Imaging, Shadows, Polarization, and Strong-Field Lensing

Horizon-scale imaging is a chain of forward problems. Spacetime bends null rays; plasma emits, absorbs, and polarizes radiation; an interferometric array samples only selected Fourier components; calibration and an inverse method turn those samples into constraints on image structure. This chapter follows that chain from metric critical curves to time-dependent, polarized VLBI data. It covers shadows, photon subrings, transfer, reconstruction, polarization, and lensing correlations, while excluding the claim that a dark reconstructed region directly images an event horizon.

Helpful background. Black-Hole Geometry and Mechanics supplies null geodesics, critical curves, and observer screens; Relativistic Matter and Compact Objects supplies magnetized emitting plasma; Black-Hole Astrophysics supplies accretion and source variability; and Computational Relativity supplies ray tracing, transfer, and synthetic-observation validation.

From spacetime to measured correlations

Three equations organize the chapter. Covariant transfer produces a screen intensity,

$$I_{\nu_{\rm obs}}(\alpha,\beta,t) =\int_{\rm ray}g^3j_{\nu_{\rm em}} e^{-\tau_\nu}\,d\ell_{\rm prop},$$

with $g=\nu_{\rm obs}/\nu_{\rm em}$. Interferometry Fourier-transforms that brightness,

$$V(u,v,t)=\int\!\!\int I(l,m,t) e^{-2\pi i(ul+vm)}\,dl\,dm,$$

and a baseline records the gain-corrupted sample

$$V^{\rm obs}_{ab}(t)=g_a(t)g_b^*(t)V(u_{ab},v_{ab},t)+n_{ab}(t).$$

Every interpretation must traverse those maps in the forward direction before inverting them. The metric does not determine $j_\nu$, the plasma image does not fill the Fourier plane, and a reconstruction does not remove gain or model uncertainty.

For Schwarzschild, the unstable circular photon orbit lies at $r=3GM/c^2$ and its critical impact parameter is $b_c=3\sqrt{3}\,GM/c^2$. That metric scale is distinct from three related observables: a capture silhouette defined relative to illumination, narrow photon subrings made by rays with extra half-orbits, and an emissivity-dependent central brightness depression. Perlick and Tsupko develop the observer-sky shadow construction in Perlick and Tsupko 2022, Open PDF, §§ II–III, especially Eq. (8); the critical-curve page explains how it differs from features of an emitting flow.

Six routes through the chapter

Read the pages in order for the complete inference chain:

1. Critical Curves, Shadows, and Photon Rings derives $b_c=3\sqrt{3}\,GM/c^2$ and separates metric, illumination, and emissivity structures.
2. Relativistic Image Formation derives invariant radiative transfer and shows how redshift, beaming, opacity, electron physics, and lensing share the image.
3. Very-Long-Baseline Interferometry maps brightness to sparse complex visibilities and constructs closure phases and amplitudes that cancel ideal station gains.
4. Image Reconstruction and Model Fitting compares CLEAN, regularized, Bayesian, geometric, and physical-library routes using blind synthetic tests and held-out data.
5. Polarimetric Black-Hole Imaging follows $Q+iU$ through synchrotron transfer, Faraday rotation, feed leakage, EVPA calibration, and magnetic-model comparison.
6. Time Variability and Strong-Field Lensing Signatures tests delayed subimages against turbulent red noise, orbiting emissivity, scattering, cadence, and look-elsewhere effects.

The conceptual seam between pages is deliberate. A critical curve becomes observable only through a transfer model. A transfer image becomes data only after sampling and calibration. An inferred feature becomes credible only after reconstruction alternatives, polarization nuisance parameters, time dependence, and synthetic counterexamples are tested.

Named data and software anchors

The chapter uses DOI-bound releases so that “the EHT data” never hides a change of epoch, Stokes product, flags, or calibration. The source status below is current through 2026-08-18.

Product	Scientific role	Reproduction boundary
M87* 2019-D01-01, DOI 10.25739/g85n-f134	Calibrated 2017 Stokes-$I$ visibilities for transfer, VLBI, reconstruction, and geometric fitting	The public Stokes-$I$ files differ slightly from the RR/LL working data used in Paper IV; flags, days, bands, and gain treatment must be retained
M87* 2019-D01-02, pinned commit 80d230e76c08edd31548e7cfe61dc7cf94300780	DIFMAP, eht-imaging, and SMILI parameter-survey pipelines	Modern dependency changes and random seeds can alter details; compare visibility predictions, not only rendered images
M87* 2023-D01-01, DOI 10.25739/q46m-m857	Public calibrated 2017 polarimetric visibilities	Feed basis, Stokes sign, D-terms, R/L gains, EVPA zero point, masks, and time averaging belong to the result
Sagittarius A* 2022-D02-01, DOI 10.25739/m140-ct59	Calibrated 2017 variable Stokes-$I$ visibilities from HOPS and rPICARD paths	Scattering, intrahour variability, cadence, averaging, and pipeline-specific calibration must be propagated
Sagittarius A* 2024-D02-01, DOI 10.25739/k39c-mh27	Public polarized data for a second horizon-scale source	A different mass scale, variability regime, and scattering screen prevent direct transfer of M87* assumptions

Paper IV’s independent imaging teams and synthetic source survey show why multiple algorithms are necessary Event Horizon Telescope Collaboration 2019, §§ 5–9. Paper VI shows why an observed emission-ring diameter must be calibrated to a metric scale through geometric and GRMHD models rather than equated with it Event Horizon Telescope Collaboration 2019, §§ 3–6 and § 8.

A defensible inference ladder

Begin with the released observables: dates, stations, baselines, bands, flags, averaging, thermal covariance, and calibration products. Next specify the forward model, including spacetime, plasma emissivity and absorptivity, source evolution, scattering, and instrumental response. Only then select a likelihood, gain and physical priors, reconstruction family, and feature statistic.

Validation must attack each link. Analytic point-source, Gaussian, and thin-ring transforms check Fourier normalization. Flat-spacetime and homogeneous-slab limits check transfer. Injected station gains check closure invariance. Blind synthetic images test whether regularizers manufacture rings. Leakage and Faraday injections test polarization recovery. Variable null movies sampled at the actual cadence calibrate lensing-correlation trials.

Statistical uncertainty is conditional on the chosen likelihood and prior. Systematic uncertainty includes calibration, sparse coverage, scattering, variability, electron distribution, regularization, and physical-library incompleteness. Some of those alternatives produce separate posterior modes rather than a symmetric error bar. The most durable conclusions are therefore expressed at the level supported by all routes: a characteristic ring-scale structure and polarized, variable emission around the named compact objects. Detailed brightness maps, three-dimensional magnetic topology, and higher-order lensing echoes remain more conditional.

Review the chapter

Model a source as an infinitesimally thin circular ring of angular diameter $d=42\,\mu{\rm as}$. Its normalized visibility is $J_0(\pi d q)$, where $q$ is baseline length in wavelengths and the first zero of $J_0$ is $2.4048$. Find the first-null baseline at $\lambda=1.3$ mm. Then compare $d$ with the Schwarzschild critical-curve diameter for $M=6.5\times10^9M_\odot$ and $D=16.8$ Mpc. Does the difference alone test general relativity?

Solution

The first null satisfies $\pi d q_1=2.4048$, so $q_1d=0.7655$. Converting the diameter,

$$d=42\times10^{-6}\ {\rm arcsec} =2.036\times10^{-10}\ {\rm rad},$$

gives

$$q_1=\frac{0.7655}{2.036\times10^{-10}} =3.76\times10^9, \qquad B_1=\lambda q_1\simeq4.89\times10^6\ {\rm m}.$$

Thus a roughly $4{,}890$ km projected baseline reaches the thin-ring null. Using $GM_\odot/c^2=1.4766$ km,

$$\theta_g=\frac{GM}{c^2D}\simeq3.82\,\mu{\rm as}, \qquad d_c=6\sqrt{3}\,\theta_g\simeq39.7\,\mu{\rm as}.$$

The model emission ring is about $5.8\%$ larger than the Schwarzschild critical-curve diameter. That difference alone is not a test of general relativity: a thin ring was assumed, the measured visibility is sparse and calibrated, and emissivity, opacity, spin, inclination, mass, and distance mediate the conversion from emission-ring size to critical-curve size. The exercise connects a Fourier feature to an angular scale; it does not bypass the forward model.

References

• Event Horizon Telescope Collaboration. “First M87 EHT Results: Calibrated Data.” Data product 2019-D01-01 (2019). DOI.
• Event Horizon Telescope Collaboration. “First M87 EHT Results: Calibrated Polarimetric Data.” Data product 2023-D01-01 (2023). DOI.
• Event Horizon Telescope Collaboration. “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole.” The Astrophysical Journal Letters 875 (2019): L4. DOI.
• Event Horizon Telescope Collaboration. “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole.” The Astrophysical Journal Letters 875 (2019): L6. DOI.
• Event Horizon Telescope Collaboration. “First Sagittarius A* EHT Results: Calibrated Data.” Data product 2022-D02-01 (2022). DOI.
• Perlick, Volker, and Oleg Yu. Tsupko. “Calculating Black Hole Shadows: Review of Analytical Studies.” Physics Reports 947 (2022): 1–39. DOI.