Autocorrelation in black hole flare movies: Departures from Kerr

The gist The autocorrelation in black hole movies encodes time delays and angular separations between gravitationally lensed images, offering a means of testing the spacetime geometry.

Setup and Methodology

The study investigates departures from Kerr geometry using the Johannsen-Psaltis (JP) spacetime with a single nonzero deformation parameter as a test case The researchers generate movies of a compact hot spot on an equatorial circular orbit and characterize the lensing-induced secondary ridge by its slope and intercept in the time-angle autocorrelation map The two-dimensional autocorrelation exhibits a reflection symmetry, i.e., C2D (T, Φ) = C2D (−T, −Φ). For numerical calculations, the continuous integrals above are replaced by discrete Riemann sums.

Secondary Ridge Observables

The secondary ridge observables are defined by approximating the center line of this branch by Φ = kT + b These parameters, k and b, are determined by minimizing χ2(k, b) = X (Ti,Φi)∈R C(Ti, Φi) [Φi − kTi − b]2, which yields specific formulas for k and b. In the face-on limit, the slope satisfies k = ωK For a finite inclination, the slope still retains part of the physical meaning of the angular velocity, corresponding to the “pattern speed” introduced in [46] and [47].

Constraining Parameters

The measured slope and intercept can jointly constrain spin and deformation The researchers construct a Kerr reference region by varying both the spin and the orbital angular velocity at fixed orbital radius and observer inclination. They find that some low-spin configurations with negative ϵ3 produce slope-intercept pairs outside this reference region, not only for Keplerian but also for sub-Keplerian and super-Keplerian orbital motions.

Distinguishing Non-Kerr Signatures

The results show that the spin and metric deformation parameter affect the ridge observables differently Increasing the prograde spin primarily raises the intercept, whereas negative ϵ3 shifts the observables toward larger slopes and smaller intercepts. For rs = 8M and θo = 17◦, low-spin configurations with negative ϵ3 extend below its lower boundary, with intercepts smaller than the Kerr minimum at the same slope. These separation occurs at each of the three investigated ratios, ω/ωK = 0.7, 1.0, and 1.3, and therefore persists when the source moves more slowly or more rapidly than the Keplerian rate. The Kerr band thus provides a direct consistency test.

Implications for Gravity Tests

The pair of ridge observables distinguishes the effects of a negative JP deformation from those of changes in the Kerr spin and orbital angular velocity within the reference region. The Kerr band thus provides a direct consistency test. A natural next step is to extend the analysis to other parametrized metrics and black hole solutions in specific theories of gravity. Future work can extend the analysis to more general trajectories and use numerical simulations to model spatially extended, evolving emission. The comparison of resulting autocorrelation signatures would reveal which features are shared by different non-Kerr metrics and which depend on the particular spacetime.

Appendix Details

The center of the hot spot follows the equatorial circular orbit xµs(t) = t, rs, π/2, ϕs(t) = ϕs,0 + ωt Its four-velocity is uµs = u t s (1, 0, 0, ω), u t s = −gtt − 2ωgtϕ − ω2 gϕϕ−1/2. The corresponding emissivity profile is J(x) = J0 exp − x2 / 2s2. The integration is performed along null geodesics traced backward from the observer’s screen. At every point along a ray, the hot-spot position is evaluated at the corresponding emission time tp. This slow-light prescription accounts for the source evolution during photon propagation and, in particular, for the different emission phases associated with the direct and higherorder images. The details of the GRRT prescription are provided in Appendix A.

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The results show that a Kerr model with a different spin and orbital angular velocity can reproduce the same pair (k, b). Such regions appear at all three displayed ratios, demonstrating that distinguishable non-Kerr signatures persist whether the motion is slower or faster than the Keplerian rate. The Kerr band thus provides a direct consistency test. Points inside the band admit a Kerr interpretation, while points below its lower edge have intercepts smaller than any Kerr model in the reference band can produce at the same slope. The latter identify departures from Kerr without requiring prior knowledge of the orbital angular velocity. Increasing ω/ωK shifts the colored regions toward larger k and smaller b. The dependence of the slope k on source motion is expected, whereas the dependence of the intercept b on ω can be understood from the different propagation times of the image orders. At a fixed observing time, the higher-order image corresponds to an earlier emission time than the direct image. Changing the orbital angular velocity therefore changes the orbital-phase separation between the source positions represented by these images, and hence their angular separation and the ridge intercept. The enlarged right panel also illustrates how the ridge observables can be used to constrain spin and deformation together. At ω/ωK = 1, the curves of constant a and constant ϵ3 show that increasing the spin mainly raises the intercept, whereas changing the deformation also shifts the slope appreciably. These different responses allow measurements of k and b to jointly constrain a and ϵ3. An independent measurement of ω can likewise supply the orbital information required for this analysis, with the corresponding ratio ω/ωK determined by each spacetime.

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The autocorrelation in black hole movies encodes time delays and angular separations between gravitationally lensed images, offering a means of testing the spacetime geometry. This paper investigates this question using the Johannsen-Psaltis spacetime [31] with a single nonzero deformation parameter ϵ3 as a test case. The researchers generate movies of a compact hot spot on an equatorial circular orbit and characterize the lensing-induced secondary ridge by its slope and intercept in the time-angle autocorrelation map. The results show that the spin and metric deformation parameter affect the ridge observables differently. With the orbital radius, inclination, and ω/ωK specified, the different responses of the slope and intercept allow joint constraints on the spin and metric deformation parameter. For rs = 8M and θo = 17◦, we constructed a Kerr reference band spanning 0 ≤ a ≤ 1 and 0.5 ≤ ω/ωK ≤ 1.5. Low-spin configurations with negative ϵ3 extend below its lower boundary, with intercepts smaller than the Kerr minimum at the same slope. Their autocorrelation signatures cannot be reproduced by changing the Kerr spin and orbital angular velocity <ref:

1. Improve black hole parameter estimation by developing a joint constraint system for spin and metric deformation using the slope (k) and intercept (b) of an approximately linear secondary ridge in the time-angle autocorrelation map. This allows AI systems to jointly constrain both parameters when the orbital radius, observer inclination, and orbital angular velocity are specified.

2. Develop a non-Kerr signature detection module that distinguishes between metric deformation and orbital variations by identifying regions where secondary peaks lie outside this reference region defined by varying spin and angular velocity. This enables the AI to identify departures from Kerr without requiring prior knowledge of the orbital angular velocity.

3. Create an automated diagnostic tool for strong-field gravity testing by comparing measured observables against a Kerr reference band constructed across varying spin and orbital angular velocity ratios (a = 0 to a = 1 and 0.5 ≤ ω/ωK ≤ 1.5). This system can classify observed (k, b) pairs as either consistent with Kerr geometry or indicative of a departure from it.

4. Implement a sub-Keplerian and super-Keplerian motion classifier that utilizes the same secondary ridge observables to distinguish non-Kerr signatures regardless of the source's orbital speed (sub-Keplerian, Keplerian, and super-Keplerian motion). This expands the applicability of the test beyond standard Keplerian assumptions.

Autocorrelation in black hole movies encodes the time delays and angular separations between gravitationally lensed images, offering a means of testing the spacetime geometry. Using an orbiting hot spot in the parametrized Johannsen-Psaltis metric with a single deformation parameter, we investigate the ability of lensing-induced correlations to distinguish departures from Kerr. From ray-traced movies, we extract the slope (k) and intercept (b) of an approximately linear secondary ridge in the time-angle autocorrelation map as potential future observables. Variations in the spin and the metric deformation parameter produce distinctive trajectories in the (k,b) plane, allowing the two observables to jointly constrain both parameters when the orbital radius, observer inclination, and orbital angular velocity are specified. By varying the orbital angular velocity, we construct a Kerr reference region in the (k,b) plane, and we find that, for some low-spin configurations with negative deformation parameters, the measured slope-intercept pairs of the secondary peaks lie outside this reference region. This feature appears for sub-Keplerian, Keplerian, and super-Keplerian motion, indicating that secondary-ridge measurements from black hole flare movies have the potential to distinguish non-Kerr signatures.