Acoustic causality in relativistic shells
  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 super-sonic flow. When Lorentz factor of the radial velocity of the flow is constant or obeys 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 inverse of Lorentz factor and logarithmically increases with respect to r, in contrast, accelerative expansions in power-law case make this angle bounded. 
Shunichiro Kinoshita, Yuuiti Sendouda, and Keitaro Takahashi