[Paper Review] A tensor formalism for transfer and Compton scattering of polarized light

Jamie Portsmouth, Edmund Bertschinger|arXiv (Cornell University)|Dec 3, 2004

Stellar, planetary, and galactic studies10 references3 citations

TL;DR

This paper introduces a covariant tensor formalism for radiative transfer and Compton scattering of partially polarized light, replacing traditional Stokes parameters with a polarization tensor $ I^{ ueta} $ that simplifies Lorentz boosting and frame transformations. The formalism enables a manifestly covariant kinetic equation for Compton scattering, derived via a semi-classical approach, and provides a more elegant and flexible alternative to the Stokes parameter method, especially in relativistic contexts like the Sunyaev-Zeldovich effect.

ABSTRACT

A novel covariant formalism for the treatment of the transfer and Compton scattering of partially polarized light is presented. This was initially developed to aid in the computation of relativistic corrections to the polarization generated by the Sunyaev-Zeldovich effect (demonstrated in a companion paper), but it is of more general utility. In this approach, the polarization state of a light beam is described by a tensor constructed from the time average of quadratic products of the electric field components in a local observer frame. This leads naturally to a covariant description which is ideal for calculations involving the boosting of polarized light beams between Lorentz frames, and is more flexible than the traditional Stokes parameter approach in which a separate set of polarization basis vectors is required for each photon. The covariant kinetic equation for Compton scattering of partially polarized light by relativistic electrons is obtained in the tensor formalism by a heuristic semi-classical line of reasoning. The kinetic equation is derived first in the electron rest frame in the Thomson limit, and then is generalized to account for electron recoil and allow for scattering from an arbitrary distribution of electrons.

Motivation & Objective

• To develop a more flexible and covariant formalism for describing the transfer and scattering of partially polarized light in relativistic astrophysical environments.
• To overcome the computational complexity of the Stokes parameter formalism when performing Lorentz boosts and frame transformations in Compton scattering calculations.
• To provide a physically motivated derivation of the kinetic equation for Compton scattering in the context of relativistic electrons and polarized radiation.
• To demonstrate the utility of the tensor formalism in computing relativistic corrections to polarization in the Sunyaev-Zeldovich effect, as shown in a companion paper.

Proposed method

• The polarization state of light is described by a symmetric, traceful, complex Hermitian tensor $ I^{ ueta} $, constructed from time averages of quadratic products of electric field components in a local observer frame.
• The formalism generalizes the Stokes parameters into a 3×3 polarization matrix for a given direction, with transversality enforced via $ n^i I_{ij} = 0 $, and extends naturally to a 4×4 covariant tensor for relativistic calculations.
• A phase space distribution function $ f^{ ueta} $ is defined as a tensor analogue to the standard distribution function, enabling a covariant kinetic equation.
• The Compton scattering cross-section is derived in the electron rest frame in the Thomson limit, then generalized to include electron recoil and arbitrary electron velocity distributions.
• The formalism avoids the need to redefine polarization basis vectors for each photon direction, eliminating the need for complex rotation operations during angular integrations in transfer equations.
• The derivation uses a semi-classical approach and ensures manifest Lorentz invariance by carefully handling polarization vector gauge freedom through projection and orthogonalization procedures.

Experimental results

Research questions

• RQ1 How can the transfer and Compton scattering of partially polarized light be described in a way that simplifies Lorentz boosting between inertial frames?
• RQ2 What is a physically motivated, covariant generalization of the Stokes parameter formalism that avoids basis vector redefinition for each photon direction?
• RQ3 How can the kinetic equation for Compton scattering be derived in a manifestly covariant form using a tensor-based description of polarization?
• RQ4 In what way does the tensor formalism simplify the computation of relativistic corrections to polarization in the Sunyaev-Zeldovich effect?
• RQ5 Can the formalism be extended to other scattering processes beyond Compton scattering with minimal modification?

Key findings

• The polarization tensor formalism replaces the four Stokes parameters with a 4×4 covariant tensor $ I^{ ueta} $, which naturally incorporates Lorentz invariance and simplifies frame transformations.
• The kinetic equation for Compton scattering is derived in a manifestly covariant form, with the scattering process described via projection matrices acting on the polarization tensor.
• The formalism eliminates the need for rotating polarization basis vectors when computing scattering between different photon directions, significantly reducing computational complexity.
• The derivation shows that the scattering amplitude remains invariant under initial and final state interchange, even when polarization vectors are in an arbitrary gauge, by using a gauge-covariant projection procedure.
• The tensor formalism provides a more elegant and computationally efficient alternative to the Stokes parameter method, particularly in relativistic scattering problems involving multiple Lorentz frames.
• The method is directly applicable to computing relativistic corrections in the Sunyaev-Zeldovich effect, as demonstrated in the companion paper (Portsmouth & Bertschinger, 2004).