  Radio mode feedback, associated with the propagation of powerful outflows in
active galaxies, is a crucial ingredient in galaxy evolution. Extragalactic
jets are well collimated and relativistic, both in terms of thermodynamics and
kinematics. They generate strong shocks in the ambient medium, associated with
observed hotspots, and carve cavities that are filled with the shocked jet
flow. In this Letter, we compare the pressure evolution in the hotspot and the
cavity generated by relativistic and classical jets. Our results show that the
classical approach underestimates the cavity pressure by a factor larger or
equal to 2 for a given shocked volume during the whole active phase. The
tension between both approaches can only be alleviated by unrealistic jet flow
densities or gigantic jet areas in the classical case. As a consequence, the
efficiency of a relativistic jet heating the ambient is typically around 20%
larger compared with a classical jet, and the heated volume is 2 to 10 times
larger during the time evolution. This conflict translates into two
substantially disparate manners, both spatially and temporal, of heating the
ambient medium. These differences are expected to have relevant implications on
the star formation rates of the host galaxies and their evolution.
