  In this paper, we theoretically propose an optomechanical scheme to explore
the possibility of simulating the propagation of the collective excitations of
the photon fluid in a curved spacetime. For this purpose, we introduce two
theoretical models for two-dimensional photon gas in a planar optomechanical
microcavity and a two-dimensional array of coupled optomechanical systems. In
the reversed dissipation regime (RDR) of cavity optomechanics where the
mechanical oscillator reaches equilibrium with its thermal reservoir much
faster than the cavity modes, the mechanical degrees of freedom can
adiabatically be eliminated. The adiabatic elimination of the mechanical mode
provides an effective nonlinear Kerr-type photon-photon interaction. Using the
nonlinear Schr\"{o}dinger equation (NLSE), we show that the phase fluctuations
in the two-dimensional photon fluid obey the Klein-Gordon equation for a
massless scalar field propagating in a curved spacetime. The results reveal
that the photon fluid as well as the corresponding metric can be controlled by
manipulating the system parameters.
