{"id":3091,"date":"2022-06-23T15:38:47","date_gmt":"2022-06-23T06:38:47","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=3091"},"modified":"2022-08-25T14:27:17","modified_gmt":"2022-08-25T05:27:17","slug":"%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%81%ab%e3%82%88%e3%82%89%e3%81%aa%e3%81%84%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%81%ae%e7%b5%b1%e4%b8%80%e7%9a%84%e7%90%86%e8%a7%a3","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%81%ae%e7%b5%b1%e4%b8%80%e7%9a%84%e7%90%86%e8%a7%a3\/%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%a4%89%e6%8f%9b%e3%81%ab%e3%82%88%e3%82%89%e3%81%aa%e3%81%84%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%81%ae%e7%b5%b1%e4%b8%80%e7%9a%84%e7%90%86%e8%a7%a3\/","title":{"rendered":"\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u306b\u3088\u3089\u306a\u3044\u8d64\u65b9\u504f\u79fb\u306e\u7d71\u4e00\u7684\u7406\u89e3"},"content":{"rendered":"

\u7d71\u4e00\u7684\u7406\u89e3\u306b\u5411\u3051\u305f\u6e96\u5099<\/h3>\n

\u3053\u3053\u3067\u306f\uff0c\u4ee5\u4e0b\u306e2\u3064\u306e\u539f\u7406\u539f\u5247\u304b\u3089\uff0c\u3053\u308c\u3089\u306e\u8d64\u65b9\u504f\u79fb\u304c\u7d71\u4e00\u7684\u306b\u7406\u89e3\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n

I<\/strong>. 4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{k}\\) \u3067\u3042\u3089\u308f\u3055\u308c\u308b\u96fb\u78c1\u6ce2\u306f\u30cc\u30eb\u6e2c\u5730\u7dda\u4e0a\u3092\u4f1d\u64ad\u3059\u308b\u3002<\/p>\n

$$ \\frac{d\\boldsymbol{k}}{dv} = \\boldsymbol{0}, \\quad \\boldsymbol{k}\\cdot\\boldsymbol{k} = 0$$<\/p>\n

<\/span>\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u4e0b\u4ed8\u6dfb\u5b57\u6210\u5206 \\(k_{\\mu} = g_{\\mu\\nu} k^{\\nu}\\) \u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\uff0c\u3053\u308c\u3092\u5229\u7528\u3059\u308b\u3002
$$\\frac{dk_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}, \\quad g_{\\alpha\\beta} k^{\\alpha} k^{\\beta} = 0$$<\/p>\n

II<\/strong>. 4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u96fb\u78c1\u6ce2\u306e\u632f\u52d5\u6570 \\(\\omega\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a4\u6b21\u5143\u30b9\u30ab\u30e9\u30fc\u3067\u66f8\u3051\u308b\u3002<\/p>\n

$$ \\omega = – \\boldsymbol{k}\\cdot\\boldsymbol{u} = – k_{\\mu} u^{\\mu}$$<\/p>\n

<\/p>\n

\u5149\u6e90\u306e\u5f8c\u9000\u901f\u5ea6\u306b\u3088\u308b\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c<\/h3>\n

\u7279\u6b8a\u76f8\u5bfe\u8ad6\u7684\u72b6\u6cc1\u306b\u304a\u3044\u3066\u306f\uff0c<\/p>\n

$$g_{\\alpha\\beta, \\mu} \\Rightarrow \\eta_{\\alpha\\beta, \\mu} = 0$$<\/p>\n

\u3067\u3042\u308b\u304b\u3089\uff0c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u3059\u3079\u3066\u306e\u6210\u5206 \\(k_{\\mu} \\equiv \\eta_{\\mu\\nu} k^{\\nu}\\) \u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002<\/p>\n

$$\\frac{dk_{\\mu}}{dv} = 0 \\ \\Rightarrow \\ k_{\\mu} = \\mbox{const.}$$<\/p>\n

\u3064\u307e\u308a\uff0c\u5149\u6e90\u304b\u3089\u653e\u305f\u308c\u305f\u3068\u304d\u3082\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u3068\u304d\u3082\u540c\u3058 \\(k_{\\mu}\\) \u3067\u3088\u3044\u3002<\/p>\n

\u3053\u308c\u3092\u4f7f\u3044\uff0c\u9759\u6b62\u89b3\u6e2c\u8005 \\(A\\) \u304c\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570\u3092<\/p>\n

$$\\omega_{\\rm obs} \\equiv – k_{\\mu} u^{\\mu}, $$<\/p>\n

\u5149\u6e90\u3068\u3068\u3082\u306b\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005 \\(B\\) \u304c\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570\u3092
$$\\omega_0 \\equiv – k_{\\mu} \\bar{u}^{\\mu} $$
\u3068\u3057\u3066\uff0c\u5149\u6e90\u304c\u904b\u52d5\u3059\u308b\u5834\u5408\u306e\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u306e\u5f0f
$$\\omega_{\\rm obs} = \\omega_0 \\frac{\\sqrt{1-V^2}}{1-V \\cos\\theta}$$
\u304c\u3059\u3067\u306b\u5225\u30da\u30fc\u30b8<\/a>\u3067\u6c42\u3081\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n

\u7279\u306b\u5149\u6e90\u304c\u9759\u6b62\u89b3\u6e2c\u8005\u304b\u3089\u9060\u3056\u304b\u308b\u5834\u5408\u306f \\(\\theta = \\pi\\) \u3067\u3042\u308a\uff0c<\/p>\n

\"\"<\/p>\n

 <\/p>\n

$$\\omega_{\\rm obs}= \\omega_0\\frac{\\sqrt{1-V^2}}{1 + V} = \\omega_0 \\sqrt{\\frac{1-V}{1+V} } < \\omega_0$$<\/p>\n

\u3068\u306a\u308a\uff0c\u632f\u52d5\u6570\u306f\u5c0f\u3055\u304f\uff0c\u3057\u305f\u304c\u3063\u3066\u6ce2\u9577\u304c\u4f38\u3073\u3066\u89b3\u6e2c\u3055\u308c\u308b\u3002<\/p>\n

\u3088\u3063\u3066\u30ed\u30fc\u30ec\u30f3\u30c4\u5909\u63db\u3092\u4f7f\u308f\u305a\u306b2\u3064\u306e\u539f\u7406\u539f\u5247\u304b\u3089\u5149\u6e90\u306e\u5f8c\u9000\u901f\u5ea6\u306b\u3088\u308b\u30c9\u30c3\u30d7\u30e9\u30fc\u52b9\u679c\u3092\u8aac\u660e\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

 <\/p>\n

\u91cd\u529b\u6e90\u8fd1\u508d\u304b\u3089\u306e\u5149\u304c\u53d7\u3051\u308b\u91cd\u529b\u8d64\u65b9\u504f\u79fb<\/h3>\n

\u91cd\u529b\u6e90\u8fd1\u508d\u306e\u6642\u7a7a\u306f\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u89e3
\n$$ds^2 =\u00a0 -\\left(1-\\frac{r_g}{r}\\right) dt^2 + \\frac{dr^2}{1-\\frac{r_g}{r}} + r^2 (d\\theta^2+\\sin^2\\theta d\\phi^2)$$
\n\u3067\u8868\u3055\u308c\u308b\u3068\u3059\u308b\uff0e\u3053\u3053\u3067 \\(r_g \\equiv 2 G M\\) \u306f\u91cd\u529b\u534a\u5f84\u3002<\/p>\n

\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306e\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306f \\(t = x^0\\) \u306b\u3088\u3089\u306a\u3044\u304b\u3089\\(g_{\\alpha\\beta, 0} = 0\\)\uff0e\u3053\u308c\u304b\u3089\u305f\u3060\u3061\u306b
\n$$\\frac{dk_{0}}{dv} =0\u00a0 \\ \\Rightarrow \\ k_0 = \\mbox{const.} \\equiv – \\omega_c$$
\n\u3068\u304a\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u4e00\u65b9\uff0c\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u9759\u6b62\u89b3\u6e2c\u8005\u306e4\u5143\u901f\u5ea6 \\(u^{\\mu}\\) \u306f\uff08\u300c\u9759\u6b62\u300d\u3060\u304b\u3089\u7a7a\u9593\u5ea7\u6a19\u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3068\u898f\u683c\u5316\u6761\u4ef6 \\(u_{\\mu} u^{\\mu} = -1\\) \u304b\u3089\uff09
\n$$u^{\\mu} = \\left(u^0, 0, 0, 0\\right) = \\left(\\frac{1}{\\sqrt{1-\\frac{r_g}{r}}}, 0, 0, 0 \\right)$$
\n\u3068\u66f8\u3051\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u3053\u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u306f
\n$$\\omega = – k_{\\mu} u^{\\mu} = – k_0 u^0 = \\frac{\\omega_c}{\\sqrt{1-\\frac{r_g}{r}}}$$
\n\u3068\u306a\u308a\uff0c\u540c\u3058\u5149\u3092\u89b3\u6e2c\u3057\u3066\u3082\uff0c\u305d\u306e\u632f\u52d5\u6570 \\(\\omega\\) \u306f\u89b3\u6e2c\u8005\u306e\u4f4d\u7f6e\u3092\u8868\u3059\u52d5\u5f84\u5ea7\u6a19 \\(r\\) \u306b\u4f9d\u5b58\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u7279\u306b\uff0c\\(r=r_1\\) \u3067\u632f\u52d5\u6570 \\(\\omega_1\\) \u306e\u5149\u3092 \\(r = r_2 > r_1\\) \u3067\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u3068\uff0c\u305d\u306e\u632f\u52d5\u6570 \\(\\omega_2\\) \u306f
\n$$\\frac{\\omega_2}{\\omega_1} = \\frac{\\sqrt{1-\\frac{r_g}{r_1}}}{\\sqrt{1-\\frac{r_g}{r_2}}} < 1, $$
\n$$\\therefore\\ \\ \\omega_2 < \\omega_1$$
\n\u3068\u306a\u308a\uff0c\u91cd\u529b\u6e90\u306b\u8fd1\u3044\u5834\u6240\u304b\u3089\u653e\u5c04\u3055\u308c\u305f\u5149\u3092\u96e2\u308c\u305f\u5834\u6240\u3067\u89b3\u6e2c\u3059\u308b\u3068\uff0c\u632f\u52d5\u6570\u306f\u5c0f\u3055\u304f\uff0c\u3057\u305f\u304c\u3063\u3066\u6ce2\u9577\u306f\u4f38\u3073\u3066\u89b3\u6e2c\u3055\u308c\u308b\u3002\u3053\u306e\u3088\u3046\u306b\u3057\u3066\uff0c2\u3064\u306e\u539f\u7406\u539f\u5247\u304b\u3089\u5149\u306e\u91cd\u529b\u8d64\u65b9\u504f\u79fb\u3092\u8aac\u660e\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\u5b87\u5b99\u81a8\u5f35\u306b\u3088\u3063\u3066\u5f15\u304d\u8d77\u3053\u3055\u308c\u308b\u5b87\u5b99\u8ad6\u7684\u8d64\u65b9\u504f\u79fb<\/h3>\n

\u4e00\u69d8\u7b49\u65b9\u306a\u81a8\u5f35\u5b87\u5b99\u306e\u8a08\u91cf\u306f \\(a d\\eta\\ = dt\\)\u00a0 \u3067\u5b9a\u7fa9\u3055\u308c\u308b\u5171\u5f62\u6642\u9593 \\(\\eta\\) \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3068\u4fbf\u5229\u3067\u3042\u308b\u3002
\n\\begin{eqnarray}
\nds^2 &=& -dt^2 + a^2(t) \\gamma_{ij} dx^i dx^j \\\\
\n&=& -a^2(\\eta) d\\eta^2 + a^2(\\eta) \\Bigl(d\\chi^2 + \\sigma^2(\\chi)\\left(d\\theta^2 + \\sin^2\\theta d\\phi^2 \\right) \\Bigr)
\n\\end{eqnarray}
\n\u3053\u3053\u3067 $\\sigma(\\chi)$ \u306f\u300c\u5b9a\u66f2\u7387\u7a7a\u9593\u8a08\u91cf\u306e\u3044\u304f\u3064\u304b\u306e\u8868\u793a\u4f8b<\/a>\u300d\u306b\u66f8\u3044\u305f\u3088\u3046\u306b\uff0c<\/p>\n

$$\\sigma(\\chi) = \\frac{\\sin \\left( \\sqrt{k} \\chi\\right)}{\\sqrt{k}}$$<\/p>\n

\u307e\u305f \\(k\\) \u306f3\u6b21\u5143\u5b9a\u66f2\u7387\u7a7a\u9593\u306e\u66f2\u7387\u5b9a\u6570\u3067\u3042\u308b\u3002\u3053\u306e\u3088\u3046\u306b\u66f8\u3044\u3066\u304a\u304f\u3068 $k < 0$ \u306e\u5834\u5408\u3082 $k = 0$ \u306e\u5834\u5408\u3082\uff08$k \\rightarrow 0$ \u306e\u6975\u9650\u3092\u3068\u308b\u3053\u3068\u3067\uff09\u4f7f\u3048\u308b\u3053\u3068\u306f\u300c\u5b9a\u66f2\u7387\u7a7a\u9593\u8a08\u91cf\u306e\u3044\u304f\u3064\u304b\u306e\u8868\u793a\u4f8b<\/a>\u300d\u3067\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n

\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306e\u7b2c\u30bc\u30ed\u6210\u5206\u306f\uff0c\u30cc\u30eb\u6761\u4ef6\u3092\u4f7f\u3046\u3068
\n$$\\frac{dk_{0}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, 0} k^{\\alpha} k^{\\beta} = \\frac{1}{a}\\frac{da}{d\\eta} g_{\\alpha\\beta} k^{\\alpha} k^{\\beta} =0$$
\n\u3068\u306a\u308a\uff0c\u305f\u3060\u3061\u306b
\n$$k_0 = \\mbox{const.} \\equiv – \\omega_c$$
\n\u3068\u304a\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u4e00\u65b9\uff0c\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3051\u308b\u5171\u52d5\u89b3\u6e2c\u8005<\/strong><\/span>\u306e4\u5143\u901f\u5ea6\u306f\uff08\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3044\u3066\u306f\u7a7a\u9593\u5ea7\u6a19\u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3092\u300c\u9759\u6b62\u300d\u3068\u306f\u8a00\u308f\u305a\u306b\u300c\u5171\u52d5\u300d\u3068\u8a00\u3046\uff09\\(g_{00}(u^0)^2 = -1\\) \u3088\u308a
\n$$u^{\\mu} = \\left(u^0, 0, 0, 0 \\right) = \\left(\\frac{1}{a}, 0, 0, 0 \\right)$$
\n\u3068\u66f8\u3051\u308b\uff0e\u3057\u305f\u304c\u3063\u3066\uff0c\u3053\u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u306f
\n$$\\omega = – k_{\\mu} u^{\\mu} = -k_0 u^0 = \\frac{\\omega_c}{a(\\eta)}$$
\n\u3068\u306a\u308a\uff0c\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570\u306f\uff0c\u5149\u304c\u653e\u51fa\u3055\u308c\u305f\u6642\u523b\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 \\(a(\\eta)\\) \u306b\u53cd\u6bd4\u4f8b\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\u7279\u306b\u6642\u523b \\(\\eta = \\eta_e\\) \u306b\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u632f\u52d5\u6570\u3092 \\(\\omega_e\\) \u3068\u6e2c\u5b9a\u3057\u305f\u5149\u3092\uff0c
\n\u73fe\u5728\u6642\u523b \\(\\eta = \\eta_0 > \\eta_e\\) \u306b\u6e2c\u5b9a\u3059\u308b\u3068\u304d\u306e\u632f\u52d5\u6570\u3092 \\(\\omega_0\\) \u3068\u3059\u308b\u3068\uff0c
\n$$\\frac{\\omega_0}{\\omega_e} = \\frac{a(\\eta_e)}{a(\\eta_0)} < 1$$
\n\u3068\u306a\u308a\uff0c\u5b87\u5b99\u521d\u671f\u306b\u653e\u5c04\u3055\u308c\u305f\u5149\u3092\u73fe\u5728\u89b3\u6e2c\u3059\u308b\u3068\uff0c\u632f\u52d5\u6570\u306f\u5c0f\u3055\u304f\uff0c\u3057\u305f\u304c\u3063\u3066\u6ce2\u9577\u306f\u4f38\u3073\u3066\u89b3\u6e2c\u3055\u308c\u308b\u3002\u3053\u306e\u3088\u3046\u306b\u3057\u3066\uff0c2\u3064\u306e\u539f\u7406\u539f\u5247 \u304b\u3089\u5149\u306e\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3051\u308b\u8d64\u65b9\u504f\u79fb\u3092\u8aac\u660e\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\u53c2\u8003\u6587\u732e<\/h3>\n

G. F. R. Ellis \u2013 Relativistic Cosmology, in \u201cGeneral Relativity and Cosmology\u201d ed. B. K. Sachs<\/strong><\/span><\/a> (Academic Press, New York, 1971) \u306e P.146 \u306e (6.10b) \u5f0f\uff1a<\/p>\n

$$1 + z = \\frac{(u^a k_a)_{\\rm emitter}}{(u^a k_a)_{\\rm observer}}$$<\/p>\n

\u3068\u66f8\u3044\u3066\u3042\u308a\uff0c\u305d\u306e\u4e0b\u306b<\/p>\n

“This relation is true no matter what the separation of emitter and observer, and holds independent of an interpretation of the red-shift as a $\\langle\\langle$ Doppler $\\rangle\\rangle$ or $\\langle\\langle$ gravitational $\\rangle\\rangle$ red-shift.” \u3068\u3042\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u7d71\u4e00\u7684\u7406\u89e3\u306b\u5411\u3051\u305f\u6e96\u5099 <\/p>\n

\u3053\u3053\u3067\u306f\uff0c\u4ee5\u4e0b\u306e2\u3064\u306e\u539f\u7406\u539f\u5247\u304b\u3089\uff0c\u3053\u308c\u3089\u306e\u8d64\u65b9\u504f\u79fb\u304c\u7d71\u4e00\u7684\u306b\u7406\u89e3\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n

I. 4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{k}\\) \u3067\u3042\u3089\u308f\u3055\u308c\u308b\u96fb\u78c1\u6ce2\u306f\u30cc\u30eb\u6e2c\u5730\u7dda\u4e0a\u3092\u4f1d\u64ad\u3059\u308b\u3002<\/p>\n

$$ \\frac{d\\boldsymbol{k}}{dv} = \\boldsymbol{0}, \\quad \\boldsymbol{k}\\cdot\\boldsymbol{k} = 0$$<\/p>\n

\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306f4\u5143\u6ce2\u6570\u30d9\u30af\u30c8\u30eb\u306e\u4e0b\u4ed8\u6dfb\u5b57\u6210\u5206 \\(k_{\\mu} = g_{\\mu\\nu} k^{\\nu}\\) \u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\uff0c\u3053\u308c\u3092\u5229\u7528\u3059\u308b\u3002$$\\frac{dk_{\\mu}}{dv} = \\frac{1}{2} g_{\\alpha\\beta, \\mu} k^{\\alpha} k^{\\beta}, \\quad g_{\\alpha\\beta} k^{\\alpha} k^{\\beta} = 0$$<\/p>\n

II. 4\u5143\u901f\u5ea6 \\(\\boldsymbol{u}\\) \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u96fb\u78c1\u6ce2\u306e\u632f\u52d5\u6570 \\(\\omega\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a4\u6b21\u5143\u30b9\u30ab\u30e9\u30fc\u3067\u66f8\u3051\u308b\u3002<\/p>\n

$$ \\omega = – \\boldsymbol{k}\\cdot\\boldsymbol{u} = – k_{\\mu} u^{\\mu}$$<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":3059,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-3091","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\/3091","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\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=3091"}],"version-history":[{"count":12,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3091\/revisions"}],"predecessor-version":[{"id":3564,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3091\/revisions\/3564"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3059"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=3091"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}