Opening
General relativity remains our best description of gravity, yet solving its field equations exactly is notoriously difficult. A new approach—the small-$c$ expansion, where $c$ is the speed of light—offers a systematic way to construct exact solutions in a limit where gravity becomes more tractable. Researchers have now shown that this expansion framework, formulated in ADM variables, yields richer and more explicit stationary solutions than previously known, potentially offering new tools for understanding rotating compact objects like black holes and neutron stars.
What they found
The authors studied the small-$c$ expansion of general relativity up to next-to-next-to-leading order (NNLO)—essentially the first three terms in a systematic expansion. Their key finding is that the stationary sector of this theory exhibits two distinct branches: a strong-gravity branch and a weak-gravity branch, each with increasingly complex structure at higher orders.
In the strong-gravity branch, they obtained exact vacuum solutions at next-to-leading order (NLO) in Carroll gravity, including the Lense–Thirring and rotating C-metric backgrounds. Extending to NNLO, they constructed corresponding Lense–Thirring-type and C-metric-type exact geometries. Notably, these solutions also emerge 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 $\mathcal{O}(J^3)$ at NNLO, where $J$ denotes angular momentum.
The weak-gravity branch proved equally productive. The team found exact Hartle–Thorne-type solutions featuring an independent quadrupole moment, along with exact spin-squared corrections and a mixed quadrupolar-rotating solution. By extending their analysis to include multipole moments up to $\ell=4$ (compared to the $\ell=0,2$ sector typically studied), they demonstrated that the full NLO/NNLO theory admits significantly richer stationary vacuum solutions than the magnetic Carroll truncation previously considered.
Why it matters
These results establish the ADM formulation as a practical framework for constructing and analyzing stationary backgrounds in the small-$c$ expansion. For researchers studying compact astrophysical objects, this matters because the solutions naturally accommodate rotational and higher-multipole deformations—precisely the features needed to model slowly rotating black holes and neutron stars. The systematic organization of these deformations could improve theoretical predictions for gravitational wave signals and electromagnetic observations from such objects.
What's next
The authors suggest their framework may prove useful for organizing rotational and multipole effects relevant to compact objects, though they do not detail specific observational predictions or follow-up calculations. Open questions include whether these exact solutions can be matched to realistic matter distributions and whether the framework extends to time-dependent scenarios relevant to mergers and transients.
Starithm continuously monitors real-time gravitational wave and electromagnetic alerts from compact object mergers and other relativistic events, making these theoretical advances directly relevant to interpreting observational data as it arrives.