Polarized Ray Tracing

Polarized ray tracing must keep three rotations conceptually separate: parallel transport of the screen basis through curved spacetime, rotation between that basis and the local plasma basis, and physical Faraday rotation or conversion within the plasma. Combining them without a recorded convention can yield plausible Stokes images with incorrect signs or angles. Verification therefore begins with basis geometry before it reaches emission physics.

Required background. Affine and Variational Geodesics supplies null rays and parallel transport.

Helpful background. Formulations of General Relativity supplies tetrads and numerical spacetime data; Differential Equations for Relativity supplies matrix transfer integration; Black-Hole Astrophysics supplies magnetized emission models; and Computational Relativity supplies the shared evidence standard.

A transported polarization basis

Along a null momentum $p^a$, choose two spacelike screen vectors $f_1^a$ and $f_2^a$ satisfying

$$p_a f_A^a=0, \qquad f_A^a f_{Ba}=\delta_{AB}, \qquad p^b\nabla_b f_A^a=0.$$

Adding a multiple of $p^a$ to a screen vector does not change the physical transverse polarization, so numerical comparisons should use projected screen components rather than raw coordinate vectors. Re-orthonormalization can control roundoff, but it must preserve orientation; an unnoticed reflection changes polarization handedness.

Define $Q\gt0$ along $f_1$ and $U\gt0$ for linear polarization at $+45^\circ$ from $f_1$ toward $f_2$. Take $(f_1,f_2,\widehat k)$ right handed in the local observer frame, where $\widehat k$ points along the future photon propagation. Use $V=I_{\mathrm R}-I_{\mathrm L}$, with right circular polarization defined by electric-field rotation in the right-hand sense about $\widehat k$ at fixed position. At the central pixel of the camera defined on the preceding page, $\widehat k=-e_{(3)}$, so choosing $f_1=e_{(1)}$ requires $f_2=-e_{(2)}$ for this handedness. This is the IEEE convention used by the source below; reversing it requires translating the circular emission, absorption, and conversion terms. Under a screen rotation by $\chi$,

$$Q+iU\longrightarrow e^{-2i\chi}(Q+iU),$$

for the convention used here, while $I$ and $V$ remain unchanged. This transformation is the clean test that separates a coordinate or basis rotation from physical Faraday rotation.

Matrix transfer in the plasma frame

The transfer treatment assumes geometric optics, a locally defined plasma response, and prescribed coefficients that do not depend on the radiation field. It is not valid in a strongly diffractive regime or when scattering couples rays and frequencies without an additional closure. Collect invariant Stokes components into $\boldsymbol{\mathcal S}=(\mathcal I,\mathcal Q,\mathcal U,\mathcal V)^{\mathsf T}$. Along the ray,

$$\frac{d\boldsymbol{\mathcal S}}{d\lambda} =\boldsymbol{\mathcal J} -\mathbf K\boldsymbol{\mathcal S},$$

With the stated Stokes convention, one explicit coefficient convention is

$$\mathbf K= \begin{pmatrix} a_I&a_Q&a_U&a_V\\ a_Q&a_I&r_V&-r_U\\ a_U&-r_V&a_I&r_Q\\ a_V&r_U&-r_Q&a_I \end{pmatrix}, \qquad a_A=\nu\alpha_{\nu,A},\quad r_A=\nu\rho_{\nu,A}.$$

Each invariant Stokes component is its specific intensity divided by $\nu^3$, and $\mathcal J_A=j_{\nu,A}/\nu^2$. Use the affine normalization and forward path direction of scalar transfer. The symmetric absorption block describes attenuation and dichroism; the antisymmetric Faraday block rotates $(Q,U,V)$. This matrix and its convention are given in Mościbrodzka and Gammie 2018, §2, Eq. (6), and §3.2, Eq. (19), PDF.

At each step, rotate the transported screen basis into the basis in which the local coefficients were evaluated. That geometric rotation is applied even in vacuum, whereas Faraday coefficients vanish with the plasma. Mościbrodzka and Gammie develop a covariant coherency-tensor and Stokes implementation in Mościbrodzka and Gammie 2018, §2, Eqs. (1)–(16), §3, and Appendix A.

Large absorption or Faraday depth makes the matrix system stiff. A formal matrix exponential or analytic constant-coefficient segment can be more stable than explicit stepping, but rapid coefficient or basis variation still limits segment length. Check positivity of $I$ and the physical bound $\sqrt{Q^2+U^2+V^2}\leq I$ within numerical tolerance.

A pure Faraday-rotation solution

Set every coefficient to zero except constant $r_V$. Then $\mathcal I'=\mathcal V'=0$, $\mathcal Q'=-r_V\mathcal U$, and $\mathcal U'=r_V\mathcal Q$, so

$$\mathcal Q+i\mathcal U =(\mathcal Q_0+i\mathcal U_0)e^{i r_V\Delta\lambda}.$$

The electric-vector position angle changes by $r_V\Delta\lambda/2$, while $\mathcal Q^2+\mathcal U^2$ is conserved. Starting at $(\mathcal I,\mathcal Q,\mathcal U,\mathcal V)=(1,0.6,0,0)$ and $r_V\Delta\lambda=\pi/2$ gives $(1,0,0.6,0)$. This tests the sign and the factor of two independently of ray bending. A passive rotation of the screen instead produces the opposite phase factor already stated above; confusing the two is a common polarization error.

Versioned polarization benchmarks

Use the published “snake” curved-ray test, Kerr transport tests, homogeneous analytic Faraday slabs, and the common GRMHD snapshot suite described by Prather and collaborators. Pin the exact input release or repository revision and checksums rather than only the paper year. Their test definitions, coefficient conventions, and cross-code comparisons are given in Prather et al. 2023, §§3.1–3.3 and §§4.1–4.4, Eqs. (12)–(19), Tables 1–3.

The dataset is synthetic. The evidence cutoff is 2026-08-18; no measured polarization likelihood is evaluated here. If an official EHT polarization release is later introduced, its data DOI, calibration pipeline version, leakage correction, and station selections must be recorded separately.

Begin with vacuum basis transport: rotate the camera screen by a known $\chi$ and recover the spin-two Stokes transformation. Then use constant coefficients with analytic pure emission, absorption, Faraday rotation, and conversion solutions, including optically and Faraday thick cases. Finally compare codes on identical metric, fluid snapshot, coefficients, camera, frequency, masks, and image alignment. Agreement after each code optimizes its own flux or orientation is not a controlled comparison.

Calibration, statistics, and systematics

Calibration records camera handedness and scale, tetrad normalization, $Q/U$ reference axis, $V$ sign, frequency and flux units, magnetic-field orientation, charge sign, and every coefficient convention. Selections fix spacetime and snapshot, inclination, field of view, frequency band, ray and pixel masks, density or magnetization cuts, and time averaging before results are inspected.

Use separate normalized image residuals for $I,Q,U,V$, plus errors in fractional linear and circular polarization and electric-vector position angle where intensity is safely above a declared threshold. State pixel weighting and alignment. These deterministic metrics are not likelihoods and have no priors. Low-intensity pixels require an absolute or masked metric because fractional polarization becomes ill-conditioned.

Systematic variations include coordinate chart, screen basis, transfer tolerance, fluid interpolation, image resolution, frequency grid, electron prescription, Faraday depth, snapshot sampling, and post-processing. Cross-code spread is meaningful only with identical inputs. A clean reproduction retains the versioned configuration, coefficients, transported basis diagnostics, Stokes images, checksums, and refinement tables. This specification does not imply that the calculation has been implemented or run.

Reject the result if vacuum creates a basis-dependent physical polarization, analytic slabs miss thin or thick limits, the polarization bound fails, Stokes signs differ after convention translation, or cross-code residuals plateau under refinement. The most common failure mode is a hidden basis or $V$-sign mismatch, not insufficient pixel count.

Check your understanding

Two codes agree exactly in $I$ and $V$, while $(Q,U)$ from one is obtained from the other by a constant rotation through twice the documented camera-basis angle. Should this be counted as a transfer disagreement?

Solution

Not before translating bases. Apply the spin-two transformation associated with the documented screen rotation and compare again. If $Q$ and $U$ then agree within refinement error, the difference was representational. If a residual remains or the required angle varies along a ray without the predicted parallel transport, inspect basis orientation, local plasma rotation, and Faraday signs before interpreting any physics.

Continue with Synthetic-Observation Validation.

References

• Mościbrodzka, Monika, and Charles F. Gammie. 2018. “ipole—Semi-Analytic Scheme for Relativistic Polarized Radiative Transport.” Monthly Notices of the Royal Astronomical Society 475, 43–54. DOI.
• Prather, Cora, et al. 2023. “Comparison of Polarized Radiative Transfer Codes Used by the Event Horizon Telescope Collaboration.” The Astrophysical Journal 950, 35. arXiv.