  We show that the event-by-event fluctuations of the transverse size of the
initial source, which follow directly from the Glauber treatment of the
earliest stage of relativistic heavy-ion collisions, cause, after hydrodynamic
evolution, fluctuations of the transverse flow velocity at hadronic freeze-out.
This in turn leads to event-by-event fluctuations of the average transverse
momentum, p_T. Simulations with GLISSANDO for the Glauber phase, followed by a
realistic hydrodynamic evolution and statistical hadronization carried out with
THERMINATOR, lead to agreement with the RHIC data. In particular, the magnitude
of the effect, its centrality dependence, and the weak dependence on the
incident energy are properly reproduced. Our results show that bulk of the
observed event-by-event p_T fluctuations may be explained by the fluctuations
of the size of the initial source.
