c^2 -\\left( 1 -\\frac{r_g}{r}\\right)\\left(c^2 + \\frac{\\ell^2}{r^2} \\right) \\tag{3}
\n\\end{eqnarray}<\/p>\n\u9759\u6b62\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u9045\u308c<\/h3>\n
\n0 &=& \\epsilon^2 c^2 -\\left( 1 -\\frac{r_g}{r}\\right) c^2 \\\\
\n\\therefore\\ \\ \\epsilon &=& \\sqrt{1 -\\frac{r_g}{r}}
\n\\end{eqnarray}<\/p>\n\u52d5\u5f84\u65b9\u5411\u306b\u81ea\u7531\u843d\u4e0b\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u9045\u308c<\/h3>\n
\n0 &=& \\epsilon^2 c^2 -\\left( 1 -\\frac{r_g}{r_i}\\right) c^2 \\\\
\n\\therefore\\ \\ \\epsilon &=& \\sqrt{1 -\\frac{r_g}{r_i}} \\\\
\n\\therefore\\ \\ d\\tau &=& dt\\, \\frac{1 -\\frac{r_g}{r}}{\\sqrt{1 -\\frac{r_g}{r_i}} }
\n\\end{eqnarray}<\/p>\n\u6709\u754c\u306a\u675f\u7e1b\u8ecc\u9053\u4e0a\u3092\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u9045\u308c<\/h3>\n
\n0 &=& \\epsilon^2 c^2 -\\left(1 \u2013\\frac{r_g}{r_{\\rm min}} \\right) \\left( c^2+ \\frac{\\ell^2}{r_{\\rm min}^2}\\right) \\\\
\n0 &=& \\epsilon^2 c^2 -\\left(1 -\\frac{r_g}{r_{\\rm max}} \\right) \\left( c^2+ \\frac{\\ell^2}{r_{\\rm max}^2}\\right) \\end{eqnarray}<\/p>\n
\n\\epsilon^2&=& \\frac{ (r_{\\rm min} + r_{\\rm max}) (r_{\\rm min} -r_g) (r_{\\rm max} -r_g)}
\n{r_{\\rm min} r_{\\rm max} (r_{\\rm min} + r_{\\rm max}) -(r_{\\rm min}^2 + r_{\\rm min}r_{\\rm max}+r_{\\rm max}^2) r_g}\\\\
\n\\ell^2&=& \\frac{c^2 r_{\\rm min}^2\u00a0 r_{\\rm max}^2\u00a0 r_g}
\n{r_{\\rm min} r_{\\rm max} (r_{\\rm min} + r_{\\rm max}) -(r_{\\rm min}^2 + r_{\\rm min}r_{\\rm max}+r_{\\rm max}^2) r_g}
\n\\end{eqnarray}<\/p>\n
\ne \\equiv \\frac{ r_{\\rm max} -r_{\\rm min}}{ r_{\\rm max} +r_{\\rm min}}$$<\/p>\n
\n\\epsilon^2&=& \\frac{2a^2(1 -e^2) \u20134 a r_g + 2r_g^2}{2a^2(1 -e^2) \u2013a(3+e^2) r_g} \\\\
\n&=&1\u00a0 \u2013\\frac{r_g}{2 a} + \\frac{ (1 -e^2) r_g^2 }{4a^2(1 -e^2)\u00a0 \u20132a(3+e^2) r_g} \\\\
\n\\ell^2&=&c^2\u00a0 \\frac{a^2 (1 -e^2)^2 r_g}{2 a (1 -e^2) \u2013(3+e^2) r_g} \\\\
\n&=& GM a (1 -e^2) \\left( 1 + \\frac{(3+e^2) r_g}{2a(1 -e^2) \u2013(3+e^2) r_g}\\right)
\n\\end{eqnarray}<\/p>\n\u7279\u306b\u5186\u8ecc\u9053\u306e\u5834\u5408<\/h4>\n
\n\\epsilon^2 &=& \\frac{2 r^2\u00a0 \u20134 r r_g + 2r_g^2}{2 r^2\u00a0 \u20133 r r_g} \\\\
\n&=& \\frac{\\left(1 -\\frac{r_g}{r}\\right)^2}{1 -\\frac{3}{2} \\frac{r_g}{r}}\\\\
\n\\therefore\\ \\ \\epsilon &=& \\frac{1 -\\frac{r_g}{r}}{\\sqrt{1 -\\frac{3}{2} \\frac{r_g}{r}}} \\\\ \\ \\\\
\n\\therefore\\ \\ d\\tau &=& dt\\, \\frac{1 -\\frac{r_g}{r}}{\\epsilon} \\\\
\n&=& dt\\, \\sqrt{1 -\\frac{3}{2} \\frac{r_g}{r}}
\n\\end{eqnarray}<\/p>\n\u5186\u8ecc\u9053\u3067\u306a\u3044\u5834\u5408\u306b\u3082\u4e00\u822c\u306b\u8fd1\u4f3c\u306a\u3057\u3067<\/h4>\n
\nd\\tau &=& dt\\, \\frac{1 -\\frac{r_g}{r}}{\\epsilon} \\\\
\n\\epsilon &=& \\sqrt{1\u00a0 \u2013\\frac{r_g}{2 a} + \\frac{(1 -e^2) r_g^2\u00a0 }{4a^2(1 -e^2) \u20132a(3+e^2) r_g}}
\n\\end{eqnarray}<\/p>\n$r_g \\ll a$ \u306e\u5834\u5408\u306e\u8fd1\u4f3c\u89e3<\/h4>\n
\n\\epsilon &\\simeq& \\sqrt{1\u00a0 \u2013\\frac{r_g}{2 a} + \\frac{ r_g^2\u00a0 }{4a^2}} \\\\
\n&\\simeq& 1 -\\frac{1}{4} \\frac{\\,r_g}{a} + \\frac{3}{32} \\left(\\frac{\\,r_g}{a}\\right)^2+ \\cdots
\n\\end{eqnarray}<\/p>\n
\n\\epsilon_{\\rm circ} = \\frac{1 -\\frac{\\,r_g}{r}}{\\sqrt{1 -\\frac{3}{2} \\frac{\\,r_g}{r}}} &\\simeq&
\n1 -\\frac{1}{4} \\frac{\\,r_g}{r} + \\frac{3}{32} \\left(\\frac{\\,r_g}{r}\\right)^2+ \\cdots \\\\
\n\\end{eqnarray}<\/p>\n\uff08\u307b\u307c\uff09\u6955\u5186\u8ecc\u9053\u306e\u5834\u5408\u306e\u6642\u9593\u306e\u9045\u308c<\/h4>\n
\n\\left\\langle \\frac{1}{\\ r_{\\scriptscriptstyle\\rm N}} \\right\\rangle &\\equiv&
\n\\frac{1}{T} \\int_0^T \\, \\frac{1}{\\ r_{\\scriptscriptstyle\\rm N}}\\, dt \\\\
\n&=& \\frac{1}{a}
\n\\end{eqnarray}<\/p>\n\n