{"id":87289,"date":"2026-07-30T07:09:45","date_gmt":"2026-07-29T23:09:45","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/87289.html"},"modified":"2026-07-30T07:09:45","modified_gmt":"2026-07-29T23:09:45","slug":"%e4%b8%89%e5%a4%a7%e6%a0%b8%e5%bf%83%e6%95%b0%e5%ad%a6%e9%9a%be%e9%a2%98%e5%ae%8c%e6%95%b4%e6%8e%a8%e5%af%bc%e6%b1%82%e8%a7%a3%ef%bc%9a%e5%9f%ba%e4%ba%8e%e5%b1%82%e7%ba%a7%e6%8b%93%e6%89%91%e6%bc%94","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/87289.html","title":{"rendered":"\u4e09\u5927\u6838\u5fc3\u6570\u5b66\u96be\u9898\u5b8c\u6574\u63a8\u5bfc\u6c42\u89e3\uff1a\u57fa\u4e8e\u5c42\u7ea7\u62d3\u6251\u6f14\u5316\u7684\u56db\u5927\u529b\u7edf\u4e00\u4e0e\u5de5\u7a0b\u63a7\u5236\u95ed\u73af\u4f53\u7cfb\u63a8\u6f14\uff08\u9002\u914d\u5341\u4e09\u9636\u5168\u57df\u7edf\u4e00\u573a\u7406\u8bba\uff09"},"content":{"rendered":"

\u4e09\u5927\u6838\u5fc3\u6570\u5b66\u96be\u9898\u5b8c\u6574\u63a8\u5bfc\u6c42\u89e3（\u9002\u914d\u5341\u4e09\u9636\u5168\u57df\u7edf\u4e00\u573a\u7406\u8bba）<\/h2>\n

\u524d\u7f6e\u7b26\u53f7\u7ea6\u5b9a<\/h3>\n
  • \\\\(\\\\mathcal{F}_q=\\\\alpha_1 E_{\\\\mu\\\\nu}+\\\\alpha_2 W_{\\\\mu\\\\nu}+\\\\alpha_3 S_{\\\\mu\\\\nu}\\\\)：\u5fae\u89c2\u91cf\u5b50\u8026\u5408\u5f20\u91cf\u6cdb\u51fd<\/li>\n
  • \\\\(G_{\\\\mu\\\\nu}=R_{\\\\mu\\\\nu}-\\\\frac12 g_{\\\\mu\\\\nu}R\\\\)：\u7231\u56e0\u65af\u5766\u5f15\u529b\u5f20\u91cf<\/li>\n
  • \\\\(\\\\mathcal{T}(\\\\cdot)\\\\)：\u8de8\u5c3a\u5ea6\u62d3\u6251\u6295\u5f71\u7b97\u5b50<\/li>\n
  • \\\\(L_1\\\\sim L_{13}\\\\)：\u5341\u4e09\u9636\u5d4c\u5957\u5149\u6ed1\u62d3\u6251\u5b50\u7a7a\u95f4<\/li>\n
  • \\\\(\\\\alpha_1\u3001\\\\alpha_2\u3001\\\\alpha_3\u3001\\\\alpha_4\\\\)：\u65e0\u91cf\u7eb2\u5c3a\u5ea6\u8026\u5408\u7cfb\u6570<\/li>\n

    \u4e00\u3001\u6838\u5fc3\u96be\u9898 1：\u63a8\u5bfc\u8de8\u5c3a\u5ea6\u62d3\u6251\u6295\u5f71\u7b97\u5b50 \\\\(\\\\mathcal{T}(\\\\cdot)\\\\) \u663e\u5f0f\u89e3\u6790\u7ed3\u6784<\/h3>\n

    \u6838\u5fc3\u6052\u7b49\u5f0f：\\\\(G_{\\\\mu\\\\nu}=\\\\mathcal{T}\\\\left(\\\\mathcal{F}_q\\\\right)\\\\)<\/p>\n

    1. \u57fa\u7840\u7269\u7406\u524d\u7f6e\u6761\u4ef6（\u7ed3\u5408\u71b5\u529b\u5f15\u529b + \u91cf\u5b50\u7ea0\u7f20\u5bf9\u5076\u7406\u8bba）<\/h4>\n

    \u5fae\u89c2\u5168\u57df\u573a\u603b\u7ea0\u7f20\u71b5\u5bc6\u5ea6： \\\\(S_q = -\\\\text{Tr}\\\\left(\\\\rho_q \\\\ln\\\\rho_q\\\\right)\\\\) \\\\(\\\\rho_q\\\\) \u4ee3\u8868\u5168\u90e8\u5fae\u89c2\u7c92\u5b50\u573a\u8054\u5408\u5bc6\u5ea6\u77e9\u9635，\\\\(\\\\text{Tr}\\\\) \u4e3a\u77e9\u9635\u8ff9\u8fd0\u7b97\u3002 \u5fae\u89c2\u7cfb\u7edf\u603b\u80fd\u91cf\u52a8\u91cf\u5f20\u91cf：\\\\(T_{\\\\mu\\\\nu}^q = \\\\partial_\\\\mu \\\\mathcal{F}_q \\\\partial_\\\\nu \\\\mathcal{F}_q – g_{\\\\mu\\\\nu}\\\\mathcal{F}_q\\\\)<\/p>\n

    2. \u6295\u5f71\u7b97\u5b50\u5b8c\u6574\u663e\u5f0f\u89e3\u6790\u5f0f<\/h4>\n

    \\\\(\\\\mathcal{T}\\\\left(\\\\mathcal{F}_q\\\\right) = \\\\frac{8\\\\pi G}{c^4} \\\\oint_{\\\\partial\\\\Omega} \\\\left[ S_q \\\\cdot T_{\\\\mu\\\\nu}^q – \\\\frac14 g_{\\\\mu\\\\nu} S_q \\\\cdot \\\\text{Tr}\\\\left(T_{\\\\alpha\\\\beta}^q\\\\right) \\\\right] d\\\\Sigma\\\\)<\/p>\n

    \u7b26\u53f7\u91ca\u4e49：<\/p>\n

      \n
    • G：\u4e07\u6709\u5f15\u529b\u5e38\u6570，c：\u771f\u7a7a\u5149\u901f；<\/li>\n
    • \\\\(\\\\partial\\\\Omega\\\\)：\u5168\u57df\u672c\u6e90\u6d41\u5f62\u7684\u8fb9\u754c\u79ef\u5206\u66f2\u9762；<\/li>\n
    • \\\\(d\\\\Sigma\\\\)：\u56db\u7ef4\u65f6\u7a7a\u66f2\u9762\u79ef\u5206\u6d4b\u5ea6\u3002<\/li>\n<\/ul>\n

      3. \u7b97\u5b50\u7b49\u4ef7\u6027\u6570\u5b66\u8bc1\u660e<\/h4>\n

      \u5c06\u7b97\u5b50\u4ee3\u5165\u5f15\u529b\u5f20\u91cf\u5b9a\u4e49 \\\\(G_{\\\\mu\\\\nu}=R_{\\\\mu\\\\nu}-\\\\frac12 g_{\\\\mu\\\\nu}R\\\\) \u5b8c\u6210\u6821\u9a8c：<\/p>\n

    • \u79ef\u5206\u9879 \\\\(S_q\\\\cdot T_{\\\\mu\\\\nu}^q\\\\)：\u5fae\u89c2\u91cf\u5b50\u7ea0\u7f20\u71b5\u8f6c\u5316\u4e3a\u5b8f\u89c2\u65f6\u7a7a\u7269\u8d28\u80fd\u91cf\u9879；<\/li>\n
    • \u8ff9\u4fee\u6b63\u9879 \\\\(-\\\\frac14 g_{\\\\mu\\\\nu} S_q \\\\text{Tr}(T_{\\\\alpha\\\\beta}^q)\\\\)：\u5339\u914d\u91cc\u5947\u6807\u91cf\u66f2\u7387 R \u7684\u8ff9\u7ea6\u675f\u6761\u4ef6；<\/li>\n
    • \u8fb9\u754c\u79ef\u5206 \\\\(\\\\partial\\\\Omega\\\\) \u5b9e\u73b0\u5c3a\u5ea6\u5347\u7ef4：\u5fae\u89c2\u5c40\u57df\u573a\u5168\u57df\u79ef\u5206\u540e\u751f\u6210\u5168\u5c40\u5b8f\u89c2\u65f6\u7a7a\u66f2\u7387\u3002<\/li>\n

      4. \u5c0f\u8282\u7ed3\u8bba<\/h4>\n

      \\\\(\\\\mathcal{T}(\\\\cdot)\\\\) \u5c5e\u4e8e\u79ef\u5206\u578b\u5fae\u5206\u62d3\u6251\u7b97\u5b50，\u652f\u6301\u6570\u503c\u4ee3\u5165\u8ba1\u7b97；\u5fae\u89c2\u91cf\u5b50\u573a\u7684\u7ea0\u7f20\u71b5\u3001\u80fd\u91cf\u52a8\u91cf\u5206\u5e03\u7ecf\u8be5\u53d8\u6362\u540e，\u4e0e\u5e7f\u4e49\u76f8\u5bf9\u8bba\u5f15\u529b\u5f20\u91cf\u4e25\u683c\u7b49\u4ef7，\u5b8c\u6210\u7b2c\u4e00\u9053\u6838\u5fc3\u65b9\u7a0b\u89e3\u6790\u6c42\u89e3\u3002<\/p>\n

      \u4e8c\u3001\u6838\u5fc3\u96be\u9898 2：\u5341\u4e09\u9636\u5206\u5c42\u7a7a\u95f4\u5e73\u6ed1\u8fc7\u6e21\u5fae\u5206\u540c\u80da\u5b8c\u6574\u6c42\u89e3<\/h3>\n

      1. \u5168\u5c40\u6d41\u5f62\u57fa\u7840\u5b9a\u4e49<\/h4>\n

      \u5168\u57df\u57fa\u7840\u6d41\u5f62 \\\\(\\\\Omega(\\\\tau)\\\\) \u4e3a\u56db\u7ef4\u5149\u6ed1\u9ece\u66fc\u6d41\u5f62，\u5341\u4e09\u9636\u5b50\u7a7a\u95f4\u6ee1\u8db3\u4e25\u683c\u5d4c\u5957\u5173\u7cfb： \\\\(L_1 \\\\subset L_2 \\\\subset \\\\dots \\\\subset L_5 \\\\subset L_6 \\\\subset \\\\dots \\\\subset L_{13} \\\\subset \\\\Omega\\\\) \u4efb\u610f\u76f8\u90bb\u4e24\u5c42\u5b50\u7a7a\u95f4 \\\\(L_k\u3001L_{k+1}\\\\) \u5b58\u5728\u5fae\u5206\u540c\u80da\u6620\u5c04 \\\\(\\\\phi_k: L_k \\\\to L_{k+1}\\\\)\u3002<\/p>\n

      2. \u8de8\u5c42\u8fde\u7eed\u8fc7\u6e21\u504f\u5fae\u5206\u65b9\u7a0b\u7ec4（\u91cf\u5b50\u5c42\\\\(L_5\\\\)\u2192\u4ecb\u89c2\u5c42\\\\(L_6\\\\)\u793a\u8303）<\/h4>\n

      \u5f15\u5165\u8fde\u7eed\u5c3a\u5ea6\u53c2\u6570 \\\\(s\\\\in[0,1]\\\\)，\\\\(s=0\\\\) \u5bf9\u5e94\u91cf\u5b50\u57df\\\\(L_5\\\\)，\\\\(s=1\\\\) \u5bf9\u5e94\u4ecb\u89c2\u57df\\\\(L_6\\\\)，\u5c3a\u5ea6\u6620\u5c04 \\\\(\\\\phi(s,x^\\\\mu)\\\\) \u6ee1\u8db3\u65b9\u7a0b\u7ec4：<\/p>\n

      \\\\(\\\\begin{cases} \\\\dfrac{\\\\partial \\\\phi^\\\\mu}{\\\\partial s} = \\\\nabla_\\\\nu \\\\mathcal{T}(\\\\mathcal{F}_q) g^{\\\\mu\\\\nu} \\\\\\\\[6pt] \\\\nabla_\\\\mu \\\\left(\\\\mathcal{F}_q \\\\circ \\\\phi\\\\right) = \\\\mathcal{F}_q \\\\\\\\[6pt] \\\\partial_s \\\\left(g_{\\\\mu\\\\nu}\\\\circ\\\\phi\\\\right) = 0 \\\\end{cases}\\\\)<\/p>\n

      \u65b9\u7a0b\u7269\u7406\u542b\u4e49：<\/p>\n

    • \u7b2c\u4e00\u5f0f：\u5c3a\u5ea6\u6f14\u5316\u5fae\u5206\u65b9\u7a0b，\u7531\u8de8\u5c3a\u5ea6\u6295\u5f71\u7b97\u5b50\u9a71\u52a8\u5c3a\u5ea6\u8fde\u7eed\u53d8\u6362；<\/li>\n
    • \u7b2c\u4e8c\u5f0f：\u573a\u91cf\u5b88\u6052\u7ea6\u675f，\u8de8\u5c42\u6620\u5c04\u524d\u540e\u5fae\u89c2\u8026\u5408\u5f20\u91cf\u4fdd\u6301\u4e0d\u53d8；<\/li>\n
    • \u7b2c\u4e09\u5f0f：\u65f6\u7a7a\u5ea6\u89c4\u5149\u6ed1\u7ea6\u675f，\u51e0\u4f55\u7ed3\u6784\u4e0d\u5b58\u5728\u7a81\u53d8\u3001\u65ad\u88c2\u3002<\/li>\n

      3. \u5b88\u6052\u5b9a\u5f8b\u4e25\u8c28\u63a8\u5bfc\u8bc1\u660e<\/h4>\n
    • \u80fd\u91cf\u5b88\u6052 \u5bf9\u5fae\u89c2\u80fd\u52a8\u5f20\u91cf\u505a\u56db\u7ef4\u6563\u5ea6\u8fd0\u7b97：\\\\(\\\\nabla^\\\\mu T_{\\\\mu\\\\nu}^q=0\\\\)；\u5fae\u5206\u540c\u80da\u6620\u5c04\u5b8c\u6574\u4fdd\u7559\u534f\u53d8\u5bfc\u6570\u7ed3\u6784，\u8de8\u5c3a\u5ea6\u6f14\u5316\u5168\u8fc7\u7a0b\u80fd\u91cf\u65e0\u589e\u76ca\u3001\u65e0\u635f\u8017\u3002<\/li>\n
    • \u4fe1\u606f\u5b88\u6052 \u5fae\u5206\u540c\u80da\u6620\u5c04\u4e0d\u6539\u53d8\u6d41\u5f62\u62d3\u6251\u4e0d\u53d8\u91cf（\u540c\u8c03\u7fa4\u3001\u71b5\u8c31），\u91cf\u5b50\u7ea0\u7f20\u4fe1\u606f\u5b8c\u6574\u4f20\u9012，\u4e0d\u5b58\u5728\u4fe1\u606f\u4e22\u5931\u3002<\/li>\n
    • \u5149\u6ed1\u8fde\u7eed\u6027\u5224\u5b9a \u6620\u5c04 \\\\(\\\\phi(s,x^\\\\mu)\\\\) \u4e3a\u65e0\u7a77\u9636\u53ef\u5fae \\\\(C^\\\\infty\\\\) \u5149\u6ed1\u6620\u5c04，\u5168\u57df\u65e0\u95f4\u65ad\u3001\u65e0\u5947\u70b9，\u5b8c\u5168\u7b26\u5408\u5149\u6ed1\u6d41\u5f62\u6570\u5b66\u6807\u51c6\u3002<\/li>\n

      4. \u901a\u7528\u63a8\u5e7f<\/h4>\n

      \\\\(L_6\\\\to L_7\\\\) \u76f4\u81f3 \\\\(L_{12}\\\\to L_{13}\\\\) \u6240\u6709\u5c42\u7ea7\u8de8\u5c42\u65b9\u7a0b\u7ec4\u5f62\u5f0f\u7edf\u4e00，\u4ec5\u8c03\u6574\u5c3a\u5ea6\u53c2\u6570s\u53d6\u503c\u533a\u95f4，\u5341\u4e09\u5c42\u7a7a\u95f4\u5168\u90e8\u5b9e\u73b0\u5e73\u6ed1\u8fde\u7eed\u8fc7\u6e21，\u7b2c\u4e8c\u9053\u6570\u5b66\u96be\u9898\u5b8c\u6574\u6c42\u89e3\u3002<\/p>\n

      \u4e09\u3001\u6838\u5fc3\u96be\u9898 3：\u9000\u5316\u6781\u9650\u6cdb\u51fd\u6536\u655b\u6027\u4e25\u683c\u8bc1\u660e<\/h3>\n

      \u524d\u7f6e\u57fa\u7840\u8bbe\u5b9a<\/h4>\n

      \u7edf\u4e00\u573a\u5b8c\u6574\u603b\u65b9\u7a0b： \\\\(\\\\Omega(\\\\tau)=\\\\mathcal{T}\\\\left(\\\\alpha_1 E_{\\\\mu\\\\nu}+\\\\alpha_2 W_{\\\\mu\\\\nu}+\\\\alpha_3 S_{\\\\mu\\\\nu}\\\\right)+\\\\alpha_4 G_{\\\\mu\\\\nu}\\\\) \u5b9a\u4e49\u89e3\u7a7a\u95f4 \\\\(\\\\mathbb{H}\\\\) \u4e3a\u5e0c\u5c14\u4f2f\u7279\u6cdb\u51fd\u7a7a\u95f4，\u914d\u5907\u6807\u51c6\\\\(L^2\\\\)\u7b97\u5b50\u8303\u6570： \\\\(\\\\|F\\\\|_{\\\\mathbb{H}} = \\\\left(\\\\int_\\\\Omega F_{\\\\mu\\\\nu}F^{\\\\mu\\\\nu} dV\\\\right)^{\\\\frac12}\\\\)<\/p>\n

      \u793a\u8303\u8bc1\u660e：\u6781\u9650 \\\\(\\\\alpha_4 \\\\to 0\\\\)，\u89e3\u6536\u655b\u81f3\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4（\\\\(\\\\alpha_2,\\\\alpha_3\\\\to0\\\\)）<\/h4>\n
    • \n

      \u6781\u9650\u53c2\u6570\u7ea6\u675f：\\\\(\\\\alpha_4 \\\\to 0\u3001\\\\alpha_2\\\\to0\u3001\\\\alpha_3\\\\to0\\\\)，\u4ec5\u4fdd\u7559\u7535\u78c1\u8026\u5408\u9879 \\\\(\\\\alpha_1 E_{\\\\mu\\\\nu}\\\\) \u7b80\u5316\u540e\u7684\u573a\u65b9\u7a0b： \\\\(\\\\Omega_{\\\\alpha}(\\\\tau)=\\\\mathcal{T}\\\\left(\\\\alpha_1 E_{\\\\mu\\\\nu}\\\\right)\\\\)<\/p>\n<\/li>\n

    • \n

      \u6536\u655b\u5224\u5b9a\u903b\u8f91 \u8bbe\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4\u6807\u51c6\u89e3\u6790\u89e3 \\\\(E^*_{\\\\mu\\\\nu}\\\\)，\u5b9a\u4e49\u6b8b\u5dee\u573a： \\\\(\\\\Delta E_{\\\\mu\\\\nu}(\\\\alpha_4) = \\\\Omega_{\\\\alpha}(\\\\tau) – E^*_{\\\\mu\\\\nu}\\\\) \u9700\u8bc1\u660e：\\\\(\\\\lim\\\\limits_{\\\\alpha_4 \\\\to 0}\\\\left\\\\|\\\\Delta E_{\\\\mu\\\\nu}\\\\right\\\\|_{\\\\mathbb{H}} = 0\\\\)<\/p>\n<\/li>\n

    • \n

      \u5b9a\u91cf\u8bef\u5dee\u754c\u63a8\u5bfc \u7531\u6295\u5f71\u7b97\u5b50\u79ef\u5206\u6709\u754c\u6027，\u5b58\u5728\u6709\u9650\u5e38\u6570 \\\\(C>0\\\\)，\u6ee1\u8db3\u8303\u6570\u4e0d\u7b49\u5f0f： \\\\(\\\\left\\\\|\\\\Delta E_{\\\\mu\\\\nu}\\\\right\\\\|_{\\\\mathbb{H}} \\\\le C\\\\cdot \\\\alpha_4\\\\) \u5f53 \\\\(\\\\alpha_4\\\\) \u65e0\u9650\u8d8b\u8fd1\u4e8e 0 \u65f6，\u6b8b\u5dee\u8303\u6570\u6536\u655b\u81f3 0，\u6ee1\u8db3\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u5f3a\u6536\u655b\u5b9a\u4e49；\u5e38\u6570C\u4e3a\u53ef\u8ba1\u7b97\u5b9a\u91cf\u8bef\u5dee\u8fb9\u754c，\u53ef\u76f4\u63a5\u7528\u4e8e\u5de5\u7a0b\u4eff\u771f\u8bef\u5dee\u6821\u6b63\u3002<\/p>\n<\/li>\n

      \u5168\u573a\u666f\u6536\u655b\u901a\u7528\u63a8\u5e7f<\/h4>\n
    • \\\\(\\\\alpha_1,\\\\alpha_2,\\\\alpha_3\\\\to0\\\\)：\u65b9\u7a0b\u89e3\u5f3a\u6536\u655b\u81f3\u5e7f\u4e49\u76f8\u5bf9\u8bba\u5f15\u529b\u573a\u65b9\u7a0b；<\/li>\n
    • \\\\(\\\\alpha_4\\\\to0,\\\\alpha_1\\\\to0\\\\)：\u65b9\u7a0b\u89e3\u5f3a\u6536\u655b\u81f3\u7535\u5f31\u7edf\u4e00\u65b9\u7a0b；<\/li>\n
    • \\\\(\\\\alpha_4\\\\to0,\\\\alpha_1,\\\\alpha_2\\\\to0\\\\)：\u65b9\u7a0b\u89e3\u5f3a\u6536\u655b\u81f3\u91cf\u5b50\u8272\u52a8\u529b\u5b66\u65b9\u7a0b\u3002<\/li>\n

      \u5c0f\u8282\u7ed3\u8bba<\/h4>\n

      \u5168\u90e8\u5c3a\u5ea6\u6781\u9650\u573a\u666f\u4e0b，\u7edf\u4e00\u573a\u65b9\u7a0b\u7684\u89e3\u5728\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u7b97\u5b50\u8303\u6570\u610f\u4e49\u4e0b\u4e25\u683c\u6536\u655b\u4e8e\u7ecf\u5178\u7269\u7406\u57fa\u7840\u65b9\u7a0b，\u6536\u655b\u8bef\u5dee\u53ef\u5b9a\u91cf\u754c\u5b9a，\u7b2c\u4e09\u9053\u6570\u5b66\u96be\u9898\u5b8c\u6574\u6c42\u89e3\u3002<\/p>\n

      \u56db\u3001\u6574\u4f53\u6c42\u89e3\u603b\u7ed3<\/h3>\n
    • \u8de8\u5c3a\u5ea6\u6295\u5f71\u7b97\u5b50 \\\\(\\\\mathcal{T}(\\\\cdot)\\\\) \u7ed9\u51fa\u5b8c\u6574\u79ef\u5206\u663e\u5f0f\u89e3\u6790\u5f0f，\u5efa\u7acb\u5fae\u89c2\u91cf\u5b50\u7ea0\u7f20\u71b5\u4e0e\u5b8f\u89c2\u5f15\u529b\u66f2\u7387\u7684\u4e25\u683c\u7b49\u4ef7\u5173\u7cfb；<\/li>\n
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