Spin-electron-acoustic waves (sometimes called spin-plasmons) can be found in degenerate electron gases if spin-up electrons and spin-down electrons move relatively each other. Here, we suggest relativistic hydrodynamics with separate spin evolution, which allows us to study linear and nonlinear spin-electron-acoustic waves, including the spin-electron-acoustic solitons. The presented hydrodynamic model is the corresponding generalization of the relativistic hydrodynamic model with the average reverse gamma factor evolution, which consists of equations for evolution of the following functions: the partial concentrations (for spin-up electrons and spin-down electrons), the partial velocity fields, the partial average reverse relativistic gamma factors, and the partial flux of the reverse relativistic gamma factors. We find that the relativistic effects decrease the phase velocity of spin-electron-acoustic waves. Numerical analysis of the changes of dispersion curves of the Langmuir wave, spin-electron-acoustic wave, and ion-acoustic wave under the change of the spin polarization of electrons is presented. It is demonstrated that dispersion curves of the Langmuir wave and spin-electron-acoustic wave get closer to each other in the relativistic limit. Spin dependence of the amplitude and width of the relativistic spin-electron-acoustic soliton is demonstrated as well. Reformation of the bright soliton of potential of the electric field into the dark soliton under the influence of the relativistic effects is found.
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