Light Ring Stability Across Spacetime Dimensions and Theories of Gravity

Abstract

The light ring theorem states that four-dimensional stationary, axisymmetric, horizonless ultracompact objects (UCOs) possess light rings (LRs) in pairs, one of which is stable if the Null Energy Condition (NEC) holds in General Relativity (GR). The existence of a stable LR may trigger nonlinear spacetime instabilities, thereby raising questions about the viability of UCOs as long-lived black hole mimickers. We extend the LR existence and stability theorem beyond four-dimensional GR to modified gravity theories, in particular, $f(R)$ gravity, scalar-tensor theories, and restricted Horndeski gravity, in ($d \geq 4$) dimensional spacetimes. Our findings show that in four-dimensional modified gravities, the stability of LRs is governed by theory-specific coupling parameters in addition to the NEC. Consequently, a stable LR is not guaranteed to occur in certain parameter regimes. In higher dimensions ($d>4$), we confirm using topological arguments that LRs still emerge in pairs; however, the topological charges are now associated with four distinct types of LRs, including two types of saddle points. The stability analysis further constrains the allowed LR pairings, implying that, unlike in four-dimensional GR, the emergent pair does not necessarily include a stable LR. These findings directly impact tests of GR using UCOs, demonstrating that LR stability is highly sensitive to both the underlying gravitational theory and spacetime dimensionality.