Light zips faster than anything else in the Universe, and that's not just a fact, it's the foundation of general relativity. But what if we mentally slow light down to a snail's pace? That's exactly the kind of game with equations the authors of a new paper have played. They took Einstein's equations, which describe how massive bodies curve spacetime, and considered the case where the speed of light speed of light is very small. This doesn't mean light actually slows down: it's just mathematically convenient to expand the formulas in a series where the leading term corresponds to 'slow light.' This trick is called a small expansion in the speed of light, and it allows peering into corners of gravity usually hidden behind the complexity of the full equations.

In the stationary case, when the field doesn't change with time, the equations split into two branches: one describes strong gravity, the other weak gravity. In the strong-gravity branch, scientists constructed exact solutions corresponding to rotating objects. First, they obtained analogs of the frame-dragging effect (Lense–Thirring effect) and rotating C-metrics — solutions describing accelerating black holes. Then, adding higher-order corrections, they found more accurate rotating geometries, up to cubic corrections in angular momentum. In the weak-gravity branch, they managed to find Hartle–Thorne-type solutions with an independent quadrupole moment (i.e., the body's non-sphericity), as well as corrections quadratic in rotation and mixed quadrupole-rotation effects. Moreover, the authors included multipoles up to fourth order, describing ever finer details of the gravitational field's shape.

These results show that the full theory at NLO and NNLO orders (next-to-leading and next-to-next-to-leading order) yields a much richer set of stationary vacuum solutions than the simplified 'magnetic' version. In other words, trimming the equations to a minimum hides many interesting possibilities. The practical value is that the formulation in ADM variables (named after Arnowitt, Deser, and Misner) becomes a convenient tool for constructing and analyzing stationary backgrounds in the small-speed-of-light expansion. This could be useful for modeling rotational and multipole deformations expected in real compact objects like neutron stars or black holes.