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

TL;DR

This work studies higher-dimensional Lifshitz-type spacetimes within $R^2$-corrected gravity by exploiting a differential-forms formalism to derive the field equations of $f(R)$ gravity with a torsion-free constraint. For the particular form $f(R) = 1/2 (α R^2+R) + Λ$, Lifshitz configurations exist when $α = 1/(8Λ)$ and $R = -4Λ$, yielding a degenerate, perfect-square action that supports static and stationary solutions with product geometries $Li_m × Ω_{(n-m)}$ and hyperscaling-violating variants for arbitrary Lifshitz exponent $z$. The paper provides explicit metric ansatzes, exponents, and horizon structures, and shows that these Lifshitz black holes generally have zero entropy (per Wald's formula) but nonzero temperature, with several extremal configurations identified. These results broaden the exact solution space in quadratic gravity and offer new backgrounds for non-relativistic holography, while suggesting extensions to other quadratic gravity models and richer thermodynamic analyses.

Abstract

In this work, we investigate higher-dimensional Lifshitz-type topological static and stationary solutions of $R^2$-corrected gravity theory using the language of differential forms. We obtain new product manifold solutions of the form $Li_{m}\times Ω_{(n-m)}$ , where $Li_m$ represents an $m$-dimensional Lifshitz type submanifold and $Ω_{(n-m)}$ denotes $(n-m)$-dimensional compact constant curvature manifold. In addition, we present hyperscaling Lifshitz solutions for arbitrary Lifshitz parameter $ z $. We also discuss some thermodynamical properties of both static and stationary solutions, including extremal cases.