Polarimetric Strong-Field Tests

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Polarization carries information that total intensity discards: projected magnetic orientation, optical depth, Faraday rotation and conversion, and relativistic transport along null rays. Those effects are entangled. An azimuthal polarization pattern is not by itself a geometric invariant, and a position-angle rotation is not automatically gravitational. A strong-field test must identify a frequency, time, or baseline dependence that source and calibration alternatives cannot reproduce. The evidence boundary is 2026-08-19.

Helpful background. Weak-Field Gravity supplies far-field transport checks; Probability and Statistical Inference supplies full-Stokes likelihoods; Black-Hole Geometry and Mechanics supplies null rays and observer tetrads; and Black-Hole Astrophysics supplies synchrotron emission, magnetic fields, Faraday transfer, and polarized imaging.

Vacuum transport and plasma transfer

In geometric optics, a photon wavevector $k^\mu$ follows a null geodesic and a transverse polarization vector $f^\mu$ is parallel transported:

k^\nu\nabla_\nu k^\mu=0, \qquad k^\nu\nabla_\nu f^\mu=0, \qquad k_\mu f^\mu=0.

The measured angle requires an observer tetrad and a screen basis. Define complex linear polarization $P=Q+iU$ and electric-vector position angle $\chi=\tfrac12\arg P$ , modulo $\pi$ . If the screen axes rotate by $\psi$ , the components transform as $P' = e^{-2i\psi}P$ and $\chi'=\chi-\psi$ ; this is a coordinate change, not a physical rotation. The linear polarization fraction $\lvert P\rvert/I$ is unchanged. Parallel transport along a fixed ray in vacuum is achromatic in standard general relativity; different observing frequencies may still sample different emitting regions and rays. Plasma Faraday rotation is approximately

\Delta\chi=\mathrm{RM}\,\lambda^2

for a simple external screen, with rotation measure RM in radians per square metre when wavelength is in metres. The observed angle is $\chi(\lambda)=\chi_0+\mathrm{RM}\lambda^2$ modulo $\pi$ ; differences must therefore be unwrapped consistently. Internal Faraday rotation, conversion between linear and circular polarization, optical depth, and relativistic electron distributions require polarized transfer. In a local plasma frame over a small path segment it has the form

\frac{d\boldsymbol S}{ds} =\boldsymbol j-\mathbf K\boldsymbol S, \qquad \boldsymbol S=(I,Q,U,V)^T.

Here $s$ is physical path length in that local frame. The emissivity vector $\boldsymbol j$ and propagation matrix $\mathbf K$ depend on magnetic strength and direction, electron distribution, density, temperature, and local frequency. This flat-frame notation must not be integrated unchanged across a curved spacetime: a relativistic calculation transports the screen basis and uses invariant Stokes intensities $\boldsymbol S_\nu/\nu^3$ , with consistently transformed emission and propagation coefficients. Geometry sets rays and transported bases; the plasma sets where polarized photons are created, absorbed, rotated, or converted. This is why a gravity parameter cannot be attached to an image-domain angle without a source model.

Full-Stokes visibility data and calibration

An interferometer measures correlations in receptor bases that are converted into Stokes visibilities. The response includes complex station gains, bandpass, feed basis, cross-hand phase, polarization leakage or D-terms, absolute angle calibration, parallactic-angle coverage, and network constraints. Scattering, time averaging, and intrinsic variability remain. Closure traces or polarimetric closure quantities reduce certain station corruptions but do not remove thermal noise, baseline errors, sparse coverage, or source evolution.

The M87 polarized-ring analysis reconstructs linear polarization with multiple imaging methods and synthetic-data controls Event Horizon Telescope Collaboration 2021, Paper VII §§3–7. Its physical interpretation compares full-polarization data with GRMHD and radiative-transfer libraries Event Horizon Telescope Collaboration 2021, Paper VIII §§3–5. These are source-and-geometry comparisons, not a source-free measurement of parallel transport.

A new analysis must pin the actual versioned full-Stokes dataset and data release, observing days, bands, stations, scans, flags, leakage solution, cross-hand convention, averaging, scattering prescription, and calibrator products. The fixed M87 release 2019-D01-01 is an essential Stokes- $I$ visibility boundary, but it must not be silently presented as the complete full-Stokes product required for a new polarization fit. Source selection includes time intervals, signal-to-noise, array subsets, image regularization, polarization debiasing, and simulation-library cuts.

For $N$ complex visibility residuals, avoid ambiguity in covariance normalization by stacking their real and imaginary parts into the real $2N$ -vector $\boldsymbol r$ . If these components have a joint Gaussian distribution with real covariance $C_R=\langle\boldsymbol r\boldsymbol r^T\rangle$ , use

\ln\mathcal L= -\frac12\boldsymbol r^T C_R^{-1}\boldsymbol r -\frac12\ln\det C_R-N\ln(2\pi).

This density uses the measure over all real and imaginary components. For one circular complex residual whose real and imaginary parts each have variance $\sigma^2$ , it reduces to $p(r)=\exp[-\lvert r\rvert^2/(2\sigma^2)]/(2\pi\sigma^2)$ ; its complex variance is $\langle\lvert r\rvert^2\rangle=2\sigma^2$ . Mixing these two covariance definitions creates a factor-of-two error. Closure statistics require their own likelihood. Gains, leakage, absolute angle, mass-to-distance ratio, magnetic topology, electron thermodynamics, Faraday depth, inclination, geometry, and variability are nuisance or tested parameters with explicit priors. Shared calibration produces covariance among Stokes products.

Reproduction requires release identifiers and checksums, calibration software, leakage and angle solutions, selection lists, GRMHD snapshot identifiers, polarized-transfer code and commit, camera basis, frequency, electron prescription, priors, sampler diagnostics, and synthetic visibility products. This page reports no new full-Stokes fit.

Results and limitations through 2026-08-19

Evidence or position	Result supported	Limitation or scope
Resolved M87 linear polarization across independent reconstructions	Establishes ordered horizon-scale polarization structure in the selected data	Leakage, sparse coverage, variability, and reconstruction regularization remain
GRMHD plus polarized-transfer comparisons	Constrain combinations of magnetic flux state, inclination, electrons, and Faraday depth	Simulation and electron-model libraries are incomplete and share Kerr geometry
Achromatic transport versus chromatic Faraday tests	Offer a principled discriminator for propagation rotations	Optical depth, internal conversion, calibration, bandwidth, and time variability can mix scalings

The conservative position is that present polarimetry strongly informs plasma and magnetic structure while only conditionally testing geometry. A competing interpretation may attribute an ordered angle pattern or rotation to spacetime transport, but magnetic topology and Faraday effects provide physically motivated alternatives. Evidence against a geometric claim includes a $\lambda^2$ dependence, sensitivity to leakage or angle calibration, or movement under electron prescriptions. No current pattern uniquely establishes a non-Kerr metric; that boundary remains unresolved and model-dependent.

The adversarial check injects a blinded relativistic-transport signature into synthetic full-Stokes visibilities, then varies magnetic geometry, optical depth, internal and external Faraday models, leakage, and variability. Closure-polarimetric and image-domain analyses should recover the same parameter, and an achromatic candidate should predict held-out frequencies and epochs. A source model trained on the tested pattern is not an independent check.

What could change the conclusion. The interpretation could change after a fixed full-Stokes visibility or calibration release, a new frequency or epoch adds a Faraday lever arm, or a blinded multi-pipeline challenge shows that a transport signature survives magnetic, plasma, and leakage alternatives.

Exercise

At wavelengths $\lambda$ and $2\lambda$ , an observed position-angle change is $5^\circ$ and $20^\circ$ . What scaling does this support, and why is it not yet a complete plasma explanation?

Solution

The angle grows by a factor of four when wavelength doubles, which follows the external-screen relation $\Delta\chi\propto\lambda^2$ . That behavior is evidence against a purely achromatic vacuum-transport rotation. It is not a complete explanation because internal Faraday rotation, optical depth, multiple components, bandwidth averaging, angle wrapping, and calibration can distort the simple law. One should therefore fit full Stokes data across frequency and verify the same rotation measure on held-out baselines or epochs.

References

Event Horizon Telescope Collaboration (2021a), “First M87 Event Horizon Telescope Results. VII. Polarization of the Ring,” Astrophysical Journal Letters 910, L12. DOI.
Event Horizon Telescope Collaboration (2021b), “First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure Near the Event Horizon,” Astrophysical Journal Letters 910, L13. DOI.