A pair of black holes circles in a dance, gradually drawing closer. They lose energy and spin — a process known as angular momentum radiation. In standard Einstein theory, only gravitational waves carry away angular momentum. But what if nature has additional fields?

The authors of the study took a more complex theory — Einstein–Maxwell–dilaton. In addition to the familiar gravitational field, there are also a scalar field (the dilaton) and an electromagnetic field. They calculated how these three channels together carry away angular momentum from a pair of non-spinning black holes moving not in circles but along elongated orbits. The calculation was done to first post-Newtonian order, like a correction to Newtonian physics for fast and massive objects.

The result doesn't hang in the air. It exactly reproduces known limits: when the orbit is nearly circular, the balance relation holds; if you remove the dilaton, you get pure general relativity; and if the fields are very weak, their contribution naturally fades away. Together with the previously calculated energy flux, these data give a complete picture of the losses — a foundation for future models of eccentric orbit evolution and precise calculation of the phases of gravitational waves.

So the dance of black holes losing spin through invisible channels becomes a little clearer. And when detectors catch a signal from a real pair, we'll be able to check — does a scalar field exist in nature, or is gravity alone?