  The traditional systems researched in condensed matter physics always have
spatial translation symmetry. However for space-time crystal systems, the
spatial translation symmetry is no longer preserved and the lattice potential
have space-time translation symmetry instead. We show that a rational
space-time crystal system is equal to a traditional floquet system. Then, we
find a way to solve the floquet equation analytically and construct an
effective Hamiltonian of the rational space-time crystal system. By this
effective Hamiltonian, we obtain the topological classification of gapped and
gap-preserving rational space-time crystal systems. Our works reveal the
correlation between space-time crystal systems and floquet systems and give a
systematic method to explore the properties of rational space-time crystal
systems.
