Acoustic causality in relativistic shells and its implication for gamma-ray bursts

The motion of sound waves propagating in the perfect fluid with inhomogeneous background flow is effectively described as a massless scalar field on a curved space-time. This effective geometry is characterized by the acoustic metric, which depends on the background flow, and null geodesics on the geometry express the acoustic causal structure. Therefore by the effective geometry we can easily study the causality on the flows. In this paper, we consider a spherically symmetric, relativistic outflow and present the maximal causally connected region for a supersonic flow, which is a simple analog of gamma-ray bursts (GRBs). Causality in GRB outflows is important when we consider the coherency of the magnetic field. When the Lorentz factor of the radial velocity of the flow is constant or obeys the power law with respect to the radial coordinate r, we can solve it analytically. As a result we show that in the constant case the maximum angle is proportional to the inverse of the Lorentz factor and logarithmically increases with respect to r; in contrast, accelerative expansions in the power-law case make this angle bounded.