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General relativity uses curved space-time to describe accelerating frames.The movement of particles in different curved space-times can be regarded asequivalent physical processes based on the covariant transformation betweendifferent frames. In this work, we use one-dimensional curved metamaterials tomimic accelerating particles in curved space-times. The different curved shapesof structures are used to mimic different accelerating frames. The differentgeometric phases along the structure are used to mimic different movements inthe frame. Using the covariant principle of general relativity, we can obtainequivalent nanostructures based on space-time transformations, such as theLorentz transformation and conformal transformation. In this way, manycovariant structures can be found which produce the same surface plasmon fieldswhen excited by spin photons. A new kind of accelerating beam, the Rindlerbeam, is obtained based on the Rindler metric in gravity. Very large effectiveindexes can be obtained in such systems based on geometric phase gradient. Thisgeneral covariant design method can be extended to many other optical media.

Controlling Surface Plasmons Through Covariant Transformation of the Spin-Dependent Geometric Phase Between Curved Metamaterials