A Relativistic Formula for the Multiple Scattering of Photons

We have discovered analytical expressions for the probability densityfunction (PDF) of photons that are multiply scattered in relativistic flows,under the assumption of isotropic and inelastic scattering. These expressionscharacterize the collective dynamics of these photons, ranging fromfree-streaming to diffusion regions. The PDF, defined within the light cone toensure the preservation of causality, is expressed in a three-dimensional spaceat a constant time surface. This expression is achieved by summing the PDFs ofphotons that have been scattered n times within four-dimensional spacetime.We have confirmed that this formulation accurately reproduces the results ofrelativistic Monte Carlo simulations.We found that the PDF in three-dimensionalspace at a constant time surface can be represented in a separable variableform. We demonstrate the behavior of the PDF in the laboratory frame across awide range of Lorentz factors for the relativistic flow. When the Lorentzfactor of the fluid is low, the behavior of scattered photons evolvessequentially from free propagation to diffusion, and then to dynamic diffusion,where the mean effective velocity of the photons equates to that of the fluid.On the other hand, when the Lorentz factor is large, the behavior evolves fromanisotropic ballistic motion, characterized by a mean effective velocityapproaching the speed of light, to dynamic diffusion.