  Non-identical particle femtoscopy measures the size of the system emitting
particles ("radius") in heavy-ion collisions as well as the difference between
mean emission space-time coordinates of two particle species ("emission
asymmetry"). The system created in such collisions at the LHC behaves
collectively and its dynamics is well described by hydrodynamic models. A
significant emission asymmetry between pions and kaons, coming from collective
flow, enhanced by contribution from flowing resonances is predicted. We present
calculations within the (3+1)D viscous hydrodynamic model coupled to
statistical hadronization code Therminator 2, corresponding to Pb--Pb
collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV. We obtain femtoscopic radii and
emission asymmetry for pion-kaon pairs as a function of collision centrality.
The radii grow linearly with cube root of particle multiplicity density. The
emission asymmetry is negative and comparable to the radius, indicating that
pions are emitted closer to the center of the system and/or later than kaons.
Recent ALICE Collaboration measurements of identical kaon femtoscopy shows that
kaons are emitted, on average, 2.1~fm/$c$ later than pions. We modify our
calculation by introducing such delay and find that the system source size is
only weakly affected. In contrast the pion-kaon emission asymmetry is directly
sensitive to such delays and the modified calculation shows significantly lower
values of asymmetry. Therefore we propose the measurement of the pion-kaon
femtoscopic correlation function as a sensitive probe of the time delays in
particle emission.
