Systems consisting of spheres rolling on elastic membranes have been used to introduce a core conceptual idea of General Relativity (GR): how curvature guides the movement of matter. However, such schemes cannot accurately represent relativistic dynamics in the laboratory because of the dominance of dissipation and external gravitational fields. Here we demonstrate that an ``active" object (a wheeled robot), which moves in a straight line on level ground and can alter its speed depending on the curvature of the deformable terrain it moves on, can exactly capture dynamics in curved relativistic spacetimes. Via the systematic study of the robot's dynamics in the radial and orbital directions, we develop a mapping of the emergent trajectories of a wheeled vehicle on a spandex membrane to the motion in a curved spacetime. Our mapping demonstrates how the driven robot's dynamics mix space and time in a metric, and 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 the programming of a desired spacetime, such as the Schwarzschild metric near a non-rotating blackhole. Our mapping and framework facilitate creation of a robophysical analog to a general relativistic system in the laboratory at low cost that can provide insights into active matter in deformable environments and robot exploration in complex landscapes.

翻译：由在弹性膜上滚动的球体构成的系统已被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，由于耗散和外部引力场的支配作用，此类方案无法在实验室中准确再现相对论动力学。本文证明，一种"主动"物体（轮式机器人）在平地上沿直线运动，并能根据其行进的可变形地形的曲率改变速度，能够精确捕捉弯曲相对论时空中的动力学。通过系统研究机器人在径向和轨道方向上的动力学，我们建立了轮式车辆在氨纶膜上的涌现轨迹与弯曲时空中运动之间的映射关系。该映射展示了受驱机器人的动力学如何在度规中混合空间与时间，并揭示了主动粒子未必遵循真实空间中的测地线，而是遵循虚拟时空中的测地线。进一步地，该映射揭示了如何通过膜弹性与瞬时速度等参数编程实现所需时空（例如非旋转黑洞附近的施瓦西度规）。我们的映射与框架为在实验室中以低成本创建广义相对论系统的机器人物理模拟提供了可能，进而为可变形环境中的主动物质以及复杂地形中的机器人探索提供新见解。