A Relativistic Formula for the Multiple Scattering of Photons

TL;DR

This paper derives analytical expressions for the probability density function of multiply scattered photons in relativistic flows, capturing their collective dynamics from free-streaming to diffusion, validated by Monte Carlo simulations.

Contribution

It introduces a novel relativistic formula for the PDF of multiply scattered photons, accounting for causality and relativistic effects, extending previous non-relativistic models.

Findings

PDF accurately reproduces Monte Carlo results

Photon behavior transitions from free-streaming to diffusion with Lorentz factor

High Lorentz factors lead to anisotropic ballistic motion approaching light speed

Abstract

We have discovered analytical expressions for the probability density function (PDF) of photons that are multiply scattered in relativistic flows, under the assumption of isotropic and inelastic scattering. These expressions characterize the collective dynamics of these photons, ranging from free-streaming to diffusion regions. The PDF, defined within the light cone to ensure the preservation of causality, is expressed in a three-dimensional space at a constant time surface. This expression is achieved by summing the PDFs of photons that have been scattered $n$ times within four-dimensional spacetime. We have confirmed that this formulation accurately reproduces the results of relativistic Monte Carlo simulations.We found that the PDF in three-dimensional space at a constant time surface can be represented in a separable variable form. We demonstrate the behavior of the PDF in the laboratory frame across a wide range of Lorentz factors for the relativistic flow. When the Lorentz factor of the fluid is low, the behavior of scattered photons evolves sequentially 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 from anisotropic ballistic motion, characterized by a mean effective velocity approaching the speed of light, to dynamic diffusion.