Revisiting Lifshitz-type solutions in $R^2$-corrected gravity

Abstract

In this study, we construct exact higher-dimensional Lifshitz-type solutions in $R^2$-corrected gravity at the critical point of the theory, where the field equations become degenerate because of the vanishing of the effective gravitational coupling. The analysis is performed on product manifolds of the form $Li_{m}\times Ω_{(n-m)}$, where $Li_{m}$ denotes an $m$-dimensional Lifshitz-type spacetime exhibiting anisotropic scaling with dynamical exponent $z$ and $Ω_{(n-m)}$ represents an $(n-m)$-dimensional space of constant curvature. This geometric decomposition allows for a unified treatment of static, stationary (rotating), and hyperscaling-violating configurations within a purely gravitational framework. We show that the theory admits new broad families of exact Lifshitz black hole and black brane solutions, including extremal configurations, whose scaling exponents and horizon structures are constrained by higher-curvature terms. The stationary solutions are interpreted as rotating Lifshitz-type black holes with well-defined Killing horizons within the appropriate parameter ranges. Owing to the critical nature of the theory, these solutions exhibit vanishing entropy and zero conserved charges despite having non-zero temperature, reflecting the degenerate structure of the field equations. Our results extend the previously known Lifshitz constructions in Einstein and higher-derivative gravity and provide a systematic higher-dimensional framework for exploring anisotropic and hyperscaling-violating geometries supported by curvature-squared interactions.