Klein-Gordon oscillators in traversable wormhole rainbow gravity spacetime: Conditional exact solvability via a throat radius and oscillator frequency correlation

TL;DR

The paper investigates KG-oscillator bound states in a (2+1)D traversable wormhole spacetime under rainbow gravity, introducing a non-minimal coupling and a conditional exact solvability (CES) mechanism via a correlation between the throat radius $r_0$ and the oscillator frequency $\Omega$. By setting $\mathcal{F}_x=\Omega x$, the radial equation reduces to a form solvable by a truncated confluent Heun function, yielding $\tilde{\mathcal{E}}=4\Omega\left(n+\tfrac{3}{4}\right)$ and a specific $\Omega r_0^2$ relation that enforces CES. The study analyzes two loop-quantum-gravity motivated rainbow function pairs, deriving energy spectra under each and showing how rainbow parameters $\varepsilon$ (or $\beta$) and the throat radius control the asymptotic bounds and spectral clustering, with $|E|$ bounded by $E_P$ in high-energy regimes. The results offer insights into quantum dynamics in curved, energy-dependent spacetimes and demonstrate how topology and rainbow gravity jointly modulate relativistic bound states.

Abstract

In this study, we discuss an analytical solution for a set of the Klein-Gordon (KG) oscillators' energies through a correlation between the frequency of the KG-oscillators and the traversable wormhole (TWH) throat radius. Under such restricted parametric correlation (hence the notion of conditionally exact solvability is unavoidable in the process), we report the effects of throat radius, rainbow parameter, disclination parameter, and oscillator frequency on the spectroscopic structure of a vast number of $\left( n,m\right) $-states (the radial and magnetic quantum numbers, respectively). In the process, we only use two loop quantum gravity motivated rainbow functions pairs. The only rainbow functions that clearly and reliably fully adhere to the rainbow gravity model and secure Planck energy $E_p$ as the maximum possible energy for particles and anti-particles alike. Near the asymptotically flat upper and lower universes connected by the TWH, i.e., for the throat radius $r_{0 }>>1 $, the energies tend to cluster around the rest mass energies, i.e., $|E_{\pm }|\sim m_{0 }$. Whereas, for $r_{0 }<<1$ the energies tend to approach $|E_{\pm }| \leq E_{P}$.

Figures (2)

• Figure 1: For different $\left( n,m\right)$-states and $m_{0 }=m_{0 }c^{2}=1$, we show the KG-oscillator's energy levels in Eq.(\ref{['34']}) (a) against the oscillator's frequency $\Omega$ at $\alpha =0.5$ and $\beta =0$ (i.e., no rainbow gravity), (b) against the oscillator's frequency $\Omega$ at $\alpha =0.5$ and $\beta =0.5$, (c) against the TWH-throat radius $r_{0 }$ at $\alpha =0.5$ and $\beta =0.5$, (d) against the disclination parameter $\alpha$ at $\beta =0.5$ and $r_{0 }=1$, (e) against the rainbow parameter $\beta =\epsilon /E_{P}$ at $\alpha =0.5$ and $r_{0 }=1$, and (f) against the rainbow parameter $\beta$ at $\alpha =0.5$ and $r_{0 }=0.2$
• Figure 2: For different $\left( n,m\right)$-states and $m_{0 }=m_{0 }c^{2}=1$, we show the KG-oscillator's energy levels in Eq.(\ref{['29']}) (a) against the oscillator's frequency $\Omega$ at $\alpha =0.5$, $m=0$, and $\gamma =0.5$, (b) against the oscillator's frequency $\Omega$ at $\alpha =0.5$, $\gamma =0.5$, and $m=0,\pm 1,\pm 2,\pm 3$, (c) against the TWH-throat radius $r_{0 }$ at $\alpha =0.5$ and $\gamma =0.5$, (d) against the disclination parameter $\alpha$ at $\gamma =0.5$ and $r_{0 }=1$ , (e) against the rainbow parameter $\gamma =\epsilon /E_{P}^{2}$ at $\alpha =0.5$ and $r_{0 }=1$, and (f) against the rainbow parameter $\gamma$ at $\alpha =0.5$ and $r_{0 }=0.2$.