We study the polarization of particles in relativistic heavy-ion collisions at very high energy along the beam direction within a relativistic hydrodynamic framework. We show that this component of the polarization decreases much slower with center-of-mass energy compared to the transverse component, even in the ideal longitudinal boost-invariant scenario with non-fluctuating initial state, and that it can be measured by taking advantage of its quadrupole structure in the transverse momentum plane. In the ideal longitudinal boost-invariant scenario, the polarization is proportional to the gradient of temperature at the hadronization and its measurement can provide important information about the cooling rate of the Quark Gluon Plasma around the critical temperature.
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