Black Strings and String Clouds Embedded in Anisotropic Quintessence: Solutions for Scalar Particles and Implications

TL;DR

The paper constructs a cylindrically symmetric AdS spacetime describing a black string enshrouded by a cloud of strings and a quintessence fluid, deriving the exact metric function $A(\rho)$ and analyzing the existence and scaling of event horizons. The metric takes the form $A(\rho)=\overline{a}+\rho^{2}/l^{2}-\rho_{S}/\rho+N_{Q}\rho^{2\alpha_{Q}}$, with parameters $\overline{a}=8\pi G a/c^{4}$, $\rho_{S}=2Gm/c^{2}$, and $N_{Q}$ encoding quintessence strength; the quintessence energy density and anisotropic pressures satisfy $\rho_{Q}=(2\alpha_{Q}+1)\bar{N}_{Q}\rho^{2\alpha_{Q}-2}$ and $p_{\varphi}=p_{z}=-\alpha_{Q}\rho_{Q}$. The horizon structure depends on $(\overline{a},\rho_{S},N_{Q},\alpha_{Q})$, with explicit forms for the lower, middle, and upper $\alpha_{Q}$ limits, and the presence of quintessence is shown to be most influential at cosmological scales or for small $\alpha_{Q}$. The Klein–Gordon equation for a spin-0 particle in this background is reduced to a radial equation solvable near the horizon via confluent Heun functions, revealing a quintessence-induced phase—termed the dark phase—in the near-horizon wavefunction. Overall, the work highlights how dark energy candidates modify both classical geometry and quantum behavior in a cylindrically symmetric AdS setting, with implications for horizon physics and observable phase-like effects.

Abstract

We analyze the spacetime metric associated with a black string surrounded by a cloud of strings and an anisotropic fluid of quintessence in cylindrically symmetric AdS spacetime. We solve Einstein's equation to obtain the explicit form of the metric, investigate typical values for its parameters, and determine their role in the event horizon formation. Within our findings, we show that the intensity of the cloud of strings regulates the size of the event horizon and, when the cloud is absent, the horizon increases drastically for larger values of the quintessence's state parameter $α_{Q}$. Additionally, the metric shows that, unless $α_{Q}$ is close to its lower bound, the contribution from the quintessence fluid is only significant at large distances from the black string. Finally, to explore the quantum implications of this dark energy candidate, we use the confluent Heun function to solve the Klein-Gordon equation for a spin-0 particle near the event horizon. Our results indicate that the presence of quintessence alters the particle's radial wave function. This modification, in principle, could give rise to an observable that we termed as \enquote{dark phase}.

Figures (8)

• Figure 1: Logarithmic plot of the quintessence parameter $N_{\scaleto{Q}{4pt}}$ as a function of $\alpha_{\scaleto{Q}{4pt}}$. According to \ref{['Estimate-N_Q']}, there is an exponential decrease in $N_{\scaleto{Q}{4pt}}$ when the state parameter $\alpha_{\scaleto{Q}{4pt}}$ increases. The vertical axis values are calculated assuming an observable universe with a radius of $\rho_{\scaleto{obs.}{4pt}} = 3.8 \times 10^{26}$ m, ensuring that this cylindrical model has the same observable volume as our actual universe.
• Figure 2: The contribution of the quintessence term, $N_{\scaleto{Q}{4pt}}\,{\rho}^{2\alpha_{\scaleto{Q}{4pt}}}$, to the spacetime metric function $A(\rho)$ was analyzed. Using an observable universe radius of $l = \rho_{\scaleto{obs.}{4pt}} = 3.8 \times 10^{26}$ m, we found that the quintessence term's significance is most pronounced for small values of the state parameter $\alpha_{\scaleto{Q}{4pt}}$, approaching zero. Conversely, as $\alpha_{\scaleto{Q}{4pt}}$ increases towards $\alpha_{\scaleto{Q}{4pt}} = 1$, the term becomes relevant only at very large radii, comparable to the radius of the observable universe $\rho_{\scaleto{S}{4pt}}$.
• Figure 3: Plot of $A(\rho)$ for different values of $\overline{a}$. In this figure, $\alpha_{\scaleto{Q}{4pt}} = 0$, $\rho_{\scaleto{S}{4pt}} = 10^{2}$ m, $l = \rho_{\scaleto{obs.}{4pt}} = 3.8 \times 10^{26}$ m, and $N_{\scaleto{Q}{4pt}} = 8.95$.
• Figure 4: Plot of $A(\rho)$ for varying $\rho_{\scaleto{S}{4pt}}$, with $\alpha_{\scaleto{Q}{4pt}} = 0$, $\overline{a} = 10$, $l = \rho_{\scaleto{obs.}{4pt}} = 3.8 \times 10^{26}$ m, and $N_{\scaleto{Q}{4pt}} = 8.95$.
• Figure 5: $A(\rho)$ for different values of the quintessence fraction $\Omega_{\scaleto{Q}{4pt}}$. The other parameters were set as $\overline{a} = 1$, $\rho_{\scaleto{S}{4pt}} = 10$ m, $l = \rho_{\scaleto{obs.}{4pt}} = 3.8 \times 10^{26}$ m. In the left image, we have the case of $\alpha_{\scaleto{Q}{4pt}} = 0$, while on the right $\alpha_{\scaleto{Q}{4pt}} = 0.02$. Note that any small increase in the value of $\alpha_{\scaleto{Q}{4pt}}$ causes the lines on the graph of $A(\rho)$ to become much closer together, reducing the role played by the quintessence fraction.