General Relativistic Polarization Transport

• General relativistic polarization transport is a covariant framework describing the evolution of electromagnetic wave polarization via Stokes parameters along null geodesics in curved spacetime.
• It incorporates microphysical processes—such as cyclo-synchrotron emission, absorption, Faraday rotation, and conversion—to model realistic polarimetric signatures.
• Advanced numerical schemes and gauge choices enable precise simulations that aid in diagnosing black hole accretion dynamics and relativistic jet properties.

General relativistic polarization transport is the fully covariant description of how the polarization state of electromagnetic waves evolves as they propagate through curved spacetime in the presence of plasma and gravitational fields. This formalism enables the calculation of observable linear and circular polarization signatures from relativistic sources such as black hole accretion flows, jets, and neutron stars, incorporating all relevant microphysical emission, absorption, and Faraday effects as well as the impact of spacetime geometry on the transport of electric field orientation. A central achievement of this framework is the ability to self-consistently model the evolution of Stokes parameters along null geodesics, allowing direct comparison between theory, numerical simulations, and observational polarimetry in strong gravity environments (Shcherbakov et al., 2010).

1. Covariant Framework and Fundamental Equations

Polarized radiative transport in general relativity is formulated in terms of the covariant evolution of the Stokes vector $S = (I, Q, U, V)^{T}$ and/or the photon coherency tensor $N^{\alpha\beta}$ along null geodesics $x^\mu(\lambda)$ in a given spacetime metric $g_{\mu\nu}$ (Shcherbakov et al., 2010, Moscibrodzka et al., 2017, Pihajoki et al., 2016). The full transfer equation in the local (plasma comoving) frame reads

$\frac{d S^A}{d \lambda} = -K^A{}_B S^B + j^A,$

where

• $j^A$ (vector) encodes invariant emissivities (cyclo-synchrotron, bremsstrahlung, etc),
• $K^A{}_B$ (matrix) incorporates absorption ($\alpha_I, \alpha_Q, \alpha_U, \alpha_V$) and Faraday mixing (rotation $\rho_V$ and conversion $\rho_Q, \rho_U$).

The governing equations for the photon four-momentum ($k^\mu$) and polarization four-vector ($f^\mu$) under parallel transport are: $$\frac{d k^\mu}{d\lambda} = -\Gamma^\mu_{\alpha\beta} k^\alpha k^\beta,\qquad \frac{D f^\mu}{d\lambda} = \frac{d f^\mu}{d\lambda} + \Gamma^\mu_{\nu\sigma} f^\nu k^\sigma = 0, \qquad k_\mu f^\mu = 0$$ where $\Gamma$ are Christoffel symbols of $g_{\mu\nu}$ (Chowdhury, 11 Nov 2025).

A local tetrad $e_{(a)}^\mu$ aligned with the plasma velocity $u^\mu$ is constructed via Gram-Schmidt orthonormalization, and is used to project coordinate frame vectors into the local Minkowski frame. Parallel transport of the polarization basis vectors and the Stokes/coherency tensor is carried out along $k^\mu$ (Shcherbakov et al., 2010, Moscibrodzka et al., 2017, Pihajoki et al., 2016).

2. Physical Coefficients: Emission, Absorption, Faraday Rotation & Conversion

The radiative process coefficients for polarized transport are derived from plasma kinetic theory and electromagnetic response functions:

• Emission/Absorption (cyclo-synchrotron):

$$j_I(\nu) = \sqrt{3} e^2 n_e \nu/(8\pi m c) \int d\gamma\, f_0(\gamma) H_I(\nu/\nu_c(\gamma)), \quad \text{analogous formulas for } Q, V$$

with $H_{I,Q,V}$ combinations of Macdonald functions, $f_0$ the electron distribution, and $B_\perp$ the magnetic field perpendicular to the ray (Shcherbakov et al., 2010).

• Faraday Rotation ($\rho_V$):

$\rho_V(\nu) \propto n_e B_\parallel \nu^{-2}$

causes mixing of $Q/U$ via different phase velocities of left/right circular modes.

• Faraday Conversion ($\rho_Q$, $\rho_U$):

$$\rho_Q(\nu), \,\, \rho_U(\nu) \propto n_e B^2_\perp \nu^{-3}$$

enables conversion between linear and circular polarization states.

All coefficients depend on local values of $n_e$ (electron density), $T_e$ (electron temperature), $B^\mu$ (magnetic field), and the photon frequency in the plasma frame (Shcherbakov et al., 2010, Tsunetoe et al., 2020).

3. Parallel Transport in Curved Spacetime and Gauge Choices

Polarization vectors are parallel-transported along null geodesics using the local tetrad $\{e_{(1)}, e_{(2)}\}$ orthogonal to $k^\mu$: $k^\nu \nabla_\nu e_{(a)}^\mu = 0$ providing the physically meaningful evolution of polarization angle in the observer's sky frame (Brodutch et al., 2011, Chowdhury, 11 Nov 2025).

Gauge choices for basis vectors (e.g., "Newton gauge" aligning basis with local free-fall acceleration) are crucial for correct interpretation of observed polarization rotation and differentiation between coordinate artifacts and physical effects. The Walker–Penrose constant provides global invariance for the parallel transport in the Petrov-D Kerr geometry (Brodutch et al., 2011, Chowdhury, 11 Nov 2025).

4. Numerical Schemes and Algorithms

Contemporary numerical implementations solve the combined ODE system for $x^\mu$, $k^\mu$, $f^\mu$ and the Stokes/coherency tensors using Runge–Kutta or split-operator methods:

• Fourth-order Runge–Kutta with cached Christoffel symbols achieves sub-degree EVPA accuracy and 5x performance gain for polarization transport in the Kerr metric (Chowdhury, 11 Nov 2025).
• Operator splitting/Strang-splitting employed in ipole and Arcmancer for stable stepwise advancement of parallel transport and polarized transfer, with analytic solutions for constant coefficients and adaptive handling of high Faraday depth (stiff mixing) (Moscibrodzka et al., 2017, Pihajoki et al., 2016).
• Monte-Carlo schemes (radpol) propagate superphotons carrying Stokes weights through coupled synchrotron, Compton, and general relativistic effects (Moscibrodzka, 2019).

All implementations require transformation into the plasma frame at each step, evaluation of physical coefficients, and careful preservation of orthonormality and transversality constraints for the polarization basis and Stokes vector.

5. Observational Applications and Diagnostics

General relativistic polarization transport underpins modern polarimetric diagnostics of black hole accretion flows, relativistic jets, and radiative signatures:

• Spin and inclination constraints: Accurate simulations of polarized emission (I, Q, U, V) and frequency-dependent EVPA enable inference of black hole spin and viewing geometry (e.g., Sgr A*, M87) (Shcherbakov et al., 2010, Tsunetoe et al., 2020).
• Faraday rotation and conversion mapping: Synthetic images quantify rotation measure (RM) gradients, EVPA swings, and ring-like patterns in circular polarization tracing coherent magnetic structures (Tsunetoe et al., 2020, Anantua et al., 2018).
• Benchmarking against current instruments: Numerical accuracy (<0.1° EVPA error) matches or exceeds IXPE/NICER tolerances and approaches EHT requirements, facilitating direct comparison to horizon scales (Chowdhury, 11 Nov 2025).

Tables mapping simulation code, geometry, and accuracy benchmarks demonstrate the achieved precision for polarimetric modeling.

Code	Geometry	EVPA Accuracy
Kerr-transport	Kerr, arbitrary spin	≤0.1°, sub-degree
Arcmancer	Arbitrary metric	≤10⁻⁶ rel. error
ipole	Covariant, any metric	≤1% difference

Polarization transport predicts and explains observed signatures such as jet limb-brightening, polarization angle swings, depolarization in optically thick plasma, and circular polarization emergence due to Faraday conversion (Anantua et al., 2018, Tsunetoe et al., 2020).

6. Extensions and Nonlinear/Quantum Generalizations

Polarization transport in nonlinear vacuum electrodynamics (e.g., Plebański class, Born–Infeld, Heisenberg–Euler) extends beyond Maxwell theory:

• Generalized Maxwell equations yield modified dispersion relations and constitutive tensors, introducing vacuum birefringence (multiple cones) and explicit coupling of polarization to background gradients (Perlick et al., 2018).
• Transport law: For multiplicity-two rays (no birefringence), the polarization plane evolves by

$K^b\nabla_b f^a + P^a{}_b f^b = 0$

with $P^a{}_b$ determined by Lagrangian derivatives and field gradients.

• Observable effects: In Born–Infeld theory, nonlinearity induces minute but measurable rotations of polarization plane, offering precision probes of QED corrections in strong fields (Perlick et al., 2018).

Quantum optics and higher-order geometrical optics corrections further introduce helicity-dependent transport, energy flux misalignments, and gravitational spin-Hall effects, suggesting measurable polarization splitting in strong-field regimes (Dolan, 2018).

7. Summary and Future Prospects

General relativistic polarization transport provides the essential physical and computational infrastructure for interpreting polarimetric signatures from astrophysical sources in strong gravity. The framework rigorously couples emission, absorption, Faraday effects, and spacetime curvature within a self-consistent covariant formalism. Ongoing advancements in algorithmic accuracy and efficiency, inclusion of nonlinear and quantum corrections, and direct integration with GRMHD simulations position this field for high-impact synergy with current and next-generation polarimetric observatories (Shcherbakov et al., 2010, Chowdhury, 11 Nov 2025, Moscibrodzka et al., 2017, Pihajoki et al., 2016, Perlick et al., 2018).