Time Variability and Strong-Field Lensing Signatures

Strong-field lensing can send radiation from one evolving source region to the observer along several null paths. The resulting subimages differ in magnification, parity, rotation, and arrival time, suggesting echoes or repeated correlations in a horizon-scale movie. The difficulty is that turbulent accretion, orbiting emissivity patterns, interstellar scattering, sparse array sampling, and calibration already produce correlated variability. A credible lensing signature must defeat those alternatives in the measured visibility data, not merely resemble a delayed feature in a reconstructed movie.

Helpful background. Black-Hole Geometry and Mechanics supplies multiple null paths and time delays; Relativistic Matter and Compact Objects supplies evolving synchrotron emissivity; Black-Hole Astrophysics supplies turbulent accretion and variability; and Computational Relativity supplies time-dependent ray tracing and synthetic-observation tests.

Delayed subimages of a variable source

Let $S_\nu(t,\boldsymbol\beta)$ describe specific surface brightness on a thin emitting screen. On the sky region occupied by image branch $n$, geometric optics gives

$$I_{\nu_{\rm obs},n}(t,\boldsymbol\theta) =g_n^3 S_{\nu_{\rm obs}/g_n}\!\left(t-\tau_n(\boldsymbol\theta), \boldsymbol\beta_n(\boldsymbol\theta)\right).$$

Here $g_n$ is the frequency ratio along the ray; absorption would add a transmission factor. The direct image is $n=0$; higher $n$ labels paths that spend progressively longer near unstable null motion. Their angular separation and solid angle generally shrink rapidly with order. There is no additional lensing magnification factor multiplying surface brightness: $I_\nu/\nu^3$ is conserved along a vacuum ray. Magnification enters the flux integral through the solid-angle Jacobian $|\det(\partial\boldsymbol\theta/\partial\boldsymbol\beta)|$. For optically thin volume emission, integrate along each ray as on Relativistic Image Formation, rather than assuming a single emitting screen.

The source coordinates $\boldsymbol\beta_n$, redshifts, and delays $\tau_n$ vary across an extended image, so a higher-order contribution is not exactly a scaled copy of the total light curve. Combining the branches on their sky domains gives the ideal image $I_{\rm ideal}$ used below.

For one compact, slowly evolving feature, the light curve can be approximated as $F(t)=s(t)+\mu s(t-\tau)$. Its autocorrelation contains terms displaced by $\pm\tau$, but intrinsic quasiperiodicity produces similar peaks. Turbulent red noise also generates long runs and broad correlations much more often than a white-noise null predicts. Photon-ring autocorrelation proposals therefore use the linked delay and azimuthal displacement expected from lensing, not a delay alone Hadar et al. 2021, §§ 2–4.

The observable is not $I_{\rm ideal}$ itself. At time $t_j$ the array samples

$$d_{ab,j}=g_{a,j}g_{b,j}^* \int I_{\rm ideal}(t_j,\boldsymbol\theta) e^{-2\pi i\boldsymbol u_{ab,j}\cdot\boldsymbol\theta} \,d^2\theta+n_{ab,j},$$

possibly after scattering convolution and time/frequency averaging. Array rotation changes $\boldsymbol u_{ab,j}$ while the source changes. Treating the samples as a static Fourier transform confounds those two motions.

A named fast-variable data set

The observational anchor is EHT data product 2022-D02-01, “First Sagittarius A* EHT Results: Calibrated Data,” DOI 10.25739/m140-ct59. It contains low- and high-band calibrated Stokes-$I$ visibilities for 2017 April 6 and 7 from both EHT-HOPS and rPICARD reduction paths. The release products are time averaged to 10 s and frequency averaged over 1875 MHz in bands centered at 227.1 and 229.1 GHz; Paper II recommends coherent averaging within 120 s as a calibration-systematics compromise Event Horizon Telescope Collaboration 2022, § 3.3 and §§ 5.1–5.2.

That cadence is data conditioning, not continuous movie coverage. A search must preserve station, scan, baseline, band, flag, coherent-averaging, and signal-to-noise selections. It must also propagate HOPS/rPICARD calibration differences, network gain normalization, the interstellar scattering kernel, and refractive uncertainty. The Sgr A* imaging analysis explicitly compared static and dynamic methods and synthetic variable data Event Horizon Telescope Collaboration 2022, §§ 3–7. It establishes a variable, ring-dominated source under those methods; it is not an observational detection of a higher-order lensing echo.

Detection statistics and adversarial nulls

A test statistic might correlate visibility-domain residuals at trial delay $\Delta t$ and trial angular rotation $\Delta\phi$, or fit a source movie with and without a delayed, magnified component. The likelihood must include correlated thermal errors and gain nuisance parameters. Priors on source correlation time, echo magnification, delay range, scattering, and movie smoothness affect a Bayes factor or posterior interval.

The null ensemble should be harder than independent Gaussian noise. Generate variable GRMHD movies or stochastic movies with the measured power spectrum and cross-baseline covariance; include orbiting hot spots without secondary images, refractive scattering, station gain drifts, and calibration correlations. Sample every null and injected movie at the actual timestamps and $(u,v)$ coordinates. Search it over the same delay, rotation, filter, baseline subset, and data-quality choices used on the observation. This automatically counts the look-elsewhere effect if the analysis choices were fixed before unblinding.

Useful discriminants are recurrence across independent observing intervals, the predicted coupling between delay and angular displacement, consistency across baselines and bands, and recovery by both direct visibility fitting and movie reconstruction. A peak produced only after one station or scan is selected is a warning. A nondetection is informative only if injection recovery demonstrates sensitivity to the predicted magnification and delay.

Results and limitations through 2026-08-18

The evidence cutoff for this active frontier is 2026-08-18.

Evidence layer	Result at the cutoff	Scope and limitation
Analytic mechanism	Multiple null paths predict correlated delays, angular shifts, parity changes, and decreasing magnification.	Extended, optically thick, evolving emission need not behave like one repeated point source.
Idealized simulations	Analytic sources and ray-traced hot spots can show glimmer or echo structure Wong 2021, §§ 2–5.	Detectability depends on source coherence, inclination, resolution, and assumed noise.
Turbulent movie simulations	Fine-grained two-point image correlations can retain extreme-lensing structure even when time averages and ordinary light-curve autocorrelations do not at expected terrestrial next-generation resolution Bezděková et al. 2026, main text and Figs. 2–5.	This is a simulation result; the movies and analysis code are available on request rather than as a DOI-bound executable release.
Individual object: Sagittarius A*	Product 2022-D02-01 supplies a rapidly variable horizon-scale visibility time series with two reduction paths.	Sparse cadence, scattering, calibration, and intrahour evolution prevent a current robust lensing-echo claim.
Individual object: M87*	Slower intrinsic timescales and multi-epoch ring persistence offer a different regime.	A few observing days do not provide a dense, continuous movie at the relevant cadence.
Population and null evidence	No population of independently confirmed strong-field lensing echoes exists; red noise, hot spots, and refractive structure remain live nulls.	Target selection favors only the largest and brightest apparent horizons, so nondetections are not a complete population constraint.

The credible competing positions are delayed higher-order subimages, orbiting emissivity patterns, turbulent red noise, refractive scattering, calibration correlations, and quasiperiodic plasma structure unrelated to photon orbits. Which dominates depends on source population, signal-to-noise, cadence, frequency range, and inference prior. An observational strong-field echo remains unresolved.

The interpretation could change after a longer or denser horizon-scale time series, added baselines, a multi-frequency scattering constraint, independent recurrence of the same delay–rotation relation, or a robust null result whose injection campaign reaches the predicted sensitivity.

Reproducible injection protocol

A defensible teaching study starts from 2022-D02-01 and declares which pipeline product, days, bands, flags, and averaging it uses. Choose a GRMHD or stochastic source movie and record its cadence, electron prescription, viewing geometry, and random seed. Ray-trace both the direct movie and delayed higher-order subimages, inject a blinded magnification and delay, convolve with the declared scattering model, and sample the result at the release’s actual time-dependent baselines. Add thermal noise and station gains drawn from release-motivated distributions.

Analyze at least three ensembles: no lensing, lensing below nominal sensitivity, and lensing above it. Report detection efficiency and false-positive rate over the full trials grid, then repeat by band, scan block, station jackknife, and calibration path. This page specifies the experiment but does not claim to have rerun EHT calibration, GRMHD evolution, or the 2026 correlation study.

Common pitfalls

A local autocorrelation peak is not a global significance. Delay, filter, angular shift, baseline subset, and data cuts all contribute trials.

Periodicity does not identify a photon orbit. A rotating hot spot, instability, or turbulent coherence can produce a preferred timescale without a delayed subimage.

A null result without injection sensitivity is weak. Scattering, sparse coverage, and low magnification may hide the signal the test purports to exclude.

Exercises

A search evaluates 100 independent trial delays. Its most prominent peak has a one-trial null probability $p_{\rm local}=0.001$. Compute the probability of at least one equally extreme peak anywhere in the scan. Is the peak a 1% global detection?

Solution

Under the stated independence approximation, the probability that no trial exceeds the peak is $(1-p_{\rm local})^{100}$. Hence

$$p_{\rm global}=1-(1-0.001)^{100} =1-0.999^{100} \simeq0.0952.$$

The global false-alarm probability is about $9.5\%$, so this is not a 1% global detection. For a 1% global threshold, one would require $p_{\rm local}=1-0.99^{1/100}\simeq1.005\times10^{-4}$. Real delay trials are correlated, so the effective number of trials should be calibrated with the complete red-noise and sampling simulation rather than assumed to be exactly 100.

References

• Bezděková, Barbora, Shahar Hadar, George N. Wong, et al. “Extreme Lensing Signatures Revealed by Correlations of Simulated Black Hole Movies.” Nature Astronomy (2026). DOI.
• Event Horizon Telescope Collaboration. “First Sagittarius A* EHT Results: Calibrated Data.” Data product 2022-D02-01 (2022). DOI.
• Event Horizon Telescope Collaboration. “First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration.” The Astrophysical Journal Letters 930 (2022): L13. DOI.
• Event Horizon Telescope Collaboration. “First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole.” The Astrophysical Journal Letters 930 (2022): L14. DOI.
• Hadar, Shahar, Michael D. Johnson, Alexandru Lupsasca, and George N. Wong. “Photon Ring Autocorrelations.” Physical Review D 103 (2021): 104038. DOI.
• Wong, George N. “Black Hole Glimmer Signatures of Mass, Spin, and Inclination.” The Astrophysical Journal 909 (2021): 217. DOI.