New Formula Cracks the Code for Measuring a Rotating Black Hole's Shadow

A team of researchers has created a new mathematical formula to quickly and precisely calculate the shadow of a rotating black hole. The model, developed by Leo P. Werneck, Francisco J. Arbelaez, and their collaborators, avoids the need for slow, complex simulations while maintaining near-perfect accuracy. Their findings were published in Classical and Quantum Gravity.

The formula, called a 'surrogate' model, predicts the equivalent diameter of a black hole's shadow with sub-percent accuracy. It accounts for the black hole's spin and the observer's viewing angle, using a 15-parameter polynomial combined with an exponential correction. This approach simplifies calculations that previously required heavy computational power.

Experiments tested the model's precision on a training grid, showing a median error of just 0.0105 percent and a maximum error of 0.782 percent. Further checks on a denser, out-of-sample dataset confirmed its reliability, with a median error of 0.023 percent and a worst-case error of 1.64 percent. The team also introduced a normalised size observable, y(a∗,i), which divides the shadow's diameter by 6√3M to focus on intrinsic properties.

Beyond shadow size, the model predicts precession frequency and phase evolution—key factors in gravitational wave astronomy. These predictions help refine parameter estimates in studies of black hole mergers and other extreme cosmic events.

The surrogate model offers a fast, accurate alternative to numerical simulations for studying black hole shadows. Its ability to handle spin, inclination, and gravitational wave effects could streamline research in astrophysics. The findings are now available for use in numerical relativity and gravitational waveform analysis.