<\/p>\n
\u91cd\u529b\u4ee5\u5916\u306e\u529b\u3092\u53d7\u3051\u305a\u306b\u91cd\u529b\u5834\u4e2d\u3092\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\uff08\u30c6\u30b9\u30c8\u7c92\u5b50\uff09\u306f\u6e2c\u5730\u7dda\u4e0a\u3092\u904b\u52d5\u3059\u308b\u3002<\/p>\n
\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306f\u7403\u5bfe\u79f0\u3067\u3042\u308b\u305f\u3081\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u904b\u52d5\u3092\u8d64\u9053\u9762\u4e0a \\(\\displaystyle \\theta = \\frac{\\pi}{2}\\) \u306b\u5236\u9650\u3067\u304d\u308b\u3002\u3053\u306e\u6761\u4ef6\u306e\u3082\u3068\uff0c\u6e2c\u5730\u7dda\u3092\u89e3\u3044\u305f\u7d50\u679c\uff0c\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306e 4\u5143\u901f\u5ea6 \\(\\bar{u}^{\\mu}\\) \u306f<\/p>\n
$$\\bar{u}^{\\mu} = \\left(\\frac{\\epsilon}{1-\\frac{r_g}{r}},\u00a0 u^1, 0, \\frac{\\ell}{r^2}\\right)$$<\/p>\n
\u898f\u683c\u5316\u6761\u4ef6 \\(g_{\\mu\\nu} \\bar{u}^{\\mu} \\bar{u}^{\\nu} = -1\\) \u3088\u308a<\/p>\n
$$\\left(\\bar{u}^1\\right)^2 = \\left(\\frac{dr}{d\\tau}\\right)^2 = \\epsilon^2 -\\left( 1-\\frac{r_g}{r}\\right) \\left( 1 + \\frac{\\ell^2}{r^2}\\right)$$<\/p>\n
\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a<\/strong><\/span>\u306b\u304a\u3044\u3066\uff0c<\/p>\n \u3068\u3059\u308b\u3002<\/p>\n \u3053\u306e\u3068\u304d\uff0c\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u9032\u307f<\/span> \\(\\varDelta \\bar{\\tau}\\)<\/strong> \u3068\uff0c\u5730\u4e0a \\(r = r_1 ( = R)\\) \u306b\u9759\u6b62\u3057\u7d9a\u3051\u3066\u3044\u308b\u6642\u8a08\u306e\u9032\u307f<\/strong> <\/span>\\(\\varDelta \\tau_1\\) \u306e\u6bd4 \\(\\displaystyle \\frac{\\varDelta \\bar{\\tau}}{\\varDelta \\tau_1}\\) \u306f\u3069\u3046\u306a\u308b\u304b\uff0c\u3068\u3044\u3046\u8a71\u3002<\/p>\n \u307e\u305a\uff0c \u7570\u306a\u308b\u5730\u70b9\u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u9032\u307f\u65b9\u306e\u6bd4\u306f\uff0c\u65e2\u306b\u8aac\u660e\u3057\u305f\u3088\u3046\u306b<\/p>\n $$\\frac{\\varDelta \\tau}{\\varDelta \\tau_1} = \\frac{\\sqrt{1 -\\frac{r_g}{r}}}{\\sqrt{1 -\\frac{r_g}{r_1}}} $$<\/p>\n \u6b21\u306b\uff0c\u540c\u3058\u52d5\u5f84\u5ea7\u6a19 \\(r\\) \u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u9032\u307f \\(\\varDelta \\tau\\) \u3068\uff0c\u901f\u3055 \\(V\\) \u3067\u904b\u52d5\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u9032\u307f \\(\\varDelta \\bar{\\tau}\\) \u306f\uff0c\u7279\u6b8a\u76f8\u5bfe\u8ad6\u306e\u5834\u5408\u3068\u540c\u69d8\u306b\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u72b6\u6cc1\u4e0b\u306b\u304a\u3044\u3066\u3082\uff0c\u30ed\u30fc\u30ec\u30f3\u30c4\u56e0\u5b50<\/strong><\/span>\u306e\u9006\u6570\u3067\u3042\u308b \\(\\sqrt{1 -V^2}\\) \u3060\u3051\u7570\u306a\u308b\u306f\u305a\u3067\u3042\u308b\u3002\uff08\\(c = 1\\)\uff09<\/p>\n \\( u^{\\mu}\\) \u306e\u9759\u6b62\u89b3\u6e2c\u8005\u304b\u3089\u307f\u305f\uff0c\\( \\bar{u}^{\\mu}\\) \u3067\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306b\u95a2\u3059\u308b\u30ed\u30fc\u30ec\u30f3\u30c4\u56e0\u5b50<\/strong><\/span> \\(\\gamma\\) \u306f<\/p>\n \\begin{eqnarray} \u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n $$ \\frac{\\varDelta \\bar{\\tau}} {\\varDelta \\tau} = \\sqrt{1 -V^2} = \\frac{\\sqrt{1-\\frac{r_g}{r}}}{\\epsilon}$$<\/p>\n <\/p>\n \u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u6700\u7d42\u7684\u306b\u306f<\/p>\n $$ \\frac{\\varDelta \\bar{\\tau}}{\\varDelta \\tau_1} = \u7279\u306b\u5186\u8ecc\u9053\u306e\u5834\u5408\u306b\u306f\uff0c<\/p>\n $$\\epsilon = \\frac{1-\\frac{r_g}{r}}{\\sqrt{1 -\\frac{3}{2}\\frac{r_g}{r}}}$$<\/p>\n \u3067\u3042\u308a\uff0c\u3053\u308c\u3092\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n $$\\frac{\\varDelta \\bar{\\tau}}{\\varDelta \\tau_1} = \\frac{\\sqrt{1 -\\frac{3}{2}\\frac{r_g}{r}}}{\\sqrt{1 -\\frac{r_g}{r_1}}}$$<\/p>\n \u5186\u8ecc\u9053\u3067\u306a\u3044\u5834\u5408\u306f\uff0c\\(r_g\\) \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067\uff0c\u307b\u307c\u6955\u5186\u8ecc\u9053\uff08\u53b3\u5bc6\u306b\u306f\u9589\u3058\u305f\u6955\u5186\u8ecc\u9053\u306b\u306f\u306a\u3089\u305a\u306b\u8fd1\u70b9\u79fb\u52d5\u3059\u308b\u304c\u8fd1\u70b9\u3068\u9060\u70b9\u306f\u5b58\u5728\u3059\u308b\uff09\u3067\u3042\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n \u3057\u305f\u304c\u3063\u3066\uff0c4\u5143\u901f\u5ea6\u306e\u898f\u683c\u5316\u6761\u4ef6<\/p>\n $$\\left(\\frac{dr}{d\\tau}\\right)^2 = \\epsilon^2 -\\left( 1-\\frac{r_g}{r}\\right) \\left( 1 + \\frac{\\ell^2}{r^2}\\right)$$<\/p>\n \u304b\u3089\uff0c\\(r = r_{\\rm max}, \\ r = r_{\\rm min}\\) \u3067 \\(\\displaystyle \\frac{dr}{d\\tau} = 0\\) \u3068\u306a\u308b\u3053\u3068\u304b\u3089<\/p>\n \\begin{eqnarray} \\(r_g\\) \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067 \\(\\epsilon\\) \u306b\u3064\u3044\u3066\u89e3\u304f\u3068\uff0c<\/p>\n $$\\epsilon \\simeq 1 -\\frac{r_g}{2\\left(r_{\\rm min} + r_{\\rm max} \\right)} \\equiv 1 -\\frac{r_g}{4 a}$$<\/p>\n \u3053\u3053\u3067<\/p>\n $$a \\equiv \\frac{r_{\\rm min} + r_{\\rm max}}{2}$$<\/p>\n \u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6\u306b\u304a\u3051\u308b\u8ecc\u9053\u9577\u534a\u5f84\u306b\u5bfe\u5fdc\u3059\u308b\u3002<\/p>\n \u3053\u308c\u3092\u4ee3\u5165\u3059\u308b\u3068\uff0c\u6642\u9593\u306e\u9032\u307f\u306f<\/p>\n \\begin{eqnarray} \u5186\u8ecc\u9053\u3067\u306a\u3044\u5834\u5408\u306f\uff0c\\(r\\) \u3082\u6642\u3005\u523b\u3005\u3068\u304b\u308f\u308b\u306e\u3067\u6642\u9593\u306e\u9032\u307f\u3082\u4e00\u5b9a\u3067\u306f\u306a\u3044\u3002\u3057\u304b\u3057\uff0c\u6642\u9593\u306e\u9032\u307f\u306b\u95a2\u3057\u3066 \\(r_g\\) \u306e1\u6b21\u307e\u3067\u306e\u8fd1\u4f3c\u3067\u3042\u308c\u3070\uff0c\\(r\\) \u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6\u306e\u7d50\u679c\uff08\u6955\u5186\u8ecc\u9053\uff09\u3092\u4f7f\u3048\u3070\u3088\u3044\u3002<\/p>\n \u7279\u306b\uff0c1\u56de\u8ee2\u306e\u9593\u306e\u6642\u9593\u5e73\u5747\u3092\u3068\u308b\u3068<\/p>\n\n
\u7570\u306a\u308b\u5730\u70b9\u306b\u9759\u6b62\u3057\u3066\u3044\u308b\u6642\u8a08\u306e\u9032\u307f\u306e\u6bd4<\/span><\/h3>\n
\n
\u30ed\u30fc\u30ec\u30f3\u30c4\u56e0\u5b50<\/span><\/h3>\n
\n
\n\\gamma\\equiv \\frac{1}{\\sqrt{1-V^2}} &\\equiv&\u00a0 -u_{\\mu} \\bar{u}^{\\mu} \\\\
\n&=& -g_{00} u^0 \\bar{u}^0 \\\\
\n&=& \\left(1-\\frac{r_g}{r}\\right) \\cdot \\frac{1}{\\sqrt{1-\\frac{r_g}{r}}}\\cdot
\n\\frac{\\epsilon}{1-\\frac{r_g}{r}} \\\\
\n&=& \\frac{\\epsilon}{\\sqrt{1-\\frac{r_g}{r}}}
\n\\end{eqnarray}<\/p>\n\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\u306e\u6642\u9593\u306e\u9032\u307f<\/h3>\n
\n\\frac{\\varDelta \\tau}{\\varDelta \\tau_1} \\frac{\\varDelta \\bar{\\tau}} {\\varDelta \\tau} =
\n\\frac{\\sqrt{1 -\\frac{r_g}{r}}}{\\sqrt{1 -\\frac{r_g}{r_1}}}\u00a0 \\frac{\\sqrt{1-\\frac{r_g}{r}}}{\\epsilon}
\n=
\n\\frac{1 -\\frac{r_g}{r}}{\\epsilon \\sqrt{1 -\\frac{r_g}{r_1}}} $$<\/p>\n\u5186\u8ecc\u9053\u306e\u5834\u5408<\/h4>\n
\n
\u5186\u8ecc\u9053\u3067\u306a\u3044\u5834\u5408<\/h3>\n
\n0 &=& \\epsilon^2 -\\left( 1-\\frac{r_g}{r_{\\rm max}}\\right) \\left( 1 + \\frac{\\ell^2}{r_{\\rm max}^2}\\right) \\\\
\n0 &=& \\epsilon^2 -\\left( 1-\\frac{r_g}{r_{\\rm min}}\\right) \\left( 1 + \\frac{\\ell^2}{r_{\\rm min}^2}\\right)
\n\\end{eqnarray}<\/p>\n
\n\\frac{\\varDelta \\bar{\\tau}}{\\varDelta \\tau_1} &\\simeq&
\n\\frac{1}{\\sqrt{1 -\\frac{r_g}{r_1}}}\\left(1 -\\frac{r_g}{r}\\right)\\left(1 + \\frac{r_g}{4 a}\\right) \\\\
\n&\\simeq& \\frac{1}{\\sqrt{1 -\\frac{r_g}{r_1}}}\\left(1 + \\frac{r_g}{4 a} -\\frac{r_g}{r}\\right)
\n\\end{eqnarray}<\/p>\n