Leo C. Stein

Chaos and fractals of the black hole photon ring

2026-03-10T00:00:00+00:00

The photon ring of a Kerr black hole decomposes into a self-similar hierarchy of subrings. Here, we show that this self-similar structure persists in phase space. Moreover, near the photon shell of bound photon orbits, dynamics are controlled by a Lyapunov exponent γ, whose role we highlight by computing the first-return map for light rays close to an unstably bound orbit. Despite an exponential $e^\gamma$ sensitivity to initial conditions, nearly bound rays do not exhibit chaotic behavior. However, as the background spacetime is deformed away from the Kerr geometry, chaos sets in, with its first onset most visible near strongly resonant bound orbits in the photon shell. We display two animations: one illustrating the emergence of chaos near the photon shell, which results in a fractal phase-space structure, and another exhibiting how this chaotic, fractal, self-similar structure is encoded in the first-return map.

Parameter matching between horizon quasi-local and point-particle definitions at 1PN for quasi-circular and non spinning BBH systems in harmonic gauge

2025-10-29T00:00:00+00:00

We investigate how commonly used parameter definitions in Post-Newtonian (PN) theory compare with those from Numerical Relativity (NR) for binary black hole (BBH) systems. In NR, masses and spins of each companion are measured quasi-locally from apparent horizon geometry, whereas in PN they are attributes of point particles defined via asymptotic matching in body zones. Although these definitions coincide in the infinite-separation limit, they could differ by finite-separation corrections that matter for precision modeling. Working entirely in harmonic gauge, we perform asymptotic matching between each companion’s inner zone metric – obtained from black hole perturbation theory – and the PN two-body metric, and construct the coordinate transformation that preserves the gauge in the strong field region. We solve perturbatively for the apparent horizon (AH) on a group of harmonic inertial time slice and compute its quasi-local areal mass from the horizon geometry. Then we establish the leading order matching between quasi-local (AH based) and PN (point-particle) parameter definitions in harmonic gauge. We find that on a horizon penetrating harmonic slicing, the AH quasi-local mass agrees with the PN point-particle mass at 1PN order. For generic harmonic slicings that deviate from the horizon penetrating condition by a 1PN order perturbation, the AH mass differs from the PN mass also by a 1PN correction. This parameter matching is crucial for hybridizing PN and NR waveforms and for providing better initial conditions in NR and Cauchy-Characteristic Evolution (CCE) simulations. The framework provides a bridge between different descriptions of BBH systems, and it can be extended to spinning and eccentric cases and more general NR gauges.

Black hole spectroscopy: from theory to experiment

2025-06-02T00:00:00+00:00

The “ringdown” radiation emitted by oscillating black holes has great scientific potential. By carefully predicting the frequencies and amplitudes of black hole quasinormal modes and comparing them with gravitational-wave data from compact binary mergers we can advance our understanding of the two-body problem in general relativity, verify the predictions of the theory in the regime of strong and dynamical gravitational fields, and search for physics beyond the Standard Model or new gravitational degrees of freedom. We summarize the state of the art in our understanding of black hole quasinormal modes in general relativity and modified gravity, their excitation, and the modeling of ringdown waveforms. We also review the status of LIGO-Virgo-KAGRA ringdown observations, data analysis techniques, and the bright prospects of the field in the era of LISA and next-generation ground-based gravitational-wave detectors.

The SXS Collaboration’s third catalog of binary black hole simulations

2025-05-19T00:00:00+00:00

We present a major update to the Simulating eXtreme Spacetimes (SXS) Collaboration’s catalog of binary black hole simulations. Using highly efficient spectral methods implemented in the Spectral Einstein Code SpEC, we have nearly doubled the total number of binary configurations from 2,018 to 3,756. The catalog now densely covers the parameter space with precessing simulations up to mass ratio $q=8$ and dimensionless spins up to $|\vec{\chi}|\le 0.8$ with near-zero eccentricity. The catalog also includes some simulations at higher mass ratios with moderate spin and more than 250 eccentric simulations. We have also deprecated and rerun some simulations from our previous catalog (e.g., simulations run with a much older version of SpEC or that had anomalously high errors in the waveform). The median waveform difference (which is similar to the mismatch) between resolutions over the simulations in the catalog is $4\times10^{-4}$ . The simulations have a median of 22 orbits, while the longest simulation has 148 orbits. We have corrected each waveform in the catalog to be in the binary’s center-of-mass frame and exhibit gravitational-wave memory. We estimate the total CPU cost of all simulations in the catalog to be 480,000,000 core-hours. We find that using spectral methods for binary black hole simulations is over 1,000 times more efficient than much shorter finite-difference simulations of comparable accuracy. The full catalog is publicly available through the sxs Python package and at https://data.black-holes.org.

GWSurrogate: A Python package for gravitational wave surrogate models

2025-03-29T00:00:00+00:00

Fast and accurate waveform models are fundamentally important to modern gravitational wave astrophysics, enabling the study of merging compact objects like black holes and neutron stars. However, generating high-fidelity gravitational waveforms through numerical relativity simulations is computationally intensive, often requiring days to months of computation time on supercomputers. Surrogate models provide a practical solution to dramatically accelerate waveform evaluations (typically tens of milliseconds per evaluation) while retaining the accuracy of computationally expensive simulations. The GWSurrogate Python package provides easy access to these gravitational wave surrogate models through a user-friendly interface. Currently, the package supports 16 surrogate models, each varying in duration, included physical effects (e.g., nonlinear memory, tidal forces, harmonic modes, eccentricity, mass ratio range, precession effects), and underlying solution methods (e.g., Effective One Body, numerical relativity, black hole perturbation theory). GWSurrogate models follow the waveform model conventions used by the LIGO-Virgo-Kagra collaboration, making the package immediately suitable for both theoretical studies and practical gravitational wave data analysis. By enabling rapid and precise waveform generation, GWSurrogate serves as a production-level tool for diverse applications, including parameter estimation, template bank generation, and tests of general relativity.

Modeling the BMS transformation induced by a binary black hole merger

2025-03-13T00:00:00+00:00

Understanding the characteristics of the remnant black hole formed in a binary black hole merger is crucial for conducting gravitational wave astronomy. Typically, models of remnant black holes provide information about their mass, spin, and kick velocity. However, other information related to the supertranslation symmetries of the BMS group, such as the memory effect, is also important for characterizing the final state of the system. In this work, we build a model of the BMS transformation that maps a binary black hole’s inspiral frame to the remnant black hole’s canonical rest frame. Training data for this model are created using high-precision numerical relativity simulations of quasi-circular systems with mass ratios $q \le 8$ and spins parallel to the orbital angular momentum with magnitudes $\chi_{1}, \chi_{2} \le 0.8$ . We use Gaussian Process Regression to model the BMS transformations over the three-dimensional parameter space $\left(q, \chi_{1}^{z}, \chi_{2}^{z}\right)$ . The physics captured by this model is strictly non-perturbative and cannot be obtained from post-Newtonian approximations alone, as it requires knowledge of the strong nonlinear effects that are sourced during the merger. Apart from providing the first model of the supertranslation induced by a binary black hole merger, we also find that the kick velocities predicted using Cauchy-characteristic evolution waveforms are, on average, $\sim5\%$ larger than the ones obtained from extrapolated waveforms. Our work has broad implications for improving gravitational wave models and studying the large-scale impact of memory, such as on the cosmological background. The fits produced in this work are available through the Python package surfinBH under the name NRSur3dq8BMSRemnant.

Named a Kavli Fellow

2025-03-08T00:00:00+00:00

I am honored to have been selected as one of this year’s Kavli Fellows! The National Academy of Sceinces names these fellows annually (since 1989). This year, eighty-eight early-career scientists were named Kavli Fellows, from across industry, academia, and government. We were invited to attend the 3-day symposium, the 2025 U.S. Kavli Frontiers of Science. This is the broadest-scope conference I’ve ever attended, and it was refreshing to hear so much science from outside my tiny niche. Thanks to the NAS and the Kavli Foundation for including me!

Length dependence of waveform mismatch: a caveat on waveform accuracy

2025-02-21T00:00:00+00:00

The Simulating eXtreme Spacetimes Collaboration’s code SpEC can now routinely simulate binary black hole mergers undergoing $\sim25$ orbits, with the longest simulations undergoing nearly $\sim180$ orbits. While this sounds impressive, the mismatch between the highest resolutions for this long simulation is $\mathcal{O}(10^{-1})$ . Meanwhile, the mismatch between resolutions for the more typical simulations tends to be $\mathcal{O}(10^{-4})$ , despite the resolutions being similar to the long simulations’. In this note, we explain why mismatch alone gives an incomplete picture of code—and waveform—quality, especially in the context of providing waveform templates for LISA and 3G detectors, which require templates with $\mathcal{O}(10^{3}) - \mathcal{O}(10^{5})$ orbits. We argue that to ready the GW community for the sensitivity of future detectors, numerical relativity groups must be aware of this caveat, and also run future simulations with at least three resolutions to properly assess waveform accuracy.

Late-time tails in nonlinear evolutions of merging black holes

2024-12-11T00:00:00+00:00

We uncover late-time gravitational-wave tails in fully nonlinear 3+1 dimensional numerical relativity simulations of merging black holes, using the highly accurate SpEC code. We achieve this result by exploiting the strong magnification of late-time tails due to binary eccentricity, recently observed in perturbative evolutions, and showcase here the tail presence in head-on configurations for several mass ratios close to unity. We validate the result through a large battery of numerical tests and detailed comparison with perturbative evolutions, which display striking agreement with full nonlinear ones. Our results offer yet another confirmation of the highly predictive power of black hole perturbation theory in the presence of a source, even when applied to nonlinear solutions. The late-time tail signal is much more prominent than anticipated until recently, and possibly within reach of gravitational-wave detectors measurements, unlocking observational investigations of an additional set of general relativistic predictions on the long-range gravitational dynamics.

Actions of spinning compact binaries: Spinning particle in Kerr matched to dynamics at 1.5 post-Newtonian order

2024-11-18T00:00:00+00:00

The motion of compact binaries is influenced by the spin of their components starting at the 1.5 post-Newtonian (PN) order. On the other hand, in the large mass ratio limit, the spin of the lighter object appears in the equations of motion at first order in the mass ratio, coinciding with the leading gravitational self-force. Frame and gauge choices make it challenging to compare between the two limits, especially for generic spin configurations. We derive novel closed formulas for the gauge-invariant actions and frequencies for the motion of spinning test particles near Kerr black holes. We use this to express the Hamiltonian perturbatively in terms of action variables up to 3PN and compare it with the 1.5 PN action-angle Hamiltonian at finite mass ratios. This allows us to match the actions across both systems, providing a new gauge-invariant dictionary for interpolation between the two limits.