爱因斯坦引力的统一扩展
A unified expansion of Einstein's gravity
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中文总结 AI 辅助
本文引入统一框架扩展爱因斯坦引力，通过对庞加莱代数收缩及参数化，系统展开爱因斯坦 - 希尔伯特作用得到统一扩展，重现多种引力理论，还构建了弦卡罗尔引力并确认黑洞近视界几何是其解，为理解黑洞视界物理提供新途径。
中文摘要 AI 辅助
非洛伦兹引力理论，最常见的是伽利略引力和卡罗尔引力，通过适当缩放从广义相对论产生。广义相对论可通过对庞加莱代数进行规范得到。非洛伦兹引力的一种便捷表述是通过将切空间中的庞加莱代数收缩到其非洛伦兹对应物，如伽利略和卡罗尔代数。现有文献中不同的非洛伦兹引力理论是分别处理的。本文引入一个单一的统一扩展框架来处理所有这些不同理论。我们表明不同缩放可统一为一个由(s,n)参数化的协变形式，以及收缩参数ε。在ε→0极限下保持这些参数不固定定义了一个统一的平坦几何及其统一代数，对于特定的s和n选择它会简化为特定的非洛伦兹几何。利用切空间中的此设置，我们系统地以ε的偶次幂展开爱因斯坦 - 希尔伯特作用，称为爱因斯坦引力的统一扩展，其领先阶理论由(s,n)确定。这重现了各类引力理论，包括爱因斯坦引力、伽利略引力、卡罗尔引力。利用该扩展，我们构建了弦卡罗尔（SC）引力，其中局部度规有两个零特征值。最近已表明一般非极端黑洞的近视界区域是SC几何。通过具体例子，我们确认这些近视界几何构成SC引力的解，为从SC引力理解一般黑洞视界附近的物理铺平了道路。
英文摘要
Non-Lorentzian theories of gravity, most common of which are Galilean and Carrollian gravity, arise from General relativity under suitable scalings. General relativity can be obtained by gauging the Poincaré algebra. A convenient formulation of non-Lorentzian gravity follows the contraction of the Poincaré algebra in the tangent space to its non-Lorentzian counterparts, e.g. Galilean and Carrollian algebras. In existing literature, different non-Lorentzian theories of gravity have been addressed separately. In this paper, we introduce a single unified framework of expansion to address all these different theories. We show that different scalings can be unified into a single covariant form parametrized by $(s,n)$, alongside the contraction parameter $ε$. Keeping these parameters unfixed in the limit $ε\to 0$ defines a $\textit{unified flat geometry}$ and its $\textit{unified algebra}$, which reduces to a specific non-Lorentzian geometry for a particular choice of $s$ and $n$. Using this setup in the tangent space, we systematically expand the Einstein-Hilbert action in even powers of $ε$, which we call a $\textit{unified expansion}$ of Einstein's gravity, whose leading-order theory is fixed by $(s,n)$. This reproduces various classes of gravitational theories, including Einstein gravity (the trivial case), Galilean gravity, Carroll gravity, all of which can be extracted from this expansion. Using the expansion, we then formulate String Carroll (SC) gravity, where the local metric has two vanishing eigenvalues. The near-horizon region of generic non-extremal black holes has been recently shown to be a SC geometry. By considering explicit examples, we confirm that these near-horizon geometries constitute solutions of SC gravity, paving the way of understanding physics near the horizon of generic black holes in terms of SC gravity.
