Correlations of Simulated Black-Hole Movies Reveal Extreme-Lensing Signatures

TL;DR

The paper addresses detecting extreme gravitational lensing near black holes when the photon ring is unresolved by leveraging the time-dependent two-point image correlation function $\mathcal{C}$. It uses a high-fidelity GRMHD BH movie with ray tracing under slow-light, fast-light, and $n=0$-only prescriptions, then analyzes correlation patterns on a reduced 3D configuration space to identify lensing signatures. The key finding is that a distinct lensing peak at $T \sim 15\ GM/c^3$ and a specific azimuthal location appears in fine-grained image correlations and persists under realistic angular resolution, while coarse observables like light-curve autocorrelations fail to reveal it; the signal's strength depends on the ray-tracing scheme. This work demonstrates a practical pathway to measure extreme lensing effects with upcoming terrestrial VLBI facilities (e.g., ngEHT, BHEX), guiding instrument design and inference strategies for black-hole spacetime tests.

Abstract

A black hole's gravitational pull can deflect light rays to an arbitrary degree. As a result, any source fluctuation near the black hole creates multiple lagged images on an observer's screen. For optically thin stochastic emission, these light echoes give rise to correlations of brightness fluctuations across time-dependent images (movies). The correlation pattern disentangles source-specific characteristics from universal features dictated by general relativity. This picture has motivated a proposal to use the two-point image correlation function as a probe of extreme gravitational lensing in upcoming black-hole imaging campaigns. In this work, we test the feasibility of this method by computing the two-point correlation function of brightness fluctuations in a black-hole movie of state-of-the-art realism. The movie is generated by ray tracing a general relativistic magnetohydrodynamic simulation, which can then be blurred to any angular resolution. At an effective resolution expected to be achieved by next-generation terrestrial very-long-baseline interferometric arrays, the lensing signatures appear in neither time-averaged images nor light-curve autocorrelations. However, we demonstrate that they are clearly visible in the more fine-grained two-point image correlation function. Our positive findings motivate a more comprehensive investigation into the instrument specifications and inference techniques needed to resolve extreme lensing effects through correlations.

Figures (6)

• Figure 1: Time-averaged images of intensity as a function of screen position, $\langle I(t,x,y) \rangle$, produced from simulated slow-light ray-traced movies. Panel (a) shows the unblurred image; other panels show time-averages of movies blurred by different Gaussian kernels, characterized by their FWHM, (b) $=5 \, \mu \hbox{as}$, (c) $=10 \, \mu \hbox{as}$, (d) $=15 \, \mu \hbox{as}$, to simulate the effect of limited instrumental resolution. Hereafter, for the assumed black hole parameters, mimicking those of M87*, $GM/c^2$ corresponds to $3.8\, \mu$as. The white horizontal lines in the bottom right corner of the blurred images show the FWHM of the applied kernels.
• Figure 2: Empirical critical curves derived for movies of different blurrings (color-coded dots) and the theoretical critical curve (black circle). The blue point, A, marks the position of our fixed point, set at $(4.478,-4.478)~GM/c^2$, and the red point, B, shows the theoretically predicted position of its secondary image (assuming an equatorial source). The color crosses show points of highest correlation located on the given empirical critical curve. The black point is our chosen pole, set at $(0,0)~GM/c^2$.
• Figure 3: Two-point correlation function of intensity fluctuations $\mathcal{C}(T,\rho_c(\varphi),\varphi,x_0',y_0')$ (see Eq. \ref{['corr_fun_6D']}) obtained with different blurring kernels (rows) and ray-tracing prescriptions (columns) as a function of the angular position on the experimental critical curve $\varphi$ and the time lag $T$. The dashed line shows the angular position $\varphi_0'=-\pi/4 \, \mathrm{rad}$ of the point A, that we fix in this analysis, located at $(x_0',y_0')=(4.478,-4.478)~GM/c^2$, and the horizontal and vertical dotted lines mark the expected positions of the secondary peak in the temporal and angular domains, respectively.
• Figure 4: Correlation maps $\mathcal{C}(T_0,x,y,x_0',y_0')$ for different choices of time lag $T_0$ and blurring kernel widths. Each map was generated from slow-light ray-traced movies. The astrophysical correlation lobe peaks near the bottom right corner of the image at $T=0 \, GM/c^3$, while the lensing correlation lobe peaks near the upper left corner of the image around $T=15 \, GM/c^3$. These correlation maps are consistent with the presence of a local maximum of $\mathcal{C}(T,x,y,x_0',y_0')$ in the 3-dimensional configuration space $\{T,x,y\}$, constituting a signature of extreme lensing.
• Figure 5: (a) Light curve of the slow-light black-hole movie. The red rectangle shows the position of the zoomed-in light curve depicted in the bottom subpanel. (b) Autocorrelation function of the light curve shown in panel (a). (c) Flux density profiles obtained from the slow-light movie, corresponding to pixels at points A (blue) and B (red); see Fig. \ref{['points_cc']}. (d) Cross-correlation functions of the flux densities at points A, B derived from the slow-light (solid line), $n=0$ (dashed line), and fast-light (dashed-dotted line) movies.