Analogue space-times (and in particular metamaterial analogue space-times) have a long varied and rather complex history. Much of the previous related work to this field has focused on spherically symmetric models; however, axial symmetry is much more relevant for mimicking astrophysically interesting systems that are typically subject to rotation. Now it is well known that physically reasonable stationary axisymmetric space-times can, under very mild technical conditions, be put into Boyer–Lindquist form. Unfortunately, a metric presented in Boyer–Lindquist form is not well adapted to the “quasi-Cartesian” metamaterial analysis that we developed in our previous articles on “bespoke analogue space-times”. In the current article, we shall first focus specifically on various space-time metrics presented in Boyer–Lindquist form, and subsequently determine a suitable set of equivalent metamaterial susceptibility tensors in a laboratory setting. We shall then turn to analyzing generic space-times, not even necessarily stationary, again determining a suitable set of equivalent metamaterial susceptibility tensors. Perhaps surprisingly, we find that the well-known ADM formalism proves to be not particularly useful, and that it is instead the dual “threaded” (Kaluza–Klein–inspired) formalism that provides much more tractable results. While the background laboratory metric is (for mathematical simplicity and physical plausibility) always taken to be Riemann flat, we will allow for arbitrary curvilinear coordinate systems on the flat background space-time. Finally, for completeness, we shall reconsider spherically symmetric space-times, but now in general spherical polar coordinates rather than quasi-Cartesian coordinates. In summary, this article provides a set of general-purpose calculational tools that can readily be adapted for mimicking various interesting (curved) space-times by using nontrivial susceptibility tensors in general (background-flat) laboratory settings.

中文翻译：

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博耶-林德奎斯特时空及超越：任意时空的超材料类似物

模拟时空（特别是超材料模拟时空）有着悠久而多样且相当复杂的历史。之前与该领域相关的大部分工作都集中在球对称模型上。然而，轴对称对于模仿通常会发生旋转的天体物理学有趣的系统更为相关。现在众所周知，物理上合理的静止轴对称时空可以在非常温和的技术条件下转化为博耶-林德奎斯特形式。不幸的是，以博耶-林德奎斯特形式提出的度量并不能很好地适应我们在之前关于“定制模拟时空”的文章中开发的“准笛卡尔”超材料分析。在本文中，我们将首先专门关注以博耶-林德奎斯特形式呈现的各种时空度量，然后在实验室环境中确定一组合适的等效超材料磁化率张量。然后，我们将转向分析通用时空，甚至不一定是静止的，再次确定一组合适的等效超材料磁化率张量。也许令人惊讶的是，我们发现众所周知的 ADM 形式主义被证明并不是特别有用，相反，双“线程”（卡鲁扎-克莱因启发）形式主义提供了更容易处理的结果。虽然背景实验室度量（为了数学简单性和物理合理性）始终被视为黎曼平坦，但我们将允许平坦背景时空上的任意曲线坐标系。最后，为了完整性，我们将重新考虑球对称时空，但现在一般是球极坐标而不是准笛卡尔坐标。总之，本文提供了一组通用计算工具，可以通过在一般（背景平坦）实验室设置中使用非平凡的磁化率张量来轻松适应模拟各种有趣的（弯曲）时空。