{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:5cac9313-3d4f-4510-8bac-b257c148accb", "labels": { "zh-CN": "希尔伯特拉格朗日量", "en": "Hilbert Lagrangian" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "96e9ad8b6ab243a4e88856b38589c963d58400b0b978de5b959a3f53eea0720e", "occurrences": [ { "concept_id": "urn:uuid:5cac9313-3d4f-4510-8bac-b257c148accb", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-18/10-variation-principles-in-space-time", "anchor": "first-585ec3f84b40b0c0f6c5-zh", "chapter": "18", "section": "18.10", "offset": 5575, "range": [ 11827, 12075 ], "source_revision": "b982eed96313e8879cd18c3a3ecff3c5e674745ed6834ded3768f3f16a774ad0", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "希尔伯特拉格朗日量(Hilbert Lagrangian)", "context": { "range": [ 11784, 12478 ], "quote": "爱因斯坦场方程可由变分原理导出，只需将拉格朗日量取为里奇标量，$L = R$。这被称为希尔伯特拉格朗日量(Hilbert Lagrangian)。对于独立变量，既可以取度量张量分量$g _ { \\mu \\nu }$，也可以取其逆张量的分量$g ^ { \\mu \\nu }$。我们将采用后者，因为它稍微更为方便（读者可以尝试针对变量$\\Phi _ { A } = g _ { \\mu \\nu }$ 调整随后的分析。我们不能直接使用欧拉–拉格朗日方程 式 (18.113)，因为$R$ 依赖于$g ^ { \\mu \\nu }$ 及其一阶和二阶导数。虽然欧拉–拉格朗日分析可以推广到包含依赖于场二阶导数的拉格朗日量（见问题18.32），但这将是一项极其复杂的计算。直接进行，我们有", "quote_sha256": "7642ca77827da7c2c5c0eab3898424106688c3c9bb376dd727b8d09d46dd9c50", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-18/10-variation-principles-in-space-time", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-979b6be045d827ea50ccacfe" }, { "concept_id": "urn:uuid:5cac9313-3d4f-4510-8bac-b257c148accb", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-18/10-variation-principles-in-space-time", "anchor": "first-585ec3f84b40b0c0f6c5-en", "chapter": "18", "section": "18.10", "offset": 7232, "range": [ 13007, 13244 ], "source_revision": "0ba07ad6e3a2b36c7b9f5a47a6adc347131b5c1a4d08dfaa06310e61aaf02546", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Hilbert Lagrangian", "context": { "range": [ 12856, 14217 ], "quote": "Einstein’s field equations may be derived from a variation principle, by setting the Lagrangian to be the Ricci scalar, $L = R$ . This is known as the Hilbert Lagrangian. For independent variables it is possible to take either the metric tensor components $g _ { \\mu \\nu }$ or those of the inverse tensor $g ^ { \\mu \\nu }$ . We will adopt the latter, as it is slightly more convenient (the reader may try to adapt the analysis that follows for the variables $\\Phi _ { A } = g _ { \\mu \\nu }$ . We cannot use the Euler–Lagrange equations Eq. (18.113) as they stand since $R$ is dependent on $g ^ { \\mu \\nu }$ and its first and second derivatives. While the Euler–Lagrange analysis can be extended to include Lagrangians that depend on second derivatives of the fields (see Problem 18.32), this would be a prohibitively complicated calculation. Proceeding directly, we have", "quote_sha256": "96def77195e6715590f2c2c12f12c4f06e31e866e871e4a4ed4fa6f96d9366a4", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-18/10-variation-principles-in-space-time", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-10287223e62fa56cdaf13480" } ], "registered_positions": [ { "route": "en/chapter-18/10-variation-principles-in-space-time", "lang": "en", "section": "18.10", "signal": "term-mark", "label": "Hilbert Lagrangian", "english": "Hilbert Lagrangian", "offset": 6850 }, { "route": "zh/chapter-18/10-variation-principles-in-space-time", "lang": "zh-CN", "section": "18.10", "signal": "term-mark", "label": "希尔伯特拉格朗日量", "english": "Hilbert Lagrangian", "offset": 5193 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#equation-18-113\"", "href": "#equation-18-113", "range": [ 12230, 12253 ], "relation": "references", "target_route": "zh/chapter-18/10-variation-principles-in-space-time", "target_anchor": "equation-18-113", "resolved": true, "reason": null, "source_revision": "b982eed96313e8879cd18c3a3ecff3c5e674745ed6834ded3768f3f16a774ad0", "source_route": "zh/chapter-18/10-variation-principles-in-space-time", "occurrence_id": "occ-979b6be045d827ea50ccacfe" }, { "raw": "href=\"#equation-18-113\"", "href": "#equation-18-113", "range": [ 13614, 13637 ], "relation": "references", "target_route": "en/chapter-18/10-variation-principles-in-space-time", "target_anchor": "equation-18-113", "resolved": true, "reason": null, "source_revision": "0ba07ad6e3a2b36c7b9f5a47a6adc347131b5c1a4d08dfaa06310e61aaf02546", "source_route": "en/chapter-18/10-variation-principles-in-space-time", "occurrence_id": "occ-10287223e62fa56cdaf13480" }, { "raw": "href=\"#problem-18-32\"", "href": "#problem-18-32", "range": [ 13987, 14008 ], "relation": "references", "target_route": "en/chapter-18/10-variation-principles-in-space-time", "target_anchor": "problem-18-32", "resolved": true, "reason": null, "source_revision": "0ba07ad6e3a2b36c7b9f5a47a6adc347131b5c1a4d08dfaa06310e61aaf02546", "source_route": "en/chapter-18/10-variation-principles-in-space-time", "occurrence_id": "occ-10287223e62fa56cdaf13480" } ], "reading_order": [ "occ-979b6be045d827ea50ccacfe", "occ-10287223e62fa56cdaf13480" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "da6bf8f6c884f63cd30b06ebd562d0dfc3f01972dd6bccfe443c409ab86dc5db", "optional_enrichments": [] }