This paper compares two exact rotating black‑hole backgrounds used to model superconductors in gauge/gravity duality. One background is a four‑dimensional asymptotically anti‑de Sitter (AdS) solution dressed by a non‑minimally coupled scalar and one linear axion. The other is a five‑dimensional Lifshitz solution with dynamical exponent z between 2 and 3, supported by a dilaton, two gauge fields, two linear axions and Chern–Simons terms. In both cases the authors study the same probe Abelian‑Higgs sector, which models the superconducting condensate without changing the background geometry (the probe limit).

The authors rewrite the two geometries in a single “stationary” form that makes their common structure clear. In this form time is mixed with one spatial direction by a shift function N^x(r). That mixing forces the temporal and spatial parts of the probe gauge field to appear together. They derive a single master scalar equation in which rotation only enters through the co‑rotating potential A_t − N^x A_x. This combination is the gauge potential measured in the local rotating frame and equals the contraction of the gauge field with the horizon generator at the horizon.

That structural result explains a technical point seen in earlier numerical work: the temporal and spatial gauge components cannot be treated independently. Using the same boundary conditions and canonical ensemble as the earlier studies, the numerical profiles reported previously show a common trend. Increasing the parameter associated with rotation (called α in the AdS case and J in the Lifshitz case) reduces the condensate amplitude in both backgrounds. In the Lifshitz family, raising the dynamical exponent z at fixed J tends to enhance the condensate instead.

Important caveats are emphasized. The parameters α and J do more than produce frame dragging: they also change the matter fields that support the exact black‑hole solutions. That means the observed trends reflect whole families of correlated backgrounds, not the isolated effect of mechanical rotation alone. The analysis is also done in the probe limit, so it does not include backreaction of the condensate on the geometry. The Lifshitz solution studied is restricted to z in (2,3), and the authors keep the original choices for scalar boundary conditions (changing them would define a different numerical problem).