We have theoretically realized a photonic time crystal in a homogeneous spinor exciton-polariton condensate subject to a cavity strain induced energy splitting between the x- and y-polarized polaritons with distinct loss rates. Numerical modeling based on the time-dependent open dissipative Gross-Pitaevskii equation results in the phase transition from linearly-polarized spinor condensate at low pump powers into a circularly-polarized one and then a photonic time crystal at higher pump powers. This phenomenon occurs only under the condition that the lower-energy x-polarized condensate has a higher loss rate than the y-polaried condensate, which can exist intrinsically in the semiconductor microcavities.
