\u4e16\u306b\u3042\u307e\u305f\u3042\u308b\u6559\u79d1\u66f8\u3067\u8aac\u660e\u3055\u308c\u3066\u3044\u308b GPS \u885b\u661f\u306e\u6642\u8a08\u306e\u9032\u307f\u3002\u4f8b\u3048\u3070\uff0c\u4ee5\u4e0b\u306e\u6559\u79d1\u66f8\uff1a<\/p>\n
\u3067\u306f\uff0c\u3069\u306e\u3088\u3046\u306b\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u304b\u3002<\/p>\n
<\/p>\n
\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306b\u304a\u3044\u3066\uff0c\u534a\u5f84 $r$ \u306e\u5186\u904b\u52d5\u3092\u3059\u308b\u89b3\u6e2c\u8005\uff08\u5929\u4f53\uff0c\u4eba\u5de5\u98db\u7fd4\u4f53\u7b49\u3044\u308d\u3044\u308d\u8a00\u3044\u65b9\u304c\u3042\u308b\u304c\uff0c\u4ee5\u4e0b\u3067\u306f\u300c\u6642\u8a08\u300d\u3068\u3057\u3088\u3046\uff09\u306e\u6642\u9593\u304c\u9045\u308c\u308b\u3053\u3068\u306f\u3088\u304f\u77e5\u3089\u308c\u3066\u304a\u308a\uff0c\u305f\u3068\u3048\u3070 Wikipedia \uff08\u82f1\u8a9e\u7248\uff09<\/a>\u3067\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u3066\u3044\u308b\u3002<\/p>\n $$t_0 = t_f\\, \\sqrt{1 – \\frac{3}{2} \\cdot \\frac{r_s}{r}}$$<\/p>\n \u3053\u3053\u3067\uff0c$t_0$ \u306f\u56fa\u6709\u6642\u9593\uff0c$t_f$ \u306f\u5341\u5206\u9060\u65b9\u306e\u89b3\u6e2c\u8005\u306e\u5ea7\u6a19\u6642\u9593\uff0c$r_s \\equiv 2GM\/c^2$ \u306f\u8cea\u91cf $M$ \u306e\u5929\u4f53\u306e\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84\u3067\u3042\u308b\u3002<\/p>\n \u3053\u3053\u3067\u306f\uff0cWikipedia \u306e\u8868\u5f0f\u3092\u305d\u306e\u307e\u307e\u5f15\u7528\u3057\u305f\u305f\u3081\uff0c\u4ee5\u4e0b\u3067\u6211\u3005\u304c\u4f7f\u3046\u8868\u8a18\u3068\u306f\u82e5\u5e72\u7570\u306a\u308b\u3053\u3068\u306f\u3054\u4e86\u627f\u304f\u3060\u3055\u3044\u3002\u672c\u30b5\u30a4\u30c8\u306e\u8868\u8a18\u306b\u3057\u305f\u304c\u3048\u3070\uff0c\u5186\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u9032\u307f\u306f\u56fa\u6709\u6642\u9593\uff08\u9593\u9694\u307e\u305f\u306f\u5909\u4f4d\uff09 $d \\tau$ \u3067\u3042\u3089\u308f\u3057\uff0c\u5ea7\u6a19\u6642\u9593\u9593\u9694\u306f $dt$\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u534a\u5f84\uff08\u91cd\u529b\u534a\u5f84\uff09\u306f $r_g$ \u3067\u3042\u3089\u308f\u3057\u3066\uff0c<\/p>\n $$d\\tau = dt \\, \\sqrt{1 – \\frac{3}{2} \\cdot \\frac{r_g}{r}}$$<\/p>\n \u3068\u306a\u308b\u3002<\/p>\n \u5730\u7403\u306e\u5468\u308a\u3092\u307e\u308f\u308b GPS \u885b\u661f\u306b\u3064\u3044\u3066\u3082\u3053\u306e\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b\u3088\u3046\u306a\u6642\u9593\u306e\u9045\u308c\u304c\u304a\u3053\u308b\u3053\u3068\u306f\u77e5\u3089\u308c\u3066\u3044\u3066\uff0c\u30a2\u30e1\u30ea\u30ab\u56fd\u9632\u7701\u306f\u4e8b\u524d\u306b\u3053\u306e\u76f8\u5bfe\u8ad6\u7684\u306a\u73fe\u8c61\u306b\u5bfe\u3059\u308b\u5bfe\u7b56\u3092\u65bd\u3057\u3066 GPS \u885b\u661f\u3092\u6253\u3061\u4e0a\u3052\uff0c\u904b\u7528\u3057\u3066\u3044\u308b\u3068\u805e\u304f\u3002<\/p>\n \u3067\u306f\uff0c\u3053\u306e\u91cd\u529b\u5834\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u9045\u308c\u306e\u5f0f\u304c\u3069\u306e\u3088\u3046\u306b\u3057\u3066\u5c0e\u51fa\u3055\u308c\u305f\u306e\u304b\u3092\u8abf\u3079\u3066\u307f\u308b\u3068\uff0c\u672c\u6765\u306f\u5186\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u8ecc\u9053\u306b\u3064\u3044\u3066\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u306a\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\uff0c\u3057\u304b\u308b\u306e\u3061\u306b\uff0c\u6642\u9593\u306e\u9032\u307f\u306e\u5b9a\u7fa9\u306b\u4ee3\u5165\u3057\u3066\u8a08\u7b97\u3059\u3079\u304d\u3068\u3053\u308d\u3092\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7c21\u5358\u306a\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u3092\u4f7f\u3063\u3066\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\n \u4e0a\u8a18\u306e\u6559\u79d1\u66f8\u3067\u306f\uff0c\u5186\u8ecc\u9053\u3092\u63cf\u3044\u3066\u904b\u52d5\u3059\u308b GPS \u885b\u661f\u306e\u6642\u8a08\u306e\u9045\u308c\u306b\u3064\u3044\u3066\uff0c\u3060\u3044\u305f\u3044\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\n \u307e\u305a\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306e\u30e1\u30c8\u30ea\u30c3\u30af\u3068\u56fa\u6709\u6642\u9593\u9593\u9694 \\(d\\tau\\) \u306e\u5b9a\u7fa9\u306f\uff0c<\/p>\n \\begin{eqnarray} ds^2 &=& g_{\\mu\\nu} dx^{\\mu} dx^{\\nu} \\\\ &=& – c^2 d\\tau^2 \\\\ &=& -\\left(1 – \\frac{r_g}{r}\u00a0 \\right) c^2 dt^2 + \\frac{1}{1 – \\frac{r_g}{r}\u00a0} dr^2 + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)\\end{eqnarray}<\/p>\n \u3053\u3053\u3067$$r_g = \\frac{2GM}{c^2 r}$$<\/p>\n \u8d64\u9053\u9762\uff08\\( \\theta = \\frac{\\pi}{2} \\) \u5f93\u3063\u3066 \\(d\\theta = 0 \\) \uff09\u4e0a\u306e\u5186\u8ecc\u9053\uff08\\(r = \\mbox{const.}\\) \u5f93\u3063\u3066 \\(dr = 0\\) \uff09\u306b\u3064\u3044\u3066\u306f<\/p>\n \\begin{eqnarray} d\\tau &=& \\sqrt{\\left( 1 – \\frac{r_g}{r} \\right) dt^2 – \\frac{1}{c^2} r^2 d\\phi^2} \\\\ \u3053\u3053\u3067 $$\\displaystyle v \\equiv r \\frac{d\\phi}{dt}$$<\/p>\n \u306f\uff0c\u5186\u904b\u52d5\u3057\u3066\u3044\u308b\u5834\u5408\u306e\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u306a\u901f\u3055\u3067\u3042\u308b\u3002<\/p>\n \u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u5186\u904b\u52d5\u3067\u306f\uff0c\u9060\u5fc3\u529b\u3068\u4e07\u6709\u5f15\u529b\u306e\u3064\u308a\u5408\u3044\u304b\u3089<\/p>\n $$ m \\frac{v^2}{r} = \\frac{GMm}{r^2}, \\quad\\therefore \\ \\ v^2 = \\frac{GM}{r}$$<\/p>\n \\begin{eqnarray} \u3053\u3053\u3067\uff0c\u300c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u5ea7\u6a19\u6642\u9593 \\(t\\) \u304c \\(dt\\) \u3060\u3051\u7d4c\u904e\u3059\u308b\u9593\u306b\uff0c\u89b3\u6e2c\u8005\u306e\u6642\u8a08\u306f \\(d\\tau\\) \u9032\u3080<\/strong><\/span>\u300d\u3068\u4eee\u5b9a\u3059\u308b\u3068\uff0c\u5186\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u9032\u307f $d\\tau$ \u3068 $r=r_1$ \u306b\u9759\u6b62\u3057\u305f\u6642\u8a08\u306e\u9032\u307f $d\\tau_1$ \u3068\u306e\u6bd4\u306f<\/p>\n $$\\frac{d\\tau}{d\\tau_1} = \\frac{dt \\sqrt{1 – \\frac{3}{2} \\frac{r_g}{r}}}{dt \\sqrt{1 – \\frac{r_g}{r_1}}} = \\frac{\\sqrt{1 – \\frac{3}{2} \\frac{r_g}{r}}}{ \\sqrt{1 – \\frac{r_g}{r_1}}}$$<\/p>\n \u3053\u308c\u304c\uff0c\u4e16\u306e\u4e2d\u306e\u4e00\u822c\u7684\u306a\u6559\u79d1\u66f8\u3067\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u3068\u601d\u308f\u308c\u308b\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u306b\u57fa\u3065\u3044\u3066\u5c0e\u304b\u308c\u305f\uff0c\u91cd\u529b\u5834\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u9045\u308c\u306e\u5f0f\u3067\u3042\u308b\u3002<\/p>\n \u3057\u304b\u3057\uff0c\u3053\u306e\u65b9\u6cd5\u306f \\(dr = 0\\) \u306e\u5186\u8ecc\u9053\u306e\u5834\u5408\u3057\u304b\u4f7f\u3048\u306a\u3044\u3057\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306e\u7d50\u679c\u3092\u4f7f\u3063\u3066\u3044\u308b\u70b9\u3067\u4f55\u3089\u304b\u306e\u8fd1\u4f3c\u304c\u7d1b\u308c\u8fbc\u3093\u3067\u3044\u308b\u306e\u3067\u306f\u306a\u3044\u304b\u3068\u8003\u3048\u3089\u308c\u308b\u3002<\/p>\n \u305d\u3053\u3067\u6211\u3005\u306f\uff0c\u3053\u306e\u3088\u3046\u306a\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u306b\u983c\u3089\u305a\u306b\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u306a\u5b9a\u5f0f\u5316\u3092\u884c\u3044\uff0c\u5186\u8ecc\u9053\u4ee5\u5916\u306b\u3082\u5fdc\u7528\u3067\u304d\u308b\u3088\u3046\u306b\u3057\u3066\uff0c\u91cd\u529b\u5834\u4e2d\u306e\u89b3\u6e2c\u8005\uff08\u6642\u8a08\uff09\u306e\u904b\u52d5\u3092\u8a18\u8ff0\u3059\u308b\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\uff0c\u6642\u9593\u306e\u9045\u308c\u306e\u5b9a\u7fa9\u5f0f\u306b\u4ee3\u5165\u3057\u3066\uff0c\u91cd\u529b\u5834\u4e2d\u3092\u5186\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u9045\u308c\u306e\u5f0f\u3092\u5c0e\u3044\u305f\u3002<\/p>\n \u305d\u306e\u6982\u8981\u306f\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u304b\u3089\u8fbf\u308c\u308b\u3002<\/p>\n \u8a73\u7d30\u306a\u8a08\u7b97\u306f\u672c\u30b5\u30a4\u30c8\u306e\u3044\u308d\u3044\u308d\u306a\u30da\u30fc\u30b8\u306b\u6563\u3089\u304b\u3063\u3066\u3044\u308b\u306e\u3067\uff0c\u6982\u8981\u3092\u3042\u3089\u305f\u3081\u3066\u307e\u3068\u3081\u308b\u3068…<\/p>\n $$ \u3053\u3053\u3067\uff0c$\\epsilon$ \u306f\u904b\u52d5\u3059\u308b\u6642\u8a08\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u305f\u969b\u306b\u5f97\u3089\u308c\u305f\u5b9a\u6570\u3067\u3042\u308a\uff0c\u3082\u3046\u4e00\u3064\u306e\u5b9a\u6570 $\\ell$ \u3068\u5171\u306b\u8868\u3059\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n $$\\frac{dt}{d\\tau} = \\frac{\\epsilon}{1 – \\frac{r_g}{r}}, \\quad \\frac{d\\phi}{d\\tau} = \\frac{\\ell}{r^2}$$<\/p>\n \u307e\u305f\uff0c4\u5143\u901f\u5ea6\u306e\u898f\u683c\u5316\u6761\u4ef6\u304b\u3089 $\\epsilon$ \u3068 $\\ell$ \u306f\u4ee5\u4e0b\u306e\u95a2\u4fc2\u3092\u6e80\u305f\u3055\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002<\/p>\n $$\\left(\\frac{dr}{d\\tau}\\right)^2 = \\epsilon^2 \u2013 \\left( 1-\\frac{r_g}{r}\\right) \\left( 1 + \\frac{\\ell^2}{r^2}\\right)$$<\/p>\n \u3053\u3053\u307e\u3067\u304c\uff08\u8ecc\u9053\u3092\u8d64\u9053\u9762\u4e0a\u306b\u9650\u3063\u305f\u3082\u306e\u306e\uff0c\u4e00\u822c\u6027\u306f\u5931\u308f\u308c\u3066\u3044\u306a\u3044\u306e\u3067\uff09\u4e00\u822c\u7684\u306a\u5f0f\u3067\u3042\u308b\u3002<\/p>\n \u7279\u306b\u5186\u8ecc\u9053\u306e\u5834\u5408\u306f\uff0c$\\epsilon$ \u304c\u7c21\u5358\u306b\uff08\u53b3\u5bc6\u306b\uff09\u89e3\u3051\u3066\uff0c<\/p>\n $$\\epsilon = \\frac{1-\\frac{r_g}{r}}{\\sqrt{1 \u2013 \\frac{3}{2}\\frac{r_g}{r}}}$$<\/p>\n \u3053\u308c\u3092\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n $$\\frac{\\Delta \\bar{t}}{\\Delta t_1} = \\frac{\\sqrt{1 \u2013 \\frac{3}{2}\\frac{r_g}{r}}}{\\sqrt{1 \u2013 \\frac{r_g}{r_1}}}$$<\/p>\n \u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u304b\u3089\u5f97\u3089\u308c\u305f\u5f0f\u3068\u6bd4\u8f03\u3059\u308b\u305f\u3081\u306b\uff0c\u3042\u3089\u305f\u3081\u3066 $\\frac{\\Delta \\bar{t}} \\rightarrow d\\tau$\uff0c$r_1 \\rightarrow \\infty$ \u3068\u3057\u3066 $\\Delta t_1 \\rightarrow dt$ \u3068\u7f6e\u304d\u63db\u3048\u308b\u3068<\/p>\n $$d\\tau = dt \\, \\sqrt{1 \u2013 \\frac{3}{2}\\frac{r_g}{r}}$$<\/p>\n \u3053\u308c\u306f\uff0c\u6211\u3005\u304c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u624b\u6cd5\u3067\uff08\u304c\u3093\u3070\u3063\u3066\uff09\u5c0e\u3044\u305f\u3053\u306e\u5f0f\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u306b\u57fa\u3065\u3044\u3066\u6c42\u3081\u305f\u5f0f\u3068\uff08\u304f\u3084\u3057\u3044\u3053\u3068\u306b\u30fb\u5f53\u7136\u306a\u304c\u3089\uff09\u4e00\u81f4\u3059\u308b\u3002<\/p>\n \u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u306e\u65b9\u304c\u7c21\u5358\u306b\u5c0e\u51fa\u3067\u304d\u305f\u3088\u3046\u306b\u898b\u3048\u308b\u304c\uff0c\u3057\u304b\u3057\uff0c\u3053\u306e\u65b9\u6cd5\u306f \\(dr = 0\\) \u306e\u5186\u8ecc\u9053\u306e\u5834\u5408\u3057\u304b\u4f7f\u3048\u306a\u3044\u3057\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306e\u7d50\u679c\u3092\u4f7f\u3063\u3066\u3044\u308b\u70b9\u3067\u4f55\u3089\u304b\u306e\u8fd1\u4f3c\u304c\u7d1b\u308c\u8fbc\u3093\u3067\u3044\u308b\u53ef\u80fd\u6027\u304c\u3042\u3063\u305f\u3002<\/p>\n \u305d\u306e\u70b9\uff0c\u6211\u3005\u306e\u65b9\u6cd5\u306f\u5186\u8ecc\u9053\u4ee5\u5916\u306e\u4e00\u822c\u7684\u8ecc\u9053\u306e\u5834\u5408\u306b\u3082\u4f7f\u3048\u308b\u3057\uff0c\u5186\u8ecc\u9053\u306e\u5834\u5408\u306b\u306f\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u306a\u53b3\u5bc6\u89e3\u3068\u3057\u3066\u5c0e\u3044\u305f\u3068\u3044\u3046\u70b9\u3067\u3082\u610f\u7fa9\u304c\u3042\u308b<\/strong><\/span>\u3068\u8003\u3048\u308b\u3002<\/p>\n \u3067\u306f\uff0c\u306a\u305c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u7684\u985e\u63a8\u306b\u57fa\u3065\u3044\u305f\u7d50\u679c\u304c\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u306b\u53b3\u5bc6\u306b\u6c42\u3081\u305f\u7d50\u679c\u3068\u4e00\u81f4\u3057\u305f\u306e\u3067\u3042\u308d\u3046\u304b\uff1f\u305d\u308c\u3092\u8abf\u3079\u308b\u306e\u304c\uff0c\u65b0\u305f\u306a\u4f7f\u547d\u3068\u306a\u3063\u305f\u308f\u3051\u3060\u304c\uff0c\u3053\u306e\u7d9a\u304d\u306f\u5225\u306e\u30da\u30fc\u30b8\u306b…<\/p>\n
\n&=& dt \\sqrt{1 – \\frac{2GM}{c^2 r} – \\frac{1}{c^2} r^2 \\left(\\frac{d\\phi}{dt}\\right)^2} \\\\
\n&=& dt \\sqrt{1 – \\frac{2GM}{c^2 r} – \\frac{v^2}{c^2}} \\end{eqnarray}<\/p>\n
\n\\therefore\\ \\ d\\tau &=& dt \\sqrt{1 – \\frac{2GM}{c^2 r} – \\frac{v^2}{c^2}} \\\\
\n&=& dt \\sqrt{1 – \\frac{2GM}{c^2 r} – \\frac{GM}{c^2 r}} \\\\
\n&=& dt \\sqrt{1 – \\frac{3GM}{c^2 r} } \\\\
\n&=& dt \\sqrt{1 – \\frac{3}{2} \\frac{r_g}{r}} \\end{eqnarray}<\/p>\n\n
\n
\n
\n
\n$$\\frac{d^2 x^{\\lambda}}{d\\tau^2} + \\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\nu} \\frac{dx^{\\mu}}{d\\tau} \\frac{dx^{\\nu}}{d\\tau} = 0, \\quad x^{\\mu} = (t, r, \\theta, \\phi)$$
\n\uff08\u3053\u3053\u3067\uff0c$\\varGamma^{\\lambda}_{\\ \\ \\ \\mu\\nu}$ \u306f\u30af\u30ea\u30b9\u30c8\u30c3\u30d5\u30a7\u30eb\u8a18\u53f7\uff09\u3092\u89e3\u3044\u3066\uff0c $\\sqrt{1-V^2}$ \u3092\u6c42\u3081\uff0c<\/li>\n<\/ul>\n<\/li>\n
\n\\frac{\\Delta \\bar{t}}{\\Delta t_1} =
\n\\frac{1 \u2013 \\frac{r_g}{r}}{\\epsilon \\sqrt{1 \u2013 \\frac{r_g}{r_1}}}
\n$$<\/p>\n