  Different from usual critical point characterized by a single length scale,
critical point with emergent symmetry exhibits intriguing critical properties
characterized by two relevant length scales, attracting long-term
investigations from both theoretical and experimental aspects. A natural
question is how the critical dynamics is affected by the presence of two
relevant length scales. Here we study the relaxation critical dynamics in the
three-dimensional ($3$D) clock model, whose critical point has emergent $U(1)$
symmetry. We find that in contrast to the magnatization $M$, whose relaxation
process is described by the usual dynamic exponent $z$ of the $3$D $XY$
universality class, the angular order parameter $\phi_q$ shows a two-stage
evolution characterized by different dynamic critical exponents. While in the
short-time stage the relaxation dynamics is governed by $z$, in the long-time
stage the dynamics is controlled by a new dynamic exponent $z'$. We also show
the off-critical-point effects in the critical relaxation. Our results may be
experimentally detected in the hexagonal RMnO$_3$ (R$=$rare earth) materials.
