  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.
