  The far-field back-scattering amplitude of an electric field from a
relativistically-moving sphere is analyzed. Contrary to prior research, we do
so by expressing the fields in the helicity basis, and we highlight here its
advantages when compared to the commonly-considered parity basis. With the
purpose of exploring specific scattering phenomena considering relativistic
effects, we identify conditions that minimize the back-scattered field, leading
to a relativistic formulation of the first Kerker condition. The requirements
to be satisfied by the sphere are expressed in terms of Mie angles, which
constitute an effective parametrization of any possible optical response a
sphere might have. We are able to identify multiple combinations of Mie angles
up to octupolar order via gradient-based optimization that satisfy our newly
formulated relativistic Kerker condition, yielding minima for the
back-scattered energy as low as 0.016% of the average scattered energy. Our
results can be extended to involve multiple particles forming a metasurface,
potentially having direct implications on the design of light sails as
considered by the Breakthrough Starshot Initiative.
