Can the refractive index be negative in gravitational settings?

We propose an effective refractive index definition n eff and show its definition may turn into a coordinate dependent definition. As a naive example, we compute it within the Friedmann Robertson Walker (FRW) metric. Afterwards, we extend n eff to a covariant definition and, remarkably, we show that coupling general relativity with nonlinear electrodynamics (NLED), in analogy to what may happen for regular black holes, permits to realize negative refraction. We clarify that a negative value obtained from a metric coefficient alone is not eough, by itself, since it is coordinate dependent. We conclude that one expects an effective metric, describing general relativity coupled with NLED, to account for negative refraction, with precise coordinate-independent criteria. Hence, in the field of general relativity only, i.e., with no couplings with further fields, dealing with negative refraction may be speculative. Last but not least, we also discuss the distinction between negative refraction and negative mass in general relativity and, finally, we outline how our refractive index construction can be generalized beyond spherical symmetry toward stationary axisymmetric configurations.