We have combined first-principles calculations and high pressure experiments to study pressure induced phase transitions in silicon nitride (Si3N4). Within the quasi-harmonic approximation, we predict that the α phase is always metastable relative to the β phase over the wide pressuretemperature range. Our lattice vibration calculations indicate that there are two significant and competing phonon-softening mechanisms in the β-Si3N4, while phonon softening in the α-Si3N4 is rather moderate.When the previously observed equilibrium high-pressure and high-temperature β→γ transition is by-passed at room temperatures (RT) due to kinetic reasons, the β phase is predicted to undergo a first-order structural transformation to a denser P6 phase above 39 GPa. The estimated enthalpy barrier height is less than 70 meV/atom, which suggests that the transition is kinetically possible around room temperatures. This predicted new high-pressure metastable phase should be classified as a "post-phenacite" phase. Our high-pressure X-ray diffraction experiment confirm this predicted room-temperature phase transition around 34 GPa. No similar RT phase transition is predicted for α-Si3N4. Furthermore, we discuss the differences in pressure dependencies of phonon modes in the α, β and γ phases and the consequences on their thermal properties. We attribute the phonon modes with negative Grüneisen ratios in the α and β phases as the cause of the predicted negative thermal expansion coefficients (TEC) at low temperatures in these two phases, and predict no negative TEC in the γ phase.