Horizon-Scale Imaging Tests

Horizon-scale very-long-baseline interferometry measures sparse samples of a source’s Fourier transform. A reconstructed ring is downstream of calibration and imaging; its relation to a photon critical curve or shadow is downstream again of plasma emission and relativistic transfer. Geometry should therefore be tested in visibility and closure-quantity space across source and reconstruction families, not read directly from one image. The evidence boundary is 2026-08-19.

Helpful background. Weak-Field Gravity supplies far-field lensing checks; Probability and Statistical Inference supplies correlated visibility likelihoods; Black-Hole Geometry and Mechanics supplies photon regions, shadows, and mass–spin scales; and Black-Hole Astrophysics supplies relativistic transfer, GRMHD flows, and image reconstruction.

Visibilities before images

For small sky angles $(\alpha,\beta)$, the complex visibility is

$$V(u,v,t,\nu)=\int d\alpha\,d\beta\, I(\alpha,\beta,t,\nu) e^{-2\pi i(u\alpha+v\beta)}.$$

A baseline between stations $i$ and $j$ measures approximately

$$V^{\rm obs}_{ij}=g_i g_j^*V_{ij}+n_{ij},$$

with station gains $g_i$ and thermal noise $n_{ij}$. Closure phase around a triangle cancels station phase, while closure amplitude combines four baselines to cancel leading station amplitudes. These invariants reduce, but do not eliminate, polarization leakage, bandpass, averaging, network-calibration, and non-closing errors.

Here $(u,v)$ is projected baseline in units of observing wavelength, $(\alpha,\beta)$ are sky angles in radians, and $I$ is specific intensity per solid angle. Each baseline samples one spatial Fourier component; sparse baseline coverage explains why reconstruction needs assumptions about unsampled components. The cancellations can be checked directly, neglecting noise:

$$\arg(V^{\rm obs}_{12}V^{\rm obs}_{23}V^{\rm obs}_{31}) =\arg(V_{12}V_{23}V_{31}), \qquad \frac{\lvert V^{\rm obs}_{12}\rvert\lvert V^{\rm obs}_{34}\rvert} {\lvert V^{\rm obs}_{13}\rvert\lvert V^{\rm obs}_{24}\rvert} =\frac{\lvert V_{12}\rvert\lvert V_{34}\rvert} {\lvert V_{13}\rvert\lvert V_{24}\rvert}.$$

The triangle gain product is the positive number $\lvert g_1g_2g_3\rvert^2$. In the amplitude ratio each station magnitude appears once above and below. These identities require simultaneous, consistently averaged station-based gains; they do not justify a Gaussian approximation for low-signal closure data.

The EHT M87 calibrated Stokes-$I$ boundary is fixed release 2019-D01-01, DOI 10.25739/g85n-f134. Paper III describes correlation, fringe fitting, a priori amplitude calibration, network calibration, and validation Event Horizon Telescope Collaboration 2019, Paper III §§4–7. Paper IV compares independent imaging pipelines and synthetic-data controls Event Horizon Telescope Collaboration 2019, Paper IV §§5–9. A reproducible analysis pins observing day, band, station, scan, flag, averaging interval, flux convention, and exact visibility or closure product.

Sgr A* has a separate fixed release, 2022-D02-01, DOI 10.25739/m140-ct59. Paper II describes data processing and calibration Event Horizon Telescope Collaboration 2022, Paper II §§3–7; Paper III treats imaging under sparse coverage, rapid variability, and interstellar scattering Event Horizon Telescope Collaboration 2022, Paper III §§4–10. M87 and Sgr A* cannot be assigned one common static-source likelihood.

Geometry through a plasma forward model

The forward chain is

$$(\text{metric},M/D) \longrightarrow \text{null rays and critical curve} \longrightarrow I_\nu(\alpha,\beta,t;\eta_{\rm plasma}) \longrightarrow V(u,v,t,\nu) \longrightarrow d.$$

The observed emission ring need not coincide with the critical curve, and the critical curve is the lensed boundary associated with unstable null motion, not an image of the event horizon. Synchrotron emissivity and absorption depend on electron density and distribution, magnetic field, temperature prescription, flow inclination, and optical depth. GRMHD libraries add magnetic-flux state, spin, simulation duration and resolution, fast- or slow-light transfer, and selection by non-EHT observables. M87 Paper V explicitly calibrates physical interpretation through such model libraries Event Horizon Telescope Collaboration 2019, Paper V §§3–6; Sgr A* Paper V likewise finds that variability and multiwavelength controls strongly filter the model set Event Horizon Telescope Collaboration 2022, Paper V §§3–5.

A geometric scale check. For a Schwarzschild black hole viewed from far away, the unstable circular photon orbit is at $r=3GM$. The critical impact parameter follows from $b=r/\sqrt{1-2GM/r}$ there: $b_c=3\sqrt3\,GM$. The critical-curve angular diameter is consequently $6\sqrt3\,GM/D$, about $10.39$ gravitational radii divided by distance, whereas the horizon’s coordinate radius is $2GM$. Restoring dimensions replaces $GM$ by $GM/c^2$. Neither the horizon radius nor twice the photon-orbit radius gives the observed critical-curve diameter; gravitational bending is essential. Plasma can shift the brightest emission away from this geometric curve, which is why its calibration is an inference step.

A visibility-domain likelihood should compare time-dependent synthetic data sampled at the observed $(u,v)$ points, with thermal noise and gain terms. Parameters include mass-to-distance ratio, orientation, total flux, variability, scattering, gains, plasma model, and geometry. Priors and covariance matter: importing $M/D$ from stellar dynamics is powerful, but the prior must be declared and not reused as though it were imaging data. Image regularization is a model choice; consistency among reconstructions is valuable only when their shared assumptions and synthetic challenges are visible.

Papers VI map observed size to geometric predictions using calibrated model libraries rather than equating ring and shadow. The M87 analysis states that calibration explicitly Event Horizon Telescope Collaboration 2019, Paper VI §§3–8, and the Sgr A* metric analysis includes both Kerr and non-Kerr simulations plus compact-surface alternatives Event Horizon Telescope Collaboration 2022, Paper VI §§3–7. Those are conditional tests of model families, not direct metric reconstructions.

Calibration, selection, and reproducibility

The dataset response includes station gains, phases, clocks, bandpass, polarization combination, array geometry, scattering, coherent averaging, and variability. Selection includes observing nights, scans, stations, signal-to-noise and flag rules, closure products, variability segmentation, external flux constraints, simulation-library cuts, and reconstruction hyperparameters. Sparse coverage couples these choices to morphology.

A normalized complex-Gaussian or closure likelihood must state its noise covariance and treatment of gain nuisance parameters. Posterior predictive checks should return visibilities and closures, not only images. Reproduction requires the fixed release DOI, file checksums, calibration pipeline version, selection list, source movie or simulation identifiers, ray-transfer code and commit, scattering kernel, Fourier-sampling code, priors, sampler diagnostics, and machine-readable predictions. This page claims no new fit.

Results and limitations through 2026-08-19

Evidence or position	Result supported	Limitation or scope
M87 2019-D01-01 visibilities, closures, and multi-pipeline imaging	Support a persistent ring-like morphology on horizon scales	Sparse coverage, gain calibration, source evolution, and reconstruction regularization remain
Sgr A* 2022-D02-01 with variability and scattering controls	Supports ring-dominated time-averaged morphology under several methods	Rapid variability, scattering, averaging, and incomplete source models broaden interpretation
M87 and Sgr A* model-library metric comparisons	Find data compatible with Kerr shadow-scale predictions in declared families	Plasma, GRMHD library, mass–distance prior, selection, and deformation parameterization bound the conclusion

The conservative position is that calibrated data support ring-like emission whose scale is consistent with GR-based models. A competing interpretation may call a reconstructed diameter a direct metric measurement; evidence against that shortcut is the shift between emission ring, critical curve, and shadow across plasma models. Flexible source structure can also fit sparse visibilities, while overly narrow source libraries can make geometry look artificially precise. The horizon itself is not resolved as a luminous surface, and the conclusion remains model-dependent.

The adversarial check generates withheld GRMHD movies and geometric sources, corrupts them with the real array response, scattering, variability, and station gains, and requires stable geometry constraints across direct visibility fits and independent reconstructions. Closure and station-split tests must pass, and alternate plasma prescriptions must not turn a null geometry residual into a deviation.

What could change the conclusion. The interpretation could change after a superseding fixed visibility or calibration release, an expanded array or multi-epoch dataset, or a blinded synthetic challenge demonstrates that scattering, variability, plasma libraries, or reconstruction choices materially move the geometry inference.

Exercise

Why can closure phases remain useful when station phases are poorly known, and why do they not make an image reconstruction calibration free?

Solution

Multiplying visibilities around a closed triangle makes each station phase appear once with each sign, so the leading station-based phase factors cancel. Therefore closure phase retains source asymmetry information without solving every station phase. It does not cancel thermal noise, baseline-dependent errors, polarization leakage, time averaging, sparse Fourier coverage, scattering, or variability. Amplitude information and absolute scale also require additional calibration. A geometry test must consequently forward-model closures and calibrated visibilities with their covariance.

References

• Event Horizon Telescope Collaboration (2019a), “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration,” Astrophysical Journal Letters 875, L3. DOI.
• Event Horizon Telescope Collaboration (2019b), “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole,” Astrophysical Journal Letters 875, L4. DOI.
• Event Horizon Telescope Collaboration (2019c), “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring,” Astrophysical Journal Letters 875, L5. DOI.
• Event Horizon Telescope Collaboration (2019d), “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole,” Astrophysical Journal Letters 875, L6. DOI.
• Event Horizon Telescope Collaboration (2022a), “First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration,” Astrophysical Journal Letters 930, L13. DOI.
• Event Horizon Telescope Collaboration (2022b), “First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole,” Astrophysical Journal Letters 930, L14. DOI.
• Event Horizon Telescope Collaboration (2022c), “First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole,” Astrophysical Journal Letters 930, L16. DOI.
• Event Horizon Telescope Collaboration (2022d), “First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric,” Astrophysical Journal Letters 930, L17. DOI.
• National Radio Astronomy Observatory (2019–2022), “Event Horizon Telescope Public Data Products,” fixed releases 2019-D01-01 and 2022-D02-01. M87 DOI, Sgr A* DOI.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Horizon-Scale Imaging Tests of Kerr