Extended Scalar Particle Solutions in Black String Spacetimes with Anisotropic Quintessence

Abstract

We present new solutions to the Klein-Gordon equation for a scalar particle in a black string spacetime immersed in an anisotropic quintessence fluid surrounded by a cloud of strings, extending the analysis presented in our previous work. These novel solutions are dependent on the quintessence state parameter, $α_{Q}$, and are now valid for a much larger domain of the radial coordinate. We investigate the cases when $α_{Q} = 0,\,1/2,\,1$, encompassing both black hole and horizonless scenarios. We express the resulting radial wave functions using the confluent and biconfluent Heun functions, with special cases represented by Bessel functions. We derive restrictions on the allowed quantum energy levels by imposing constraints on the Heun parameters to ensure polynomial solutions. Furthermore, we investigate the emergence of "dark phases" associated with the radial wave function, focusing on the interesting case of $α_{Q} = 1$. Our findings provide insights into the dynamics of scalar particles in this complex spacetime and the potential impact of dark energy on quantum systems.

Figures (6)

• Figure 1: Comparison of $A(x)/\overline{a}$ for $x \in [1, 100]$, with a fixed value of $\Delta_{\scaleto{NSA}{4pt}} = N_{\scaleto{QL}{4pt}}\,\rho_{\scaleto{S}{4pt}}^{2}/\overline{a}^{3} = 6.2 \times 10^{-5}$. The solid line shows the exact case of \ref{['A(x)_Upper_Bound_Complete']}, the dashed line represents the case without quintessence, and the dash-dotted line neglects the $1/x$ term. This comparison highlights the impact of quintessence and the $1/x$ term on the behavior of $A(x)/\overline{a}$. We stress that, as the value of $N_{\scaleto{QL}{4pt}}\,\rho_{\scaleto{S}{4pt}}^{2}/\overline{a}^{3}$ decreases, it becomes more difficult to distinguish these lines.
• Figure 2: Real part of the non-normalized radial wave function ($R_{\scaleto{ST}{4pt}}^{\scaleto{(0)}{6pt}}(x)$) as a function of the dimensionless radial coordinate $x = \rho/\rho_{\scaleto{+}{4pt}}$ for the standard scenario of $\alpha_{\scaleto{Q}{4pt}} = 0$. This result is obtained from \ref{['ST-0-Dark-phase']}. The parameters are fixed at $\rho_{\scaleto{S}{4pt}} = 10^{2}$ m, $\overline{a} = 10^{-3}$, $\epsilon = n\,\overline{m}_{\phi}$, with $n = 4$, and $\overline{m}_{\phi} = 1$ m$^{-1}$ (chosen for visual clarity). For realistic values of $\overline{m}_{\phi} = m_{\phi} c/\hbar$, the oscillations would be much more rapid, making the distinction between different $N_{\scaleto{Q}{4pt}}$ values visible only at extremely small distances. Furthermore, note that besides the compression of these oscillations as $x\to 1^{+}$, due to $\ln (x-1)$, the wavelength increases with increasing $N_{\scaleto{Q}{4pt}}$.
• Figure 3: Real part of the radial wave function $R_{\scaleto{ST}{4pt}}^{\scaleto{\,(1)}{6pt}}(x)$ near the event horizon (derived from \ref{['R_WF_ST_1_NH', 'Dark_Phase_Alpha_Q_1_ST']}, with $c_{+} = 0$ and $c_{-} = R_{0}$), as a function of the dimensionless radial coordinate $x=\rho/\rho_{\scaleto{+}{4pt}}$ for the standard scenario of $\alpha_{\scaleto{Q}{4pt}} = 1$ (the upper bound). The different curves correspond to various values of $\epsilon$. We considered $N_{\scaleto{Q}{4pt}} = 10^{-52}$ m$^{-2}$, $\kappa = 1$ (indicating the particle does not move along the $z-$axis), $\rho_{\scaleto{S}{4pt}} = 2\times 10^{10}$ m, and $\overline{a} = 10^{-6}$. The parameter $\overline{m}_{\phi}$ was set to $\overline{m}_{\phi} = 2.23 \times 10^{-25}$ m$^{-1}$ for visual clarity; realistic values of $\overline{m}_{\phi}$ would require us infinitely smaller distances to discern the phase shift.
• Figure 4: Real part of the radial wave function $R_{\scaleto{ST}{4pt}}^{\scaleto{\,(1)}{6pt}}(x)$, given by \ref{['R_WF_ST_1_NH', 'Dark_Phase_Alpha_Q_1_ST']} (with $c_{+} = 0$ and $c_{-} = R_{0}$), as a function of the dimensionless distance from the event horizon $x-1$. The figure represents the standard scenario of $\alpha_{\scaleto{Q}{4pt}} = 1$ (the upper bound) near the event horizon. We considered $\kappa = 1$, implying the particle does not move in the vertical direction. The remaining parameters are $\rho_{\scaleto{S}{4pt}} = 10^{15}$ m, $\overline{a} = 10^{-6}$, $\overline{m}_{\phi} = 2.23 \times 10^{-25}$ m$^{-1}$ (chosen for visual clarity), and $\epsilon = 5\,\overline{m}_{\phi}$. The two curves illustrate the cases with $N_{\scaleto{QL}{4pt}} = 0$ m$^{-2}$ (absence of quintessence) and $N_{\scaleto{QL}{4pt}} = 10^{-52}$ m$^{-2}$ (presence of quintessence). Notably, a quintessence-induced phase shift distinguishes a particle in the presence of dark energy. As with previous figures, a convenient value for $\overline{m}_{\phi}$ was used, as realistic values would imply the phase shift would be noticeable only at much smaller distances.
• Figure 5: Real part of the radial wave function $R_{\scaleto{HL}{4pt}}^{\scaleto{(0)}{6pt}}(\rho)$ (given by \ref{['R_WF_HL_0_NO', 'Dark-phase-HL-0']}) in the horizonless scenario of $\alpha_{\scaleto{Q}{4pt}} = 0$, for various values of the quintessence parameter $N_{\scaleto{Q}{4pt}}$. Since there is no event horizon, this behavior is valid near the origin (at $\rho = 0$). For simplicity, we chose $\kappa = 1$, which describes a spin$-0$ particle moving in a plane of fixed $z = z_{0}$. The other parameters are set to $\rho_{\scaleto{S}{4pt}} = 10^{5}$ m, $\overline{a} = 1$, $\overline{m}_{\phi} = 1$ m$^{-1}$, $\epsilon = 4\,\overline{m}_{\phi}$. As with the previous figures, $\overline{m}_{\phi}$ is chosen to facilitate the visualization.