Black string immersed in perfect fluid dark matter

Abstract

We present an exact four-dimensional black string solution immersed in perfect fluid dark matter within an anti-de Sitter background. By solving the Einstein field equations for an anisotropic fluid, we obtain a metric function that modifies the standard black string geometry through a logarithmic term governed by the dark matter parameter $α$. The event horizon radii are analytically determined using the Lambert $W$ function, and the Kretschmann scalar confirms a genuine curvature singularity at the origin alongside the expected asymptotic behavior. Furthermore, we evaluate the thermodynamic properties of the solution. The heat capacity diverges at a critical horizon radius for $α>0$, a behavior commonly associated with a thermodynamic phase transition in a regime where the weak energy condition is violated.