Lower-dimensional Gauss-Bonnet gravity black holes with quintessence

In this paper, we study the $D\to3$ limit of Gauss-Bonnet gravity with quintessential matter, obtaining exact solutions that extend the BTZ metric through higher-curvature terms and quintessence coupling. The solutions exhibit a single event horizon whose radius decreases with increasing quintessence parameter $ω_q$, while developing a curvature singularity at the origin for non-vanishing quintessence. The geodesic analysis reveals stable circular photon orbits exist exclusively for phantom-like quintessence ($ω_q < -1$). Thermodynamically, the system is stable, since the specific heat is positive, and with evaporation it evolves to stable remnants whose characteristic size decreases as $ω_q$ increases, with complete evaporation prevented by quintessence effects. Furthermore, we find that all physical quantities intrinsically depend on the parameter $α$ of the Gauss-Bonnet extension.These results demonstrate the profound influence of quintessential matter on both geometric and thermodynamic properties of (2+1)-dimensional black holes, offering new perspectives on gravitational theories in lower dimensions and black hole final states.

💡 Research Summary

The paper investigates black‑hole solutions in a three‑dimensional (2+1) spacetime obtained by taking the D→3 limit of Gauss‑Bonnet (GB) gravity and coupling it to a quintessential fluid. By employing the dimensional‑regularisation trick (rescaling the GB coupling as (D‑4)α→α) the authors retain a non‑trivial GB term in three dimensions. A scalar field ϕ is introduced with the logarithmic profile ϕ=ln(r/ℓ), which satisfies the scalar field equation and ensures the consistency of the modified field equations.

The energy‑momentum tensor of the quintessence fluid follows the usual equation of state p=ω_q ρ, leading to T^t_t=T^r_r=−ω_q κ r^{-2(ω_q+1)} and T^φ_φ=−(2ω_q+1) ω_q κ r^{-2(ω_q+1)}. Solving the coupled Einstein‑GB‑scalar system yields two branches for the metric function f(r). The physically relevant “‑” branch reduces to the standard BTZ solution when the quintessence charge q≡|κ ω_q| vanishes. For generic q the solution reads

f(r)=−r²/(2α)