  We present detailed neutron scattering studies of the static and dynamic
stripes in an optimally doped high-temperature superconductor,
La$_2$CuO$_{4+y}$. We find that the dynamic stripes do not disperse towards the
static stripes in the limit of vanishing energy transfer. We conclude that the
dynamic stripes observed in neutron scattering experiments are not the
Goldstone modes associated with the broken symmetry of the simultaneously
observed static stripes, but rather that the signals originate from different
domains in the sample. These domains may be related by structural twinning, or
may be entirely different phases, where the static stripes in one phase are
pinned versions of the dynamic stripes in the other. Our results explain
earlier observations of unusual dispersions in underdoped
La$_{2-x}$Sr$_x$CuO$_{4}$ ($x=0.07$) and La$_{2-x}$Ba$_x$CuO$_{4}$ ($x=0.095$).
Our findings are relevant for all compounds exhibiting magnetic stripes, and
may thus be a vital part in unveiling the nature of high temperature
superconductivity.
