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.
as written by the authors arXiv:2604.23677
We study the small-$$c$$ expansion of general relativity in ADM variables up to next-to-next-to-leading order (NNLO). We show that, in the stationary sector, this formulation renders the field equations more tractable for explicit solution building. The stationary sector exhibits both strong-gravity and weak-field branches, whose structure becomes richer at NNLO. In the strong-gravity branch, we first obtain exact vacuum solutions of NLO Carroll gravity, including the Lense--Thirring and rotating C-metric backgrounds. At NNLO, we then construct the corresponding Lense--Thirring-type and C-metric-type exact vacuum geometries. These solutions also arise from the small-$$c$$ expansion of the Kerr and rotating C-metric geometries around the strong-gravity background, up to $$\mathcal{O}(J)$$ at NLO and up to $$\mathcal{O}(J^3)$$ at NNLO. In the weak-field branch, we find exact Hartle--Thorne-type solutions with an independent quadrupole moment, together with exact spin-squared corrections and a mixed quadrupolar-rotating solution. We further extend the $$\ell=0,2$$ sector by including higher multipoles up to $$\ell=4$$, where $$\ell$$ denotes the multipole index. These results show that the full NLO/NNLO theory admits a richer stationary vacuum sector than the magnetic Carroll truncation. More broadly, the ADM formulation provides a practical framework for constructing and analyzing stationary backgrounds in the small-$$c$$ expansion of general relativity, and may also offer a useful framework for organizing rotational and higher-multipole deformations motivated by relativistic compact-object geometries.
The Lense–Thirring effect, predicted by general relativity, was confirmed with satellites: Earth's rotation indeed slightly twists space around it, dragging satellite orbits along.
In Ted Chiang's novella 'Exhalation', the slowing of light is used as a metaphor for changing perception of time, echoing the idea of expanding equations at low speed of light.