# Acoustical spacetime geometry and shock formation

# Acoustical spacetime geometry and shock formation

In 2007 I published a monograph which treated the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. In this monograph I considered initial data which outside a sphere coincide with the data corresponding to a constant state. Under a suitable restriction on the size of the initial departure from the constant state, I established theorems which gave a complete description of the maximal classical development.. In particular, I showed that the boundary of the domain of the maximal classical development has a singular part where the inverse density of the wave fronts vanishes, signaling shock formation. In fact, the theorems which I established give a complete picture of shock formation in three-dimensional irrotational fluids . In my talk I shall give a simplified presentation of these results and of their proof. The approach is geometric, the central concept being that of the acoustical space time manifold.