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.

• The paper analytically derives a novel four-dimensional AdS black string solution incorporating perfect fluid dark matter via a logarithmic deformation.
• It rigorously examines causal, geometric, and thermodynamic properties, including horizon structure, curvature singularities, and the role of the Lambert W function.
• The study reveals that exotic PFDM regimes (α > 0) induce phase transitions and local instability, while conventional regimes (α < 0) ensure consistent thermodynamic stability.

Exact AdS Black String Solutions Immersed in Perfect Fluid Dark Matter

Introduction

The paper "Black string immersed in perfect fluid dark matter" (2605.26198) constructs an exact four-dimensional black string solution within anti-de Sitter (AdS) spacetime incorporating perfect fluid dark matter (PFDM). This study fills a gap in the literature by analytically deriving the metric deformation introduced by PFDM via a logarithmic term and thoroughly analyzing the causal, geometric, and thermodynamic features of such a system. The investigation leverages the structure of cylindrically symmetric AdS black strings as introduced by Lemos and extends them through an anisotropic PFDM source, revealing novel phase behavior and stability regimes.

Black String Solution in PFDM–AdS Geometry

The authors solve the Einstein equations for a static cylindrically symmetric spacetime with a negative cosmological constant and a PFDM energy-momentum tensor. The resultant metric function is

$$f(r) = \frac{r^2}{\ell^2} - \frac{2M}{r} + \frac{\alpha}{r} \ln\left(\frac{r}{|\alpha|}\right)$$

with $\ell$ the AdS length scale, $M$ the mass parameter, and $\alpha$ an integration constant characterizing the PFDM density. The equation of state is specified as $p = \rho/2$, where the sign of $\alpha$ controls violation or satisfaction of the weak energy condition. For $\alpha > 0$, the density and pressure are negative, representing exotic matter; for $\alpha < 0$, they are positive and conventional.

Horizon Structure and Curvature Singularities

The horizon radius $r_+$ is obtained by solving the condition $f(r_+) = 0$, leading to a transcendental equation involving logarithmic terms. The roots are expressed analytically using the Lambert $\ell$0 function, admitting distinct branches for different $\ell$1 signs. The curvature properties are examined through the Kretschmann scalar, confirming a central curvature singularity at $\ell$2, hidden within the horizon for admissible parameters.

Thermodynamic Properties and Phase Structure

The Hawking temperature associated with the horizon is given by

$\ell$3

which explicitly depends on the PFDM parameter $\ell$4 and the horizon radius $\ell$5. The heat capacity,

$\ell$6

exhibits a divergence at $\ell$7 for $\ell$8, marking a thermodynamic phase transition associated with a change in local stability.

Figure 1: Hawking temperature $\ell$9 versus event horizon radius $M$0 for selected values of $M$1.

The phase transition is absent in the $M$2 case, as the heat capacity remains strictly positive for all admissible horizon radii. This leads to an extremal configuration at a minimum radius where the temperature vanishes, with all physical horizons above this point being locally stable.

Figure 2: Hawking temperature $M$3 versus event horizon radius $M$4 for selected values of $M$5.

Figure 3: Heat capacity $M$6 versus event horizon radius $M$7 for selected values of $M$8.

Figure 4: Heat capacity $M$9 versus event horizon radius $\alpha$0 for selected values of $\alpha$1.

These thermodynamic and phase characteristics are tightly linked to the PFDM-induced modifications to the geometry and energy conditions.

Implications and Prospects

The analytic results demonstrate that PFDM leads to qualitatively distinct black string behavior in AdS backgrounds, including phase transitions for exotic matter regimes and robust stability for conventional matter. The use of the Lambert $\alpha$2 function in horizon determination extends applicability to other PFDM-coupled systems, as recently explored in black hole contexts (Hamil et al., 2024, Hamil et al., 2024, Xu et al., 2016), and supports further study of horizon branching in modified gravity settings.

The violation of the weak energy condition in the $\alpha$3 sector, and its link to phase transitions, raises questions regarding self-consistent embeddings of PFDM and its macroscopic thermodynamic effects. The results have relevance for AdS/CFT scenarios where horizon branching and thermal instabilities may influence dual field theory features, and for analogue gravity models where the causal and thermodynamic structure is closely tied to matter content.

Potential future directions include the extension to charged and rotating black strings, the incorporation of PFDM in higher-dimensional and scalar-tensor models, and the probing of dynamical stability and perturbative spectra. The analytic tractability of the solution is conducive to generalization to $\alpha$4 modified gravity and other exotic backgrounds (Santos et al., 24 Feb 2026), as well as to more complex matter couplings.

Conclusion

This paper provides a comprehensive analytic framework for PFDM-immersed AdS black strings, delineating the role of dark matter parameters in controlling horizon structure, singularity formation, and thermodynamic phase behavior. The existence of phase transitions and stability regimes intimately tied to energy condition violations highlights the rich phenomenology arising from PFDM sources in extended objects and motivates further investigation into their astrophysical and holographic implications.