  In this paper, we carry out a semi-analytic general relativistic study of a
Gamma-Ray Bursts (GRB) jet that is breaking out of a cocoon or stellar
envelope. We solve hydrodynamic equations with the relativistic equation of
state that takes care of fluid composition. In short GRBs, a general
relativistic approach is required to account for curved spacetime in strong
gravity. The piercing of the jet through the cocoon resembles a de Laval nozzle
and the jet may go through recollimation shock transitions. We show that the
possibility of shock transition and the shock properties are sensitive to the
matter composition and the cocoon strength. Obtained Lorentz factors in
thermally driven jets comfortably reach a few $\times$10.
