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        "quote": "\n> [!theorem] 拓展 · 上指标与下指标\n>\n> *在许多物理学文献中，在讨论狭义相对论时，习惯上使用上指标和下指标来书写公式，二者通过以下关系相联系：*\n>\n> $$\n> x _ {\\mu} = \\sum_ {\\nu = 0} ^ {3} \\eta_ {\\mu \\nu} x ^ {\\nu} = \\eta_ {\\mu \\nu} x ^ {\\nu}\n> $$\n>\n> *其中最后一个形式使用了[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/einstein-summation-convention\" data-concept-identity=\"urn:uuid:aebcd7da-6aaa-47db-831d-2be25a85038a\" data-concept-route=\"/concept-identities#concept-aebcd7da-6aaa-47db-831d-2be25a85038a\" id=\"first-85cf4063ce0a0f6f11cd-zh\">Einstein求和约定(Einstein summation convention)</u>](../../../../../concepts/linear-algebra/einstein-summation-convention)。*\n>\n> *引入上指标与下指标的一个动机是：狭义相对论是广义相对论的极限情形，而广义相对论是一种完全几何化的理论，其基础是把时空视为一个流形 $M$，并带有一个逐点变化的度量 $g$。在这样的理论中，重要的是区分点 $x \\in M$ 处的切空间 $T _ { x } ( M )$ 的元素与其对偶余切空间 $T _ { x } ^ { * } ( M )$ 的元素，同时利用度量 $g$ 在 $T _ { x } ( M )$ 上给出一个内积，从而给出一个同构 $T _ { x } ( M ) \\simeq T _ { x } ^ { * } ( M )$ 这一事实。在狭义相对论的情形中，$T _ { x } ( M )$ 与 $T _ { x } ^ { * } ( M )$ 之间的这种区分仅仅归结为一个符号问题，但上指标与下指标记号对于跟踪这些符号很有用。*\n>\n> *第二个动机是位置与动量自然地生活在对偶向量空间中，因此人们希望区分位置向量空间$M ^ { 4 }$与动量的对偶向量空间。然而，对于像$M ^ { 4 }$这样带有固定内积$\\eta _ { \\mu \\nu }$的向量空间，这个内积给出了$M ^ { 4 }$与其对偶的一个固定等同，这个等同同时也是作为洛伦兹群的表示的等同。鉴于这个固定的等同，我们在此不会试图通过记号来区分一个向量是在$M ^ { 4 }$中还是在其对偶中，因此将只使用下指标，而不是同时使用上指标和下指标。*\n",
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