  General relativity uses curved space-time to describe accelerating frames.
The movement of particles in different curved space-times can be regarded as
equivalent physical processes based on the covariant transformation between
different frames. In this work, we use one-dimensional curved metamaterials to
mimic accelerating particles in curved space-times. The different curved shapes
of structures are used to mimic different accelerating frames. The different
geometric phases along the structure are used to mimic different movements in
the frame. Using the covariant principle of general relativity, we can obtain
equivalent nanostructures based on space-time transformations, such as the
Lorentz transformation and conformal transformation. In this way, many
covariant structures can be found which produce the same surface plasmon fields
when excited by spin photons. A new kind of accelerating beam, the Rindler
beam, is obtained based on the Rindler metric in gravity. Very large effective
indexes can be obtained in such systems based on geometric phase gradient. This
general covariant design method can be extended to many other optical media.
