Black strings and BTZ black holes sourced by a Dekel-Zhao dark matter profile

TL;DR

The work interrogates how a Dekel-Zhao dark matter profile sourced in both (3+1) and (2+1) dimensions reshapes black string and BTZ spacetimes. By solving Einstein equations with a DZ density (using $b=2$, $g=7/2$) they obtain analytic $f(r)$ terms involving ${}_2F_1$ functions, revealing horizon behavior controlled by the inner slope $a$; horizons exist for $0<a<3$ in the black string while $a\ge2$ can erase horizons in the BTZ case, yielding naked singularities. Curvature invariants $R(r)$ and $K(r)$ develop DM-enhanced singularities for $a>0$, and the energy conditions show NEC/WEC/SEC hold but DEC is violated in the BTZ setup due to large tangential stresses. Thermodynamically, DM shifts $T_H$, entropy, and free energy but preserves local stability (positive heat capacity) and global stability (negative free energy), while transforming constant-curvature BTZ into a singular geometry, underscoring DM’s capacity to alter causal structure and cosmic censorship considerations in non-standard black hole spacetimes.

Abstract

In this work, we obtain analytical solutions for a $(3+1)$-dimensional black string and a BTZ black hole, both sourced by the Dekel-Zhao dark matter (DM) density profile. Our results indicate that the event horizon radius is sensitive to the inner slope parameter $a$; specifically, beyond a critical threshold, the horizon vanishes, leading to the formation of naked singularities. We find that the DM environment induces curvature singularities in the Ricci and Kretschmann scalars, which are absent in the vacuum BTZ case. Furthermore, an analysis of the effective energy-momentum tensor shows that while the null, weak, and strong energy conditions are strictly satisfied, the dominant energy condition is violated in the BTZ scenario due to the high tangential pressure gradient. We also observe that DM modifies the Hawking temperature and free energy without compromising local or global stability. Notably, the DM distribution transforms the originally constant-curvature BTZ spacetime into a singular one, suggesting that a inherent stiffness of the DM profile is a determinant factor in the causal structure of these solutions.

Figures (20)

• Figure 1: DZ dark matter density profile for some values of $a$. We fixed $\mu=1$, $\ell=1/2$, $\rho_{ch}=6/5$, $g=7/2$, $b=2$, and $\kappa=8\pi$.
• Figure 2: Lateral pressure $p_L$ for several values of $a$. We fixed $\mu=1$, $\ell=1/2$, $\rho_{ch}=6/5$, $g=7/2$, $b=2$, and $\kappa=8\pi$.
• Figure 3: (Left) Metric function $f(r)$ in the presence of DZ dark matter profile for $a<3$. (Right) Metric function for $a>3$ indicating no horizon at all, which can indicate naked singularities. We fixed $\mu=1$, $\ell=1/2$, $\rho_{ch}=6/5$, $g=7/2$, $b=2$, and $\kappa=8\pi$.
• Figure 4: NEC for the parameters $\mu=1$, $\ell=1/2$, $\rho_{ch}=6/5$, $g=7/2$, $b=2$, $\kappa=8\pi$. We use $a =1, 1.5,2, 2.5, \text{and } 3$. The case $\rho + p_r =0$ (trivially null).
• Figure 5: Continuous family of curves confirming DEC, $\rho - |p_L| \ge 0$, as a function of the radial coordinate $r$, for the parameter range $0 \le a \le 4$. We fixed $\mu=1$, $\ell=1/2$, $\rho_{ch}=6/5$, $g=7/2$, and $b=2$.