{"id":1580,"date":"2022-01-31T12:34:25","date_gmt":"2022-01-31T03:34:25","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1580"},"modified":"2024-01-15T15:11:39","modified_gmt":"2024-01-15T06:11:39","slug":"%e6%9b%b2%e3%81%8c%e3%81%a3%e3%81%9f%e6%99%82%e7%a9%ba%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e5%b9%be%e4%bd%95%e5%85%89%e5%ad%a6%e8%bf%91%e4%bc%bc","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%ae%87%e5%ae%99%e8%ab%96\/%e6%9b%b2%e3%81%8c%e3%81%a3%e3%81%9f%e6%99%82%e7%a9%ba%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e5%b9%be%e4%bd%95%e5%85%89%e5%ad%a6%e8%bf%91%e4%bc%bc\/","title":{"rendered":"\u66f2\u304c\u3063\u305f\u6642\u7a7a\u306b\u304a\u3051\u308b\u5e7e\u4f55\u5149\u5b66\u8fd1\u4f3c"},"content":{"rendered":"

\u89d2\u5f84\u8ddd\u96e2\u3084\u5149\u5ea6\u8ddd\u96e2\u306e\u5b9a\u7fa9\u306b\u5fc5\u8981\u306a\uff0c\u66f2\u304c\u3063\u305f\u6642\u7a7a\u306b\u304a\u3051\u308b\u5e7e\u4f55\u5149\u5b66\u8fd1\u4f3c<\/strong><\/span>\u306b\u3064\u3044\u3066\u304a\u3055\u3089\u3057\u3066\u304a\u304f\u3002<\/p>\n

\u96fb\u78c1\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb<\/h3>\n

\u96fb\u78c1\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb\u306f\uff0cSI \u5358\u4f4d\u7cfb\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002\uff08$c = 1$\uff09<\/p>\n

$$T_{\\mu\\nu} = \\varepsilon_0 \\left\\{F_{\\mu\\alpha} F_{\\nu}^{\\ \\ \\alpha}
\n– \\frac{1}{4} g_{\\mu\\nu} F_{\\alpha\\beta} F^{\\alpha\\beta}\\right\\}$$<\/p>\n

\u3053\u3053\u3067\uff0c$\\varepsilon_0$ \u306f\u771f\u7a7a\u306e\u8a98\u96fb\u7387\u3002\u6614\u306e\u5358\u4f4d\u7cfb\uff08CGS-\u30ac\u30a6\u30b9\u5358\u4f4d\u7cfb\uff09\u3092\u4f7f\u3063\u3066\u3044\u308b\u30c6\u30ad\u30b9\u30c8\u3067\u306f\uff0c\u524d\u306e\u5b9a\u6570\u90e8\u5206\u304c $\\displaystyle \\varepsilon_0 \\rightarrow \\frac{1}{4\\pi}$ \u3068\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n

\u96fb\u78c1\u30c6\u30f3\u30bd\u30eb $F_{\\mu\\nu}$ \u306f\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb $A_{\\mu}$ \u3092\u4f7f\u3063\u3066
\n$$F_{\\mu\\nu} = A_{\\nu, \\mu} – A_{\\mu, \\nu} = A_{\\nu;\\mu} – A_{\\mu; \\nu} $$<\/p>\n

\u3053\u3053\u3067\uff0c${\\ }_{, \\nu}$ \u306f $x^{\\nu}$ \u306b\u3088\u308b\u504f\u5fae\u5206\uff0c${\\ }_{; \\nu}$ \u306f\u5171\u5909\u5fae\u5206\u3092\u3042\u3089\u308f\u3059\u3002<\/p>\n

\u96fb\u8377\u5bc6\u5ea6 $\\rho$ \u3084\u96fb\u6d41\u5bc6\u5ea6 $\\boldsymbol{J}$ \u304c\u306a\u3044\u5834\u5408\u306e\u30de\u30af\u30b9\u30a6\u30a7\u30eb\u65b9\u7a0b\u5f0f\u306f
\n$$F^{\\mu\\nu}_{\\ \\ \\ \\ ;\\nu} = 0$$\u3067\u3042\u308a\uff0c\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb $A_{\\mu}$ \u306b\u5bfe\u3057\u3066\u306f\u30ed\u30fc\u30ec\u30f3\u30c4\u30b2\u30fc\u30b8\u6761\u4ef6\u3092\u63a1\u7528\u3059\u308b\u3053\u3068\u306b\u3057\u3088\u3046\u3002
\n$$A^{\\mu}_{\\ \\ ;\\mu} = 0$$<\/p>\n

\u66f2\u304c\u3063\u305f\u6642\u7a7a\u306b\u304a\u3051\u308b\u5e7e\u4f55\u5149\u5b66\u8fd1\u4f3c<\/h3>\n

\u66f2\u304c\u3063\u305f\u6642\u7a7a\u306b\u304a\u3051\u308b\u5e7e\u4f55\u5149\u5b66\u3067\u306f\uff0c\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb $A_{\\nu}$ \u304c\u96fb\u78c1\u6ce2\u306e\u6ce2\u9577\u306e\u30b9\u30b1\u30fc\u30eb\u3067\u3059\u3070\u3084\u304f\u5909\u5316\u3059\u308b\u90e8\u5206 $S(\\varphi)$ \uff08$\\varphi$ \u306f\u96fb\u78c1\u6ce2\u306e\u4f4d\u76f8\uff09\u3068\uff0c\u6642\u7a7a\u306e\u66f2\u7387\u534a\u5f84\u306e\u30b9\u30b1\u30fc\u30eb\u3067\u304d\u308f\u3081\u3066\u3086\u3063\u304f\u308a\u3068\u5909\u5316\u3059\u308b\u90e8\u5206 $\\cal{A}_{\\nu}$ \u3092\u4f7f\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3068\u3059\u308b\u3002<\/p>\n

$$A_{\\nu} = S(\\varphi) {\\cal{A}}_{\\nu}$$<\/p>\n

\u5e7e\u4f55\u5149\u5b66\u300c\u8fd1\u4f3c\u300d\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3059\u308b\u3002
\n\\begin{eqnarray}
\nA_{\\nu, \\mu} &=& \\left(S(\\varphi) {\\cal{A}}_{\\nu}\\right)_{,\\mu} \\\\
\n&=& S_{,\\mu} {\\cal{A}}_{\\nu} + S {\\cal{A}}_{\\nu, \\mu} \\\\
\n&\\simeq& S_{,\\mu} {\\cal{A}}_{\\nu} \\\\
\n&=& S’ \\varphi_{,\\mu} {\\cal{A}}_{\\nu}
\n\\end{eqnarray}<\/p>\n

$S(\\varphi)$\u306e\u5909\u5316\u306e\u30b9\u30b1\u30fc\u30eb\u306b\u6bd4\u3079\u308c\u3070 ${\\cal{A}}_{\\nu}$ \u306f\u304d\u308f\u3081\u3066\u3086\u3063\u304f\u308a\u3068\u5909\u5316\u3059\u308b\u306e\u3067\uff0c${\\cal{A}}_{\\nu, \\mu}$ \u306e\u9805\u306f\u7121\u8996\u3057\u3088\u3046\u3068\u3044\u3046\u8fd1\u4f3c\u3067\u3042\u308b\u3002<\/p>\n

 <\/p>\n

\u4f4d\u76f8\u4e00\u5b9a\u9762 $\\varphi = \\mbox{const.}$ \u306e\u6cd5\u7dda\u30d9\u30af\u30c8\u30eb\u304c\u5149\u306e\u4f1d\u64ad\u3092\u8868\u30594\u5143\u30d9\u30af\u30c8\u30eb $k^{\\mu}$ \u3068\u306a\u308b\u306e\u3067\uff0c<\/p>\n

$$\\varphi_{,\\mu} \\equiv k_{\\mu}, \\quad\\therefore\\ \\ k_{\\mu, \\nu} – k_{\\nu, \\mu} = k_{\\mu; \\nu} – k_{\\nu; \\mu} =0$$<\/p>\n

4\u5143\u30d9\u30af\u30c8\u30eb $k^{\\mu}$ \u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068<\/p>\n

$$A_{\\nu;\\mu} = S’ k_{\\mu} {\\cal{A}}_{\\nu}$$<\/p>\n

$$F_{\\mu\\nu} = S’ \\left( k_{\\mu} {\\cal{A}}_{\\nu} – k_{\\nu} {\\cal{A}}_{\\mu}\\right)$$<\/p>\n

 <\/p>\n

\u30ed\u30fc\u30ec\u30f3\u30c4\u30b2\u30fc\u30b8\u6761\u4ef6\u304b\u3089<\/p>\n

$$A^{\\mu}_{\\ \\ ;\\mu} = S’ k_{\\mu} {\\cal{A}}^{\\mu} = 0, \\quad\\therefore\\ \\ k_{\\mu} {\\cal{A}}^{\\mu} = 0$$<\/p>\n

\u307e\u305f\uff0c<\/p>\n

\\begin{eqnarray}
\n0 &=& F^{\\mu\\nu}_{\\ \\ \\ ;\\nu} \\\\
\n&=& \\left\\{S’ \\left( k^{\\mu} {\\cal{A}}^{\\nu} – k^{\\nu} {\\cal{A}}^{\\mu}\\right) \\right\\}_{;\\nu} \\\\
\n&=& S^{”} k_{\\nu} \\left( k^{\\mu} {\\cal{A}}^{\\nu} – k^{\\nu} {\\cal{A}}^{\\mu}\\right)
\n+ S’ \\left( k^{\\mu} {\\cal{A}}^{\\nu} – k^{\\nu} {\\cal{A}}^{\\mu}\\right)_{;\\nu} \\\\
\n&=& S^{”}\u00a0 \\left(0\u00a0 – k_{\\nu}k^{\\nu} {\\cal{A}}^{\\mu}\\right)
\n+ S’ \\left( k^{\\mu} {\\cal{A}}^{\\nu} – k^{\\nu} {\\cal{A}}^{\\mu}\\right)_{;\\nu}
\n\\end{eqnarray}<\/p>\n

$S^{”} $ \u306e\u4fc2\u6570\u306f\u72ec\u7acb\u306b\u30bc\u30ed\u306b\u306a\u3089\u306a\u3044\u3068\u3044\u3051\u306a\u3044\u306e\u3067\uff0c
\n$$k_{\\nu}k^{\\nu} = 0$$
\n\u3068\u3044\u3046\u30cc\u30eb\u6761\u4ef6\u304c\u51fa\u3066\u304f\u308b\u3002<\/p>\n

\u3053\u306e\u30cc\u30eb\u6761\u4ef6\u3068 “curl-free” \uff08\u300c\u3046\u305a\u306a\u3057<\/strong><\/span>\u300d\u306e\u6c17\u53d6\u3063\u305f\u8868\u73fe\uff09\u6761\u4ef6 $k_{\\mu; \\nu} – k_{\\nu; \\mu} =0$ \u3092\u4f7f\u3046\u3068\uff0c$k^{\\nu}$ \u304c\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u306b\u5f93\u3046\u3053\u3068\u304c\u5c0e\u304b\u308c\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n0 &=& \\left( k_{\\mu} k^{\\mu}\\right)_{;\\nu} \\\\
\n&=& 2 k_{\\mu; \\nu} k^{\\mu}\\\\
\n&=& 2 k_{\\nu; \\mu} k^{\\mu}\\\\
\n\\therefore\\ \\\u00a0 k^{\\nu}_{\\ \\ ; \\mu} k^{\\mu} &=& 0
\n\\end{eqnarray}<\/p>\n

\u5e7e\u4f55\u5149\u5b66\u8fd1\u4f3c\u3067\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb<\/h3>\n

\u6700\u7d42\u7684\u306b\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb\u306f \uff08$k_{\\mu} k^{\\mu}=0$\u00a0 \u3068 $k_{\\mu} {\\cal{A}}^{\\mu}=0$ \u304b\u3089 $F_{\\alpha\\beta} F^{\\alpha\\beta} = 0$\u306f\u3059\u3050\u308f\u304b\u308b\u306e\u3067\uff09<\/p>\n

\\begin{eqnarray}
\n\\frac{1}{\\varepsilon_0} T_{\\mu\\nu} &=& F_{\\mu\\alpha} F_{\\nu}^{\\ \\ \\alpha}
\n– \\frac{1}{4} g_{\\mu\\nu} F_{\\alpha\\beta} F^{\\alpha\\beta} \\\\
\n&=& (S’)^2 \\left( k_{\\mu} {\\cal{A}}_{\\alpha} – k_{\\alpha} {\\cal{A}}_{\\mu}\\right) \\left( k_{\\nu} {\\cal{A}}^{\\alpha} – k^{\\alpha} {\\cal{A}}_{\\nu}\\right) \\\\
\n&=& (S’)^2 {\\cal{A}}_{\\alpha}{\\cal{A}}^{\\alpha}k_{\\mu}k_{\\nu}
\n\\end{eqnarray}<\/p>\n

$A^2 \\equiv \\varepsilon_0 (S’)^2 {\\cal{A}}_{\\alpha}{\\cal{A}}^{\\alpha}$ \u3068\u304a\u304f\u3068<\/p>\n

$$T^{\\mu\\nu} = A^2 k^{\\mu}k^{\\nu}$$<\/p>\n

\u3053\u308c\u304c4\u5143\u30d9\u30af\u30c8\u30eb $k^{\\mu}$ \u3067\u8868\u3055\u308c\u308b\u300c\u5358\u8272\u5149\u300d\u306e\u5149\u6e90\u5929\u4f53\u304b\u3089\u653e\u51fa\u3055\u308c\u308b\uff0c\u96fb\u78c1\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb\u3067\u3042\u308b\u3002<\/p>\n

\\begin{eqnarray}
\nT^{\\mu\\nu}_{\\ \\ \\ ;\\nu} &=&\\left\\{ \\left(A^2\\right)_{,\\nu} k^{\\nu}\u00a0 + A^2 k^{\\nu}_{\\ \\ ;\\nu} \\right\\} k^{\\mu}
\n+ A^2 k^{\\mu}_{\\ \\ ;\\nu} k^{\\nu} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\u3088\u308a\uff0c\u65e2\u51fa\u306e\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f $k^{\\mu}_{\\ \\ ;\\nu} k^{\\nu} =0$ \u3068
\n$$\\left(A^2\\right)_{,\\nu} k^{\\nu}\u00a0 + A^2 k^{\\nu}_{\\ \\ ;\\nu} = 0$$\u3042\u308b\u3044\u306f\u5c11\u3057\u66f8\u304d\u6574\u3048\u3066
\n$$A_{,\\nu} k^{\\nu}\u00a0 + \\frac{1}{2} A k^{\\nu}_{\\ \\ ;\\nu} = 0$$\u304c\u5f97\u3089\u308c\u308b\u3002\u5149\u306e\u4e16\u754c\u7dda\u3092\u30d1\u30e9\u30e1\u30c8\u30e9\u30a4\u30ba\u3059\u308b\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf $v$ \u3092\u4f7f\u3048\u3070\uff0c$\\displaystyle k^{\\nu} = \\frac{dx^{\\nu}}{dv}$ \u3067\u3042\u308b\u304b\u3089<\/p>\n

$$A_{,\\nu} k^{\\nu} = \\frac{dx^{\\nu}}{dv} \\frac{\\partial A}{\\partial x^{\\nu}} = \\frac{dA}{dv}$$\u3092\u4f7f\u3046\u3068<\/p>\n

$$\\frac{dA}{dv}\u00a0 + \\frac{1}{2} A k^{\\nu}_{\\ \\ ;\\nu} = 0$$\u3068\u66f8\u3044\u3066\u3082\u3088\u3044\u3002<\/p>\n

\u307e\u3068\u3081<\/h3>\n

4\u5143\u30d9\u30af\u30c8\u30eb $k^{\\mu}$ \u3067\u8868\u3055\u308c\u308b\u300c\u5358\u8272\u5149\u300d\u306e\u5149\u6e90\u5929\u4f53\u304b\u3089\u653e\u51fa\u3055\u308c\u308b\uff0c\u96fb\u78c1\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u904b\u52d5\u91cf\u30c6\u30f3\u30bd\u30eb\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n

$$T^{\\mu\\nu} = A^2 k^{\\mu}k^{\\nu}$$<\/p>\n

$T^{\\mu\\nu}_{\\ \\ \\ ;\\nu}=0$ \u3088\u308a
\n$$A_{,\\nu} k^{\\nu}\u00a0 + \\frac{1}{2} A k^{\\nu}_{\\ \\ ;\\nu} = 0$$<\/p>\n

$k^{\\mu}$ \u306f\u3046\u305a\u306a\u3057\u306e\u30cc\u30eb\u6e2c\u5730\u7dda\u3067\u3042\u308b\u3002<\/p>\n

$$
\nk_{\\mu, \\nu} – k_{\\nu, \\mu} = k_{\\mu; \\nu} – k_{\\nu; \\mu} =0
\n$$
\n$$ k_{\\mu} k^{\\mu} = 0, \\quad k^{\\mu}_{\\ \\ ;\\nu} k^{\\nu} = 0$$<\/p>\n

 <\/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><\/a> (Academic Press, New York, 1971) \u306e P.144 \u3042\u305f\u308a\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u89d2\u5f84\u8ddd\u96e2\u3084\u5149\u5ea6\u8ddd\u96e2\u306e\u5b9a\u7fa9\u306b\u5fc5\u8981\u306a\uff0c\u66f2\u304c\u3063\u305f\u6642\u7a7a\u306b\u304a\u3051\u308b\u5e7e\u4f55\u5149\u5b66\u8fd1\u4f3c\u306b\u3064\u3044\u3066\u304a\u3055\u3089\u3057\u3066\u304a\u304f\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":1430,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1580","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\/1580","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=1580"}],"version-history":[{"count":40,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1580\/revisions"}],"predecessor-version":[{"id":7331,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1580\/revisions\/7331"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1430"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=1580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}