Systems consisting of spheres rolling on elastic membranes have been used as educational tools to introduce a core conceptual idea of General Relativity (GR): how curvature guides the movement of matter. However, previous studies have revealed that such schemes cannot accurately represent relativistic dynamics in the laboratory. Dissipative forces cause the initially GR-like dynamics to be transient and consequently restrict experimental study to only the beginnings of trajectories; dominance of Earth's gravity forbids the difference between spatial and temporal spacetime curvatures. Here by developing a mapping between dynamics of a wheeled vehicle on a spandex membrane, we demonstrate that an active object that can prescribe its speed can not only obtain steady-state orbits, but also use the additional parameters such as speed to tune the orbits towards relativistic dynamics. Our mapping demonstrates how activity mixes space and time in a metric, shows how active particles do not necessarily follow geodesics in the real space but instead follow geodesics in a fiducial spacetime. The mapping further reveals how parameters such as the membrane elasticity and instantaneous speed allow programming a desired spacetime such as the Schwarzschild metric near a non-rotating black hole. Our mapping and framework point the way to the possibility to create a robophysical analog gravity system in the laboratory at low cost and provide insights into active matter in deformable environments and robot exploration in complex landscapes.

翻译：由弹性膜上滚动的球体组成的系统已被用作教学工具，用于引入广义相对论（GR）的核心概念：曲率如何引导物质运动。然而，先前的研究揭示，这类方案无法在实验室中精确表征相对论动力学。耗散力导致最初类似广义相对论的动力学行为具有瞬态特性，从而将实验研究限制在轨迹的起始阶段；地球引力的主导作用则禁止了空间曲率与时空曲率之间的差异。本文通过建立一种在弹力布膜上行进的轮式车辆动力学映射，证明能够自主规定速度的主动物体不仅可获得稳态轨道，还能利用速度等额外参数将轨道调谐至相对论动力学。我们的映射展示了活性如何将空间与时间混合进度量张量，揭示了活性粒子并非必然遵循真实空间中的测地线，而是遵循基准时空中的测地线。该映射进一步揭示了膜弹性与瞬时速度等参数如何允许编程实现所需的时空结构（例如无自转黑洞附近的史瓦西度量）。我们的映射与框架指明了在实验室中以低成本创建机器人物理模拟引力系统的可能性，并为可变形环境中的活性物质研究及复杂地形下的机器人探索提供了新见解。