Stationary Stars Are Axisymmetric in Higher Curvature Gravity

TL;DR

This work proves that axisymmetry for stationary rotating stars, a hallmark of general relativity, remains valid in a broad class of diffeomorphism-invariant, higher-curvature gravity theories, including Lovelock gravity. The authors show that thermodynamic equilibrium inside the star yields a timelike Killing vector $\xi^a$, which extends uniquely to the exterior by solving $\Box\xi^a = -R^a{}_b\xi^b$, ensuring exterior axisymmetry. They provide a concrete Lovelock-theory argument via the deformation tensor $t_{ab}$ and Holmgren’s uniqueness, and extend the logic to generic higher-derivative theories, establishing two commuting Killing vectors and axisymmetry under asymptotic flatness. The result highlights a universal rigidity of stationary stellar configurations across broad gravitational theories and offers observational implications for gravitational waves and high-resolution imaging, while outlining directions for extensions to non-flat backgrounds.

Abstract

The final equilibrium stage of stellar evolution can result in either a black hole or a compact object such as a white dwarf or neutron star. In general relativity, both stationary black holes and stationary stellar configurations are known to be axisymmetric, and black hole rigidity has been extended to several higher curvature modifications of gravity. In contrast, no comparable result had previously been established for stationary stars beyond general relativity. In this work we extend the stellar axisymmetry theorem to a broad class of diffeomorphism invariant metric theories. Assuming asymptotic flatness and standard smoothness requirements, we show that the Killing symmetry implied by thermodynamic equilibrium inside the star uniquely extends to the exterior region, thereby enforcing rotational invariance. This demonstrates that axisymmetry of stationary stellar configurations is not a feature peculiar to Einstein gravity but a universal property of generally covariant gravitational theories, persisting even in the presence of higher curvature corrections.